Why Statistics Are Important
In this
introductory chapter we begin by providing our general theoretical
position for this book - that social work research, including quantitative
research methods, should be carried out from
a
strucfural and anti-oppressive
perspective. We provide examples of how a quantitative approach has been
throughout the history of social work practice in order to address
important structural issues. We also introduce the basic statistics terminology
used
and conclude with
an
overview of the book.
Over 10% of First Nations children in three
sample provinces were in child
of 2005 versus 3% for Metis children and 0.5% for
non-Aboriginal children. Overall, it has been estimated that there are three
times as many First Nations children placed in out-of-home care today than
in residential schools at the height of the residential school movement.
(Trocme, MacLaurin, Fallon, et al. 2005: 16)
welfare
care as
of May
The above quote, from a studyentitledWelfare
Understandi
ngthe
Overrepresentati
on ofFirofst
System,
offers
dramatic example
Nations Children in Canada’s Child
a
providing quantitative evidence to bring attention to important social justice issues,
in this case the deplorable state of the Aboriginal child welfare system in Canada.
Victor Thiessen (1993), in his book Arguing with Numbers, states that convincing
funders, the government and the general public to accept arguments requires hard
evidence in the form of numbers. Whether it is
the needs of clients
to
show how well
advocate for social justice causes,
a
program serves
social workers need
to provide statistical evidence to defend their arguments. In order to present this
kind of information effectively, social workers must have a good understanding of
quantitative research methods and statistical analyses. The aim of this book is to
lay the foundation for this knowledge and provide an introduction to statistical
concepts as they relate to social work, and to do so using a social justice lens.
Victor Thiessen’s book reinforces the point that those in the field of social work
need to be familiar with quantitative research methods. This is in spite of the fact
that progressive social workers have moved away from a positivist approach to
or
to
2
Statistics for Social Justice
research and
rely instead on a post-positivist, or interpretist, approach. Positivism
paradigm that holds that social behaviour can be studied and under¬
stood in a rational, objective and scientific manner. The term was coined by early
French sociologist Auguste Comte and the most influential advocates were a group
of philosophers who were collectively known as the “Vienna Circle of Logical
Positivists.” Their approach “amounted to the methodological assertion that any
variable which cannot be directly represented by a measurement operation has no
place in science” (Ford 1975, cited in Lincoln and Guba 1985: 45).
Post-positivist writers, on the other hand, argue that science subjugates
knowledge and that approaches to social work knowledge should “install the client
as an
important site ofknowledge” (Rossiter 2000: 27). For interpretist research¬
ers (another term for
post-positivist researchers), scientific methods are seen as
limiting the multiple voices that can contribute to the construction of social work
knowledge. They argue that the rigour and rigidity of experimental methods cannot
account for the complexity of human relations and interventions. These researchers
claim that an interpretive approach, which relies primarily on qualitative methods,
rebalances empiricist methods “with subjective, intuitive and inductive approaches,
thus lending support to new paradigms which integrate theorizing, practice and
research as part ofholistic experience” (Fook 1996: 197).
While quantitative methods best fit within the positivist paradigm, rather than
the interpretivist paradigm, and in spite of the legitimacy of the above critiques
of positivism, social work research does need to rely on quantitative methods in
order to advocate for meaningful social change (van de Sande and Schwartz 2011).
Progressive social workers, including those who adopt a structural perspective, need
to be familiar with both qualitative and quantitative methods. What distinguishes
is
a
research
the structural social work researcher is
purpose
of the research
-
not
the research method chosen but the
promoting social justice.
THE STRUCTURAL PERSPECTIVE
The
“structural”
originated in the United States in the 1970s. Ruth R.
Goldberg introduced this term to show how social workers
could intervene “to improve the quality of the relationship between people and
their social environment by bringing to bear, changing, or creating social structures”
(1974: 32). In Canada, Maurice Moreau and his colleagues at Carleton University
utilized feminist and Marxist principles to develop this approach.
Bob Mullaly’s work on the structural continued through the 1990s. He pub¬
lished a series of three books (1993,1997, 2007) which provide a comprehensive
framework for the approach. Mullaly proposes that the structural approach “seeks
to change the social system and not the individual who receives, through no fault
term
Middleman and Gale
Why Statistics Are Important
3
of their own, the results of defective social arrangements” (2007: 245 ). He argues
that within the sociological literature there are two competing perspectives — order
and conflict
with regard to how society functions. The order perspective, which
—
Mullaly argues is consistent with neoliberal ideology, sees society as basically func¬
tioning in an orderly fashion. Neoliberal ideology holds that capitalism is a sound
economic system and that the wealth accumulated by corporations and individuals
will trickle down to lower income members of society. Those who believe in the
order perspective see social problems as caused by individuals who do not respect
the rules of society. Mainstream social work has adopted the order perspective and
is concerned primarily with helping individuals adjust to the expectations of society.
Donna Baines notes: “In social work, social problems are often depoliticized by
defining them as the failings and shortcomings of individuals ... One of the ways
that wider social problems are individualized and depoliticized is by giving them
medical or psychiatric diagnoses or criminal labels” (2007: 5).
Mullaly believes that the social work profession should adopt the conflict
perspective, in keeping with a social democratic and Marxist ideology, which is
based on socialist theory and a radical analysis of society. Those who subscribe to
the conflict perspective believe that the more power a group possesses, the more
it is able to pursue its own self-interests and oppress others through coercion
and subjugation. Mullaly characterizes society as a struggle between competing
interest groups, and adds that “the goal of structural social work is twofold: (l)
to alleviate the negative effects on people of an exploitative and alienating social
order; and (2) to transform the conditions and social structures that cause these
negative effects” (2007: 245).
To explain how the structural approach translates into practice and research,
Moreau (1989) identifies the following five practice methods.
1.
2.
Defence of the Client: Social workers using the structural approach help to de¬
fend their clients against an oppressive system. Quite often, clients are not fa¬
miliar with their rights and require someone to advocate on their behalf. This
includes writing letters, attending meetings and, if needed, subverting agency
policy.
Collectivization: Clients need to know that they are not alone in their strug¬
gles. It is common for clients to feel that the problems they face are the result
of their individual shortcomings. An important role of the structural social
worker is to connect clients to support networks and reduce their felt sense of
isolation and alienation.
3.
Materialization: A
of the structural
approach is the materialist
analysis. Many of the personal problems experienced by clients are a direct
result of material deprivation; for example, a single parent on social assistance
cornerstone
4
Statistics for Social Justice
may
struggle with depression and feelings of inadequacy. Rather than focus¬
ing on the mental health issues experienced by clients, social workers using
a material
analysis will help the clients make the connection between their
poverty and mental health issues and assure them they are not to blame for
structural
4.
5.
problems beyond their control.
Increasing Client Power in the Worker-Client Relationship: Clients coming for
assistance typically experience feelings of powerlessness. Part of the work of a
structural social worker is to increase the power of clients in the worker-client
relationship by clear contracting, avoiding jargon, sharing rationales behind
proposed interventions and ensuring that clients see what is in their files. In
this way, clients will view themselves as being in control of their own prob¬
lems and the possible solutions.
Enhancing Client Power Through Personal Change: A challenge for the struc¬
tural social worker is to maximize the client’s potential for personal change
of thoughts and behaviours that are self-destructive or destructive to others
without judging or blaming. This is done by focusing on clients’ strengths and
helping them make the connection between their thoughts and behaviours
and their social context.
Moreau’s practice
methods also relate to research. For instance, with respect to
materialization, researchers have found a strong correlation between poverty and
mental illness;
the lower a person’s socio-economic status the greater their chance
of suffering from
mental distress. Studies have shown that poverty, unemployment
diagnosis of mental illness and psychiatric
hospitalization and are, therefore, causal factors (Hudson 2005). Helping clients
acquire resources such as safe and stable housing, employment and adequate food
is of utmost importance in the prevention and reliefof mental distress. Studies that
document discrimination can be used in the process of collectivization to help
clients understand the effects of social injustice (Schwartz and O’Brien 2010).
Involving clients at all stages of the research, including in quantitative studies,
is one way of empowering the client. This practice is encouraged in participatory
action research, an activist approach to research that seeks to engage and empower
the community. Proponents of participatory action research and community
empowerment research argue for participant empowerment and conscientization
to be reflected in research studies. In this way, the funding organization is made
aware of what clients feel
they need and has the opportunity to be more respon¬
sive. Furthermore, it is empowering for clients to see their ideas reflected in the
and lack of affordable housing precede a
results of the
study.
Why Statistics Are Important
5
ANTI-OPPRESSIVE PRINCIPLES
(2011) identifies ten anti-oppressive practice (aop) principles, some
directly relate to social work research. One that stands out is: “Social work
is not a neutral, caring profession, but an active political process” (5). While most
mainstream social work research texts emphasize the importance of being neutral
and objective, progressive social workers argue that social work research should
not be neutral; it should actively pursue social justice and social change. Another
important aop principle is that “participatory approaches between practitioners
and clients are necessary” (21). This means that the service user, typically the focus
of the research, should be actively involved in the design of the research. Another
aop
principle is that “self-reflexive practice and ongoing social analysis are essential
components of social justice oriented social work practice” (22). Self-reflexive
Donna Baines
ofwhich
exercises
our
are
biases
objective,
on our
as
or
also essential in social work research. We need
to
become
aware
of
they relate to our research — such as the pressure to be neutral and
biases as a result of our privilege — in order to best limit their effect
research.
Some authors suggest
that research should only be undertaken when there
good fit between the researcher’s theoretical approach to practice and their
approach to research (Westhues, Cadell, Karabanow, et al. 1999). A good part
of students’ anxiety about conducting anti-oppressive research may result from
a lack of
logical fit. Research questions, methods of data collection and ways of
interpreting data need to be consistent with the researcher’s theoretical analysis.
“Only by knowing that oppression is a social construction can social work embark
on a deconstruction of
oppressive practices and reconstruction of society charac¬
terized by true social equality” (Mullaly 2007: 284).
The difference between a traditional quantitative approach and a progressive
approach lies in the underlying purpose ofthe research. We need to remember that
the end goal of carrying out research from a social justice perspective is structural
change. For example, for almost two decades, Campaign 2000, a group established
in 1991, has been conducting research on child poverty in Canada. In 1989, the
House of Commons unanimously adopted a resolution to work on the elimination
of child poverty in Canada by the year 2000. Every year, Campaign 2000 publishes
a
report card to let the federal government know how it is doing in terms of meet¬
ing its promise. Clearly, the government has not achieved its goal. Nevertheless,
the research conducted by Campaign 2000 serves as a constant reminder to the
government to continue in its efforts. On the twentieth anniversary of the adoption
of the resolution, the government renewed its commitment to reduce and eventu¬
ally eliminate child poverty in Canada. Our point here is that Campaign 2000 uses
traditional quantitative methods of analysis — e.g., data from Statistics Canada
is
6
—
Statistics for Social Justice
and
produces frequency tables and graphs commonly found in mainstream,
quantitative research with the goal of advocacy and creating change.
PROGRESSIVE QUANTITATIVE SOCIAL WORK RESEARCH
One of the earliest
examples of quantitative research to promote structural change
study done by Dorothea Dix in the Unites States in the early 1840s. Dix
conducted a detailed examination of the treatment of people with mental illness
who were held in prison. With evidence in hand, she drafted a petition to the
Massachusetts Legislature.
was a
I
the advocate of the
helpless, forgotten, insane, and idiotic,
of beings sunk to a condition from which the most
unconcerned would start with real horror; of beings wretched in prisons,
come as
men
and
and women;
wretched in
almshouses.... If my
pictures are displeasing,
subjects, it must be recollected, offer no tranquil,
refined, or composing features, (quoted in Snyder 1975: 68)
more
coarse
our
and severe, my
Dix used the results of her
findings to increase awareness of the inhumane
and to advocate for improved conditions. She is credited
with helping to establish state hospitals for mentally ill patients both nationally
and internationally (Snyder 1975).
Later, in 1889, Jane Addams and Ellen Starr established Hull House in Chicago.
The success of their work was the result of a careful study on the needs ofthe popula¬
tion in the Chicago tenement neighbourhoods. Addams and Starr carried out both
quantitative and qualitative research that included community mapping as well
treatment
as
of these people
observational and interview data. The results of their research led
the state, federal and international levels. In her
Addams recalls the
There was, at
to
reforms
at
book Twenty Years at Hull House,
importance ofstatistical evidence in promoting social change:
that time,
statistical information
on Chicago industrial
early resident of Hull House,
suggested to the Illinois State Bureau of Labor that they investigate the
sweating system in Chicago with its attendant child labor. The head of
the Bureau adopted this suggestion and engaged Mrs. Kelley to make the
investigation. When the report was presented to the Illinois Legislature,
a
special committee was appointed to look into the Chicago conditions.
I well recall that on the Sunday the members of this commission came
to dine at Hull-House, our hopes ran high, and we believed that at last
some of the worst ills under which our
neighbors were suffering would
be brought to an end. (1912: 202)
no
conditions, and Mrs. Florence Kelley,
an
Why Statistics Are Important
7
(1997) describes research carried out in Canada during the lat¬
part of the nineteenth century by Sir Herbert Brown Ames, who was a wealthy
manufacturer living in Montreal. Ames felt that his elite position carried with it
a
responsibility for the welfare of the working class. During the fall and winter
of 1896, he surveyed each home in an area of one square mile, which included
38,000 homes in a working-class neighbourhood in Montreal. He gathered data
on
employment, family income, housing conditions and rental charges. His report,
published in 1897, challenged conventional attitudes about the causes of poverty:
Dennis Guest
ter
As to the
of poverty,
chief among them is insufficient employment.
nothing is earned, although there are such
subsisting more or less worthily upon charity. Almost without excep¬
tion each family has a wage-earner, often more than one, and upon the
regularity with which the wage-earner secures employment depends on
the scale of living for the family. (Ames 1897: 52)
Few
are
causes
the families where
Ames also selected
the poorest
causes
of the
a
poor,
smaller
sample of 323 families, which he described as
a more complete explanation of the
in order to provide
of poverty.
With 109
families,
the reply was “irregularity of work.”
without vocation but their employment was
intermittent and often work ceased altogether for considerable periods.
With 87 families, or 28 per cent, the answer was that the wage earner has
no work whatsoever, nor did there seem to be
any immediate prospect
of getting any. With 27 families, or 9 per cent, old age has unfitted and
with a like number sickness had prevented the workers from earning the
or
34 per cent,
The wage-earners were not
requisite support. (Ames 1897: 55)
As
a
result of his research, Ames was
that poverty was
instrumental in helping Canadians understand
“largely rooted in economic and social arrangement” (Guest
1997:31).
Another Canadian example described by Guest
(1997) was the work by Leonard
Marsh, former director of an interdisciplinary social science research program at
McGill
University. Based largely on his extensive research work as well as the work
of other social scientists, Marsh
published the Report on Social Security jor Canada
report became the blueprint for health and social security programs
developed in Canada during the 1940s, 1950s and 1960s.
in 1943. This
8
Statistics for Social Justice
BASIC CONCEPTS
Many of the terms used in quantitative research methods have specific
mean¬
ings. This section provides brief definitions of some of the basic concepts used
in statistics.
Many other terms
are
used in statistics, which
are
introduced in the
following chapters.
Data and Information
The
data is used in both
qualitative and quantitative research; when used in
normally in the form of
numbers or scores. Data are the raw scores or numbers obtained using question¬
naires and other methodology. Note that data is the plural form, while datum is
singular. The data are analyzed and interpreted, and the results of this analysis
and interpretation is the information. Conclusions are based on the informa¬
tion resulting from analysis. For instance, if we were to conduct an evaluation of
a
program designed to help female survivors of abuse improve their self-esteem,
we
might use a standardized questionnaire that measures self-esteem — such
as the
Rosenberg Self-Esteem Scale. We would administer the questionnaire at
the start of the program and again at the end, with the women’s scores being the
data. We would then analyze the data using a statistical test to determine if the
scores obtained at the end of the
program are significantly higher. The results of
this analysis is the information on which we can base our conclusion about the
term
statistics, data refers to the results of the measurements,
effectiveness of the program.
Variables and Constants
In its
simplest term, a variable is a characteristic that varies. Examples ofvariables
gender, race, income, education level,
religion and type of employment. A constant is a characteristic that is the same
for the people or objects that are the focus the research. Referring to our example
of the program for survivors of abuse, the constants for the women in our study
are l) that they are female, 2) that
they have experienced abuse and 3) that they
are all
participants in the program.
often found in social work research include
Conceptualization and Operationalization
Quantitative research methods rely on precise definitions and measurement, and
the
conceptualization refers to the
used in choosing and clearly
defining the variables included in the study. This precise conceptualization allows
others to replicate the study and allows readers of the research to be clear on what
is being researched. In the above example, the variable that we are interested in
is the self-esteem of the women in our study. We must therefore define what we
mean
by self-esteem. The term operationalization refers to the method used to
term
process
Why Statistics Are Important
9
measure the variable. In the case of our study, after we have defined what we mean
by self-esteem, we must show how we plan to measure this variable — in this case,
that we intend to use a standardized instrument, the Rosenberg Self-Esteem Scale.
We must demonstrate that this scale will provide the accurate and reliable data that
will help us in our analysis.
Reliability and Validity
The
degree to which the measurement instrument provides consistent results
time is called reliability. While it is impossible for any instrument, especially
those used to measure psychological characteristics, to provide perfectly consistent
measures, we need to provide evidence that the instrument we have chosen will
provide reasonably consistent results. We explain in a later chapter how this is done.
Validity refers to the degree that our chosen instrument will truly measure what
it is supposed to measure, and not something else. The questionnaire mentioned
above may provide consistent results, but it may not measure what it is intended
to measure. There are accepted ways in which validity is determined, which are
explained later in the book. In our example, the Rosenberg Self-Esteem Scale has
been tested for reliability and validity, which means that we can be safe in assuming
that it will provide reliable and valid data.
over
Levels of Measurement
There
are
four levels of measurement: nominal, ordinal, interval and ratio. A good
remember the order is that the first letter of each word spells the French
classifying observations
into mutually exclusive categories, with no inherent order or rank. For example, if
we were to ask “Are
you employed?” as a yes or no question, this would be at the
nominal level of measurement. The ordinal level is used when classifying observa¬
tions that are mutually exclusive and have an inherent order to them. An example
is level of education: elementary school, high school or university. The third is the
interval level and involves classifying observations that are mutually exclusive,
have an inherent order and have equal spacing between categories. The fourth is
the ratio level and involves classifying observations that are mutually exclusive,
have an inherent order, have equal spacing and reflect the absolute magnitude. A lot
of quantitative data is at the interval/ratio level of measurement. Typical examples
way to
word “noir” The nominal level of measurement involves
are
variables such
as
is income and
scores on
questionnaires.
OVERVIEW OF THE BOOK
Quantitative statistical analysis is normally divided into two broad categories:
descriptive and inferential statistics. While Chapter 2 of this book provides an
historical overview of empiricism, the epistemic basis of quantitative methods,
10
Statistics for Social Justice
Chapters 3 to 6 introduce descriptive statistics. Descriptive statistics are used
describe the characteristics of a sample or population. Most research projects
carried out by social workers involve descriptive analyses. For instance, in the case
of needs assessments, the data analysis can be presented using frequency distribu¬
tions, bar charts, and graphs that describe the characteristics and needs ofthe service
users. In the case of client satisfaction surveys, the analysis may result in
percentage
distribution tables indicating the percentage of service users who are satisfied with
the program. Chapter 3 introduces frequency distributions and graphs.
If we want to describe a typical case in our sample, we would use measures of
central tendency. The three common measures of central tendency are mode,
median and mean. The mode is the value that appears most often, the median is
the halfway point between the range of values, and the mean is the average of all
the values. If we want to describe to what extent the values vary, the most common
method is the standard deviation, which describes the average distance of all the
values from the mean. Chapter 4 provides a more complete explanation of central
tendency and variability.
The next three chapters introduce the basic principles of statistics. Chapter 5
focuses on the properties of the normal distribution. This type of distribution is
important because some ofthe more common statistical tests can only be done ifthe
variables are normally distributed within the population of interest. Chapter 6 looks
at the principles of hypothesis testing, which are fundamental to determining if
the results of the study are statistically significant. Chapter 7 introduces sampling
distributions and the various statistical tests used in quantitative methods.
to
The remainder of the book focuses
on
inferential statistics, which are
used to
generalize findings to a larger population based on a sample of cases. If the sample
selected using a probability approach, meaning that each individual within
the population of interest had an equal chance ofbeing selected, then we can infer
the characteristics of the population from the characteristics of the sample.
Inferential statistical tests can also show if there is a relationship between two
or more variables. In the case of a summative
program evaluation (discussed in
Chapter 6), we would use an explanatory approach, which will tell us if there is
a cause and effect
relationship between the program (the independent variable)
and the effect on clients (the dependent variable). For instance, did our program
for women who are survivors of abuse cause the improvement in self-esteem? If
we were able to use a classic
experimental design and involve a control group, we
would be able to compare the scores obtained by the program (or experimental)
group with those obtained by the control group. We would start with a hypoth¬
esis: a testable statement describing the relationship between the independent
and dependent variables. In our example of the summative program evaluation,
the hypothesis would be: “Participants in the experimental group will score higher
was
Why Statistics Are Important
than participants
II
in the control group.” For this type ofinferential statistical analysis
(described in Chapter 8). The t test is a parametric test,
dependent (outcome) variable is at the interval
or ratio level, and the data are
normally distributed.
On the other hand, if we are only able to obtain data at the nominal level, we can
still carry out a program evaluation using a non-parametric test: the chi-square test
(introduced in Chapter 9). Let’s say, we are looking at a program to help women
we
would
which
use
means
the t test
that the data for the
survivors of abuse become
more
assertive. We want to know if women who attend
the program are
less likely to return to abusive situations. We would still have two
independent variable would be the program attendance, and the data
would be at the nominal level with two possible scores; attend or did not attend.
The data for the dependent or outcome variable would also be at the nominal level
variables: the
and have
two
scores:
return
or
did not return to
an
abusive situation.
Another typical statistical
method is testing for correlation (covered in Chapter
10). If we want to find out ifthere is a relationship between years of service and job
satisfaction scores, we would find the correlation coefficient for these two variables.
This
test
requires that the data for both variables be at the interval or ratio level. If we
find that we have
a
very strong
correlation, we can use this information to develop
simple linear regression, which we can use to predict what outcome score the
participant will have, such as job satisfaction, based on the level of the predictor
variable, years of service. In Chapter 11, we review how the concepts presented in
a
the
preceding chapters can help us achieve meaningful structural change. Chapter
report writing and a brief introduction to a commonly used statistical
software package called SPSS.
12
covers
SUMMARY
The
important points to remember from this chapter are that social work does and
should conduct research
to
using both quantitative and qualitative methods as a way
promote social justice. Quantitative social work research should be carried out
within
structural and anti-oppressive practice perspective.
We provided several
examples of social workers who used quantitative methods to argue for
social change. Finally, we defined a number of terms used in statistics that we need
to understand in order to conduct research and promote social justice.
a
historical
REVIEW QUESTIONS
1.
Describe the
“positivism.” How is it different from post-positivism? Why
paradigms important to social workers carrying out research?
Compare and contrast the “order perspective” and the “conflict perspective.”
are
2.
both
term
12
Statistics for Social Justice
Discuss how each relates to
4.
5.
structural
approach to social work.
you feel is most important for a
social worker conducting research from a structural perspective and why?
Provide an example of a research study that uses traditional quantitative
methods whose goal is to promote social change. Where did you hear about
this research? How does it relate to what was discussed in this chapter?
Discuss how a research study might conceptualize and operationalize the vari¬
3. Which of Moreau’s five
a
practice methods do
able of “income.”
6.
Self-reflexive exercise: Take
a
few minutes
research and statistics. Was this
tistics done
a
to
think about your
history with
positive experience? Was this research/sta-
framework? How did it make you feel?
might your past experiences with research and/or statistics be shaping
the way you view or feel about this course?
How
using
a
structural
or aop
2
The
History of Empiricism
In this chapter, we trace the history of empiricism from the early writings
of Aristotle to contributions made by British empiricists. We also show that,
while empiricism was developed primarily by white male philosophers, it has
recently been modified by feminist scholars, whose work illustrates that
more
empiricism can be employed within a structural and anti-oppressive approach
ARISTOTLE
For Aristotlphysics
e, commonl
y consiand
deredthetohuman
be the fisciences,
rst empiricist,
naturalandsciethics,
ences,
and biology,
such the
politics
such
as
have the
as
level of validity as
mathematics. Because of his devotion to science,
Aristotle is generally credited with being the Father of Modern Science. His influ¬
ence in the area of
logic and scientific investigation on Western thinking cannot
be overstated; empiricism became the dominant scientific method for acquiring
knowledge. From Aristotle, the Western world learned about the importance of
observation and the structure of logic and of deductive reasoning, which are at the
heart of the scientific method. Aristotle’s approach shows us the promise and peril
of empirical science, as well as the important distinction between craft and science.
Although Aristotle introduced empiricism, it was primarily the seventeenthcentury British philosophers who established it as the dominant scientific method
of the Western world. The British empiricists were a product of the modern age,
when science gradually replaced religion as the chief source of knowledge. In some
respects, because of unquestioned and almost dogmatic faith, science became the
new
religion. Social work was part of Western culture, which was convinced that
science was capable of solving the ills of the world, including disease and poverty.
This is why, even today, so many in our profession are reluctant to choose post¬
positivist and interpretist methods instead of empirical methods as important
ways of knowing.
While the academic part of social work has largely moved beyond the positivist
view and its faith in empiricism, the professional part of social work is as commit¬
ted as ever. We see this in the extent to which evidence-based practice (ebp)
same
13
Statistics for Social Justice
14
has been
adopted by our profession. The underlying principle of evidence-based
use knowledge that is gathered and tested
empirically to guide their practice. In social work research, especially in the area
of program evaluations, funders often insist on empirically tested outcomes. In
fact, social work practitioners are still told to develop intervention plans with clear
measurable goals and objectives.
practice is that social workers should
THE BRITISH EMPIRICISTS
Thomas Hobbes
born in
Malmesbury, England, in 1588, the year of the
Spanish Armada. He was educated at Oxford and served as a tutor to the Cavendish
family, and as a secretary and clerk to Francis Bacon. In 1640, he published The
Elements of Law, Natural and Politic, in which he described the principles of his
philosophy on human nature and human society. That same year, he fled to Paris
in anticipation of the civil war in England, and he remained there for more than
ten years. During this period, Hobbes became tutor to the future King Charles
II. He also became familiar with Descartes’ work and wrote a critique on his
philosophy of the mind. Hobbes established himself as a materialist and believed
that there was no such thing as a non-material mind. Historians view Hobbes and
Descartes as founders of two opposing schools of philosophy: British empiricism
was
and continental rationalism.
IfThomas Hobbes is described as the founder ofBritish empiricism, John Locke
the
influential. He
born in Somerset,
England, the son of a gentle¬
fought in the parliamentary cavalry. He attended school at Westminster,
where he studied Latin, Greek and Hebrew. He obtained his master of arts degree
from Christ Church, Oxford and, upon graduation, became interested in chemistry
and physiology and spent the next several years studying medicine. In 1667, Locke
became physician and political advisor to the Earl of Shaftesbury, a member of King
Charles II’s inner cabinet. However, when Shaftesbury was found to be involved in
a
plot to exclude James II, the Catholic brother of the king, from his rightful claim
to the throne, he had to flee to Holland. Locke was likewise
obliged to flee. It was
during his stay in Holland, in 1690, that Locke produced his greatest philosophical
work, An Essay Concerning Human Understanding, which is regarded as one of the
world’s classics. Upon his return from exile, Locke obtained various posts in the
civil service, and in 1704, after years of ill health, he passed away.
Locke lived during a time of great political and religious upheaval. Like his
predecessors Descartes and Hobbes, Locke’s ideas were revolutionary and chal¬
lenged the religious dogma of the day, but unlike them, society was more ready
and able to accept his work. An Essay Concerning Human Understanding built upon
Hobbes’s theories on sensations and proposed that all knowledge is derived from
was
man
most
who
was
The History of Empiricism
15
experience. He argued that the mind of a child is like a “tabula rasa," a blank slate
upon
which experience writes.
Let us then suppose
the mind to be, as we say, white paper void of all char¬
How comes it to be furnished ? Whence comes it
by that vast store which the busy and boundless fancy of man has painted
on it with an almost endless
variety? Whence has it all the materials of
reason and
knowledge? To this I answer, in one word, from experience;
in that all our knowledge is founded, and from that it ultimately derives
itself. Our observation, employed either about external sensible objects,
or about the internal
operations of our minds perceived and reflected on
by ourselves, is that which supplies our understanding with all materials
of thinking. These two are the fountains of knowledge, from whence all
the ideas we have, or can naturally have, do spring. (Locke 1993: 45)
acters, without any ideas.
As
part ofhis theory of knowledge, Locke introduced the concept of probability.
suggested that human reason could be divided into two parts, those of which an
individual is certain and those of which “it is wise to accept” but which only have
the probability ofbeing true. He explained that since there is very little in our world
that we know with certainty to be true, we need to act based on the probability of
something to be true (Russell 1972). He believed that the real essence of things are
unknown to us; we cannot have true knowledge of items in the natural world but
only a probable belief. We can only have knowledge of things within the bounds
of our sensations, and the “love of truth” should keep us from going beyond this
point (Kenny 2004). In this respect, Locke anticipated the importance of prob¬
ability, particularly as it is used in quantitative analysis. Locke greatly advanced
the philosophy of knowledge. While many of his ideas were challenged because
of their inconsistencies, during their time they were revolutionary and greatly
influenced philosophy in England and throughout Europe. For his contribution,
He
he is considered the Father of Empiricism.
The
member of the British
discuss, George Berkeley,
Kilkenny, Ireland. When he was fifteen, he attended
Trinity College in Dublin, where in 1704 he obtained a bachelor of arts. On the
strength of two mathematical papers, he became a fellow of the College. His most
famous works were published when he was still quite young. In 1709, when only
twenty-one, he wrote An Essay Towards a New Theory of Vision, in which he gave
an account of how we
judge distances. Distances and sizes, he argued, we judge
by vision, while shapes are judged by touch. We learn to judge these qualities by
means of experience. In 1710, he wrote The
Principles of Human Knowledge, where
he proposed and ingeniously defended that there is no such thing as matter. Matter
was
next
born in 1685
near
empiricists
we
16
Statistics for Social Justice
experienced by means of the senses is only an idea and does not actually exist
reality (Kenny 2006).
In 1728, he sailed to Newport, Rhode Island, in the Unites States, hoping to
establish a college. However, when a promised grant did not materialize, he returned
to
England. Nevertheless, citizens of the United States were impressed by his
commitment to education and named a college at Yale after him and, much later,
a
university town in California. In 1734, he became the Bishop of Cloyne and, in
addition to his pastoral duties, continued to write and study until his death in 1753.
While accepting the basics of learning through experience, he believed that all
the qualities exist in the mind of the individual from information derived through
the senses. He suggested that the senses only perceive light, sound and smell but
make no inferences about them. He distinguished between ideas and perception,
stating “to be is to be perceived” (“esse ispercipi”). He proposed that the only reality
is the mind, since it is impossible for something to be thought about, without its
existing, even ifonly as a mental image. He agreed with Locke’s beliefthat complex
ideas are formed in the mind by the association of simple ideas. He also believed
that, through experience, the mind is able to create associations of ideas.
The last of the British empiricists we consider took empiricist principles to an
extreme. David Hume was born in Edinburgh, Scotland, in 1711, into a junior
branch of a noble Scottish family. He was the youngest son of a widowed mother
and consequently had to learn to make his own way in the world. From age twelve
to fifteen, he studied literature and philosophy at the University of Edinburgh. He
attempted to enter the legal profession but gave up because of an irresistible inter¬
est in philosophy. His attempts to enter the business world resulted in the same
conclusion. Instead, he decided to move to France and live frugally on a small
inheritance. He attended La Fleche College, where Descartes had studied over a
hundred years earlier. Hume’s first major work, A Treatise of Human Nature, was
published in 1739, when he was just twenty-eight. While Treatise initially received
little attention, it became the main target of criticism of the German idealists. While
struggling to get recognition as a scholar, he worked as a diplomat and in various
government services. He nevertheless continued to write, and, in the 1750s, when
his publications finally began to sell, he at last enjoyed some prosperity and rec¬
ognition. Hume retired in 1769 and returned to Edinburgh, where he lived until
as
in
his death in 1776
(Kenny 2006).
exploring the question of the mind, and going a step fur¬
ther than Berkeley, he completely did away with the mind as an entity beyond
the sensible qualities available in experience and proposed instead that what is
believed to be the mind is “the flow of ideas, memories, imagination and feelings”
(Chaplin and Krawiec 1968: 21). The mind is simply a bundle of such processes;
it is not observable itself, but only through its action in perception and thought.
Hume continued
The History
of Empiricism
17
Hume
distinguished between various types of ideas and suggested that ideas of
closely related to the original perception while ideas of imagination
are less distinct. In this
way he acknowledged that there are mental processes at
work organizing these various ideas into mental constructs. He also agreed with
his empiricist predecessors that complex ideas are combinations of simple ideas
formed by association.
memory are
We find
by experience, that when any impression has been present with
again makes its appearance there as an idea; and this it may
do after two different ways: either when in its new appearance it retains
a considerable
degree of the first vivacity, and is somewhat intermediate
betwixt an impression and an idea. The faculty, by which we repeat our
impressions in the first manner, is called the memory, and the other the
imagination. ’Tis evident at first sight, that the ideas of the memory are
much more lively and strong than those of the imagination, and the former
faculty paints its objects in distinct colours, than any which are employ’d
by the latter. (Hume 2003: IS)
the mind, it
pointed out by Chaplin and Krawiec (1968), Hume had a reductionistic and
possible a more contemporary, positivist
paradigm. By giving such prominence to primary perceptions and minimizing the
existence of even simple concepts of knowledge, the British empiricists radically
changed the course of Western philosophy.
As
mechanistic view of the mind that made
FEMINIST EMPIRICISM
Until
recently, the dominant view of the Western scientific world
that only
possessed the necessary qualities to engage in science. Linda Jean Shepherd,
biochemist and author of Lifting the Veil: The Feminine Face of Science, noted that
was
men
when the institutions of science were
forming during the mid-seventeenth century,
Royal Society of London believed that its business was “to raise a Masculine
Philosophy” (1993: 19). Shepherd argued that the commonly held view was that
scientists were to be rational, neutral and objective (and male).
During the twentieth century, the traditional view of science and philosophy
as
solely the domain of men was challenged. Very slowly, a few influential female
scientists paved the way for a feminist epistemology. Some notable examples
include Marie Curie, who won a Nobel Prize in physics in 1903, and in chemistry
in 1911, for her work on radioactivity. Dorothy Crowfoot Hodgkin won a Nobel
Prize in 1964 in chemistry for her work on the structures ofbiochemical substances.
In 1983, Barbara McClintock won a Nobel Prize in physiology of medicine for
her work in genetics. As of 2012, only forty-three women had been awarded a
the
Statistics for Social Justice
18
Nobel Prize, out
of 862 people and organizations that had been named laureates
(Connelly 2012).
Feminist
empiricists accept the basic principles of traditional empiricism —
only knowledge based on direct observation through the senses should be
accepted as scientific fact. They also uphold the empiricist view that science is
about formulating hypotheses that must be tested against experience. However,
they believe that traditional empiricism can be improved upon by making certain
modifications. They suggest that science is not value-free and that the scientific
that
method is
not
sufficient
to
screen
out
Alessandra Tanesini states that there
all of the influence of values.
are
two main criticisms
of traditional “male”
epistemology; the first concerns individualism. Philosophers such as Descartes and
Locke felt that to achieve true knowledge an individual must free himself from the
influences of society.
Stemming from this, individualism holds that knowledge
only achievable by a fully autonomous and separate individual and rejects the
notion that social factors play a role in the production of knowledge.
Feminist epistemologists severely criticize the notion that knowledge is achiev¬
able only by an autonomous and separate individual knower. Instead they believe
is
that social factors
both relevant to, and among
the causes of, knowledge, and
bearing on what they know. With respect
to feminist work, Gayle Letherby states: “It is important that we
recognize the
importance of our intellectual biography” (2003: 8). Some feminists suggest that
it is not the individual who has knowledge, but the community. Lynn Hankinson
Nelson, for example, argues: “It is the communities that construct and acquire
knowledge” (1993: 123).
A second criticism looks at the concept of the knower. The traditional subject
was
emotionally detached, objective and value-neutral. As such, scholars were
expected to write using the third person passive. Those of us trained in the tra¬
ditional scientific method were expected to say “it was found that” rather than
“I found that.” Letherby argues that feminist researchers should write in the first
person. The characteristics of emotional detachment and objectivity were associ¬
ated with “maleness” and contrasted the more “feminine” characteristics, including
nurturing, receptivity, cooperation and intuition (Shepherd 1993). It becomes
evident in the literature that while the traditional epistemic subject was supposed
to represent the universal subject, in reality, the subject was male (Tanesini 1999).
Feminist epistemologists believe that detachment and value-neutrality is nei¬
ther possible nor desirable. They posit that values cannot be turned on or off like
a switch and
suggest that emotions and values should be acknowledged. Once
acknowledged, emotions and values can enrich an investigation. Letherby argues:
“The ‘value freedom’ of traditional research is challenged but not the empiricist
goals” (2003: 44).
are
that the social location ofthe knower has
a
The
Feminist
History of Empiricism
19
empiricism is described by Sandra Harding as an attempt to rectify
the sexism and androcentrism of current science
by arguing that these are “social
by stricter adherence to the existing methodological norms of
scientific inquiry” (1987: 24). There are two ways in which these biases can be
eliminated or at least minimized. The first is by recognizing that the “context of
discovery” is just as important and the “context of justification.” The context
of discovery refers to the elements that play a role in the discovery of theories.
These elements can include, among others, 1) research hypotheses, 2) theoretical
and conceptual framework and 3) method of analysis. The context ofjustification
refers to the choice of evidence used to argue in favour or against a theory. In other
biases correctable
words, choices are made in terms of what evidence is examined and what evidence
(Tanesini 1999).
which the impact of biases can be eliminated or reduced is by
focusing on the social location of the researcher.
is deemed irrelevant
Another way in
Traditional empiricism
does not direct the researcher to locate themselves
plane as their subject matters. Consequently, when
non-feminist researchers gather evidence for or against hypotheses,
scientific method bereft of such a method is impotent to locate and
eradicate the androcentrism that shapes the research processes. (Harding
1987:184)
in the
same
critical
Because traditional
that they are neutral and objective, they
acknowledge that social factors influence and shape their research.
There are two kinds of feminist empiricism: contextual feminist empiricism,
developed by Helen Longino (1990); and naturalized feminist empiricism,
developed by Lynn Nelson (1990). We examine the key concepts ofeach by focus¬
ing on the contributions of each of these scholars.
are
not
able
empiricists
assume
to
Helen Longino
Helen
professor of philosophy at the University of Minnesota. She
began her career as an activist in the anti-war and women’s liberation movements
of the 1960s and 1970s. She was chair of the American Philosophical Association s
Committee on the Status of Women in the Profession and has written extensively
on feminist
epistemology.
Longino first began working on her book Science as Social Knowledge (1990)
out of frustration that traditional philosophy of science had not
acknowledged
the relationship between social values and scientific inquiry. As an alternative to
the “value-free” concept of science, her book provides an analysis that reconciles
social values with the objectivity of science (1990: ix). Longino begins by making
Longino is
a
the distinction between the “constitutive” values of science
—
that is the values
or
20
Statistics for Social Justice
rules that determine what is
with the “contextual” values,
philosophy of science main¬
tains that these two are distinct and independent of one another. Longino, on the
other hand, states that the “influence exerted by social and cultural context on the
directions of scientific development have led many observer-critics of science to
reject the value-freedom of science” (6). She believes that scientific knowledge
must be viewed within its political, social and cultural context and that scientific
knowledge is social knowledge and can be achieved only by individuals working
which
are
within
a
acceptable science
—
the social and cultural values. Traditional
community context.
In her first
chapter in Good Science, Bad Science, Longino covers what she
principles of epistemology. First, she states that science means
knowledge and true knowledge is achieved by scientific investigation. Second, the
philosophy of science relies on general criteria such as truth, accuracy, simplicity
and predictability; she believes that these criteria should produce “good” science.
However, while traditional epistemology insists that allowing the influence of
social and cultural values results in “bad” science, Longino argues “not only that
the science practices and content on the one hand and social needs and values on
the other are in dynamic interaction but that the logical and cognitive structures
of scientific inquiry require such interaction” (1990: 5).
Longino concludes by proposing that scientific inquiry should never stop. The
solution to making science less traditional is that science should be ongoing and
embrace multiple social contexts — even those that are incompatible. It should
always be open to divergent ideas, including those which are radically different than
the commonly held ideas of a community. By acknowledging the social context
of science, Longino makes an important contribution to feminist empiricism and
feminist epistemology more generally; however, some feminist scholars, such as
Lynn Harkinson-Nelson, suggest that it does not go far enough.
believes
are
basic
Lynn Hankinson-Nelson
professor of philosophy at the University of
Washington. She obtained a B.A. in 1980 from Rutgers University in New Jersey
and a Ph.D. in 1987 from Temple University in Philadelphia. Her areas of expertise
include feminist epistemology, feminist philosophy of science and philosophy of
biology and the social sciences.
Like Longino, Nelson accepts the basic principles of empiricism and also agrees
that science should acknowledge the influence ofvalues (Nelson 1990). Where she
differs from Longino is in her beliefthat science is not only influenced by social and
Lynn Hankinson-Nelson is
a
cultural values but that science is a social endeavour. Naturalized feminist empiricism
shares the contextual feminist empiricist view that science
is not value-free and must
be understood within its political, social and cultural context.
However, it goes further
The
History of Empiricism
21
in
challenging traditional empiricism by suggesting that even the theories upon which
are based are socially constructed. In traditional
empiricism, a
scientific investigation begins with a theoretical framework upon which a research
hypothesis is formulated. Next, data are collected and analyzed, which will prove or
disprove the hypothesis. The problem is that if the theory is socially constructed, the
data are analyzed and conclusions drawn in the context of that socially constructed
theory. The same data could result in very different conclusions if the investigation
was based on a different
socially constructed theory. Looking at our social work
theories as examples, and comparing psychodynamic theories to structural theories,
the former emphasizes ego functioning whereas the latter focuses on the structural
causes of the
problems faced by individuals (Lundy 2004). The naturalized feminist
empiricist argues that data are not independent of the theory, but that theory are
influenced by how the data are observed, measured and understood.
Lynn Nelson accepts “that there is a world that shapes and constrains what is
reasonable to believe, and that it does so by impinging on our sensory receptors”
(1990: 20). She adds that “there probably is no logical coherent way to doubt
(this) thesis” (20). The difficulty is, as Nelson points out, that once our mind has
processed this sensory information, the end product bears little resemblance to the
original sensory information. The process of conceptualizing the “raw ” sensory data
requires language that is shaped by history and culture and involves psychological
processes, which act as a filter. Any attempt to strip away these cultural and psycho¬
logical processes to return to the original sensory data is bound to meet with failure.
What is required is an extra-theory which explains how we theorize and identify
the connection between the “natural world” and our conceptualization of that
world (Quine 1972). Traditional empiricists believe that epistemology, the study
of how we understand knowledge, provides the “extra-scientific” justification of
science. Nelson, who relies heavily on the work of WV.O. Quine, states that no
such “extra-scientific” justification is possible and that epistemology is “within”
scientific discoveries
science.
are
Nevertheless, Nelson insists that
made and the link between sensory
an account
of how scientific discoveries
evidence and scientific discoveries, while
difficult, should be possible.
Nelson identifies the main feminist criticisms of traditional
empiricism, point¬
ing out that traditional empiricists have ignored the sex/gender issue as it relates
Traditional
empiricists insist that their work focuses
the “human
experience” and that it includes women; however, there is ample evidence that
the experiences of women and men are substantially different. If this is the case,
Nelson asks,-why has traditional science refused to include sex/gender issues as a
variable in its investigation? One of the fundamental principles of science is that
its conclusions are always tentative. As new theories and discoveries are made,
they replace old, less adequate ones. If this is true, traditional science should be
to science.
on
22
Statistics for Social Justice
willing to take seriously the criticism that sex/gender issues have not been dealt
with adequately as intervening variables.
Another important criticism is the “individualism” of science. Traditional
empiricists hold that science is an individual pursuit and is not connected to any
particular community. Nelson argues that there is a scientific community that it is
very much part of the larger social community, and its values are reflected in the
scientific work being conducted. Since the values of society have been historically
androcentric,
Feminist
so
have the values of the scientific community.
empiricism is different than traditional empiricism in
number of
feminist
empiricism is not individualist but acknowledges that science is a community
activity and that it reflects the values of society. Science is also done with subjects
and that the participation ofthe subject should be acknowledged. Finally, a feminist
empiricist recognizes the "who” that is theorizing matters and believes that they
are not and cannot be neutral or objective.
areas.
a
First, feminist empiricism incorporates feminist insights. Second,
IMPLICATIONS FOR SOCIAL WORK KNOWLEDGE
The introduction of empiricism
marked a turning point in the Western scientific
complete is our belief in the scientific method based on empiricism that,
for most of us, it has become the unquestioned assumption on how to achieve true
knowledge. In our Western world, only phenomena whose existence can be proven
scientifically should be accepted as fact; true knowledge must be empirically verifi¬
able. The scientific method based on empiricism heavily influences social work,
just as it does all disciplines.
The most important contribution of feminist empiricists was that they chal¬
lenged the dogmatic position that science is individualistic, objective and value-free.
While upholding some of the basic premises of empiricism — that knowledge
must be based on empirical evidence — feminist empiricists successfully tested
the individualism of science and provided sound arguments to prove that knowl¬
edge should always be viewed within a social context. They also demonstrated
that science can never be completely objective and that being value-free is neither
possible nor desirable.
This affirms our belief that basic social work values as expressed in our codes of
world. So
ethics must be followed even as we conduct research. For instance, the International
Federation of Social Workers
(ifsw) states: “Social workers have a responsibility
promote social justice, in relation to society generally, and in relation to the
people with whom they work” (2012). Potts and Brown explain: “The purpose of
anti-oppressive research is not only to produce knowledge but also to examine,
unsettle and shift the power relations” (2005: 5).
to
The History
As
of Empiricism
23
stated, feminist empiricists uphold the notion that the basic principles of
sound. For social work, this suggests that we should continue to
including, but not exclusively, quantitative methods.
Like other empiricists, feminist empiricists believe that the scientific method is
still the most effective method available for acquiring knowledge; that is, they
believe that there is a world out there that we can discover through our senses. They
point to the tremendous scientific achievements of the last centuries as proof, all
the while acknowledging that improvements to the scientific method are needed.
In social work, the type of research approach we choose should depend on the
nature of the topic being researched. A qualitative approach may be more suit¬
able for the investigation of new phenomena about which little is known. On the
empiricism
are
teach basic research methods,
other
hand, when
program,
The
we
want to
know if a certain
cause,
such
as a new
intervention
produces the desired effect, we may want to use a quantitative approach.
question is not which approach is better but which approach will best provide
the information
research
question. In many cases it makes sense to
qualitative and quantitative methods in order to take advantage of the
best of both approaches
Contrary to the beliefheld by much of the Western scientific world, the knower
is not simply an individual, but rather is an individual who is part of a larger com¬
munity. An important contribution of feminist epistemologists was their ability to
demonstrate that science is a social endeavour that it is usually conducted within a
scientific community, which in turn is part of a larger society and which is subject
to the social, cultural and political values of the time and place. An important
aspect of social work training is reflexivity. Before we engage in practice with oth¬
ers, through introspection and intersubjective reflection, we critically examine our
values and biases and the impact they have on our practice. The same is true with
to
answer our
combine
social work research. Before
we can
values and biases and how
our
conduct and what
to
hold
become
a
knower,
we
should be
social location affects the type
be
aware
of our
of research that we
knowledge.
Finally, and most importantly, feminist empiricists have affirmed the concept
that the knower is not simply male. It is safe to say, that while patriarchy is still
prevalent within the scientific community, in social work at least, both women and
men are
engaged in scientific research. As a profession committed to eradicating all
forms of oppression, including the oppression of women, the social work profes¬
sion sees this development as long overdue.
we
true
24
Statistics for Social Justice
SUMMARY
In this
chapter, we explored the history of empiricism and the roots of the scientific
early writings of Aristotle and examined the contribu¬
tions made by the British empiricists. We also looked at the important modifications
to empiricism proposed by feminist empiricists, whose work brought it to a point
where it is now more in keeping with a structural and anti-oppressive approach.
method. We discussed the
REVIEW QUESTIONS
1.
2.
Who
of the
key players in bringing empiricism into Western think¬
ing? Discuss their individual contributions.
Discuss this statement: “While the academic part of social work has largely
moved beyond the positivist view and its faith in empiricism, the professional
part of social work is as committed as ever.” Has this been true in your experi¬
were some
ence?
3. What
were
the contributions of feminist
empiricism? Discuss how their ideas
to
are
empiricists to
our
understanding of
in line with the structural approach
social work.
4.
Describe how naturalized feminist empiricism is in
search from a structural and aop perspective.
5.
“Science is
a
social endeavour.” Discuss what this
How does it relate to research conducted from
a
line with conducting
re¬
statement means to you.
structural
perspective?
3
Frequency Distributions and Graphs
In this chapter we focus on descriptive statistics and explain various methods
used to describe data. We begin by introducing different kinds of frequency
distributions, including absolute, cumulative and percentage distributions. We
describe different types of graphs, such as bar graphs, pie charts, histograms,
frequency polygons and scatterplots. We also provide guidance on how to best
use graphs, how to avoid common mistakes and how to maximize the impact
of descriptions for the purpose of meeting social justice goals. Ihe chapter
concludes with examples of studies using frequency distributions
As stated in Chapter
1, quantitative
are normal
ly divMost
ided
categories:
descriptiveresearch
statisticsmethods
and inferential
statistics.
into two broad
research
projects carried out by social workers include some form of descriptive
analysis and may also involve inferential analysis. Often, descriptive statistics are all
that are required to answer important research questions, such as, what percentage
of the population in our community is living below the poverty line? In Chapter
1, we described Campaign 2000, the national coalition dedicated to the reduction
in child poverty in Canada. It publishes a government report card every year in
which it makes extensive use of descriptive statistics to show the progress, or lack
thereof, on reducing child poverty. The following hypothetical example illustrates
how descriptive statistics can be used.
June Edwards is a young social worker who has been hired to work in the Community
Housing Department for her city. The mission of Community Housing is to provide
safe affordable housingfor individuals and families whose income is insufficient to pay
market level rent. June is aware that government cutbacks in social assistance rates and
in subsidized housing have made it impossible for Community Housing to meet the
increasing demands from low-income people. As a result, the waiting listfor housing is
growing quickly. While she accepts the fact that she has limited control over government
funding, she and her colleagues feel that, at the very least, they should document the
extent and characteristics ofpeople on the waiting list in order to determine which ones
are most in need. June is also
hoping that this information will lead to public pressure
to increase government funding.
25
26
Statistics for Social Justice
How should June
and her colleagues proceed? In what way should they describe
people on the waiting list for subsidized housing? What
format will provide the most useful information and have the greatest impact in
terms of
increasing public awareness and interest in the problem? In Chapter 1,
we stated that
descriptive statistics are used to describe the characteristics of a
sample or population, while inferential statistics are used to show a relationship
between two variables or to generalize findings to a larger population based on a
sample of cases.
the characteristics of the
FREQUENCY DISTRIBUTIONS
The
simplest way to describe the sample (or the population) under investigation is
by using frequency distributions. These are tables that display the number of times
a
specific value appears for a particular variable. A variable is any characteristic in
a
given population or sample that varies, and the values are the different measure¬
ments that a variable can take. In June s study, one of the variables that they would
naturally be interested in is income, and the values would be the different income
levels (e.g., <$10,000, $10,001-$20,000, $20,001-$30,000, etc.).
To construct a frequency distribution, we start by creating an array, which dis¬
plays all the values that exist within the data in an orderly manner, from the lowest
value to the highest value. In our hypothetical example, suppose June noticed that
there are a growing number of single-female-headed families with children on the
waiting list for subsidized housing. Let’s say that she is interested in finding out
about the number of children in these families. This information would be
because it
impor¬
indication of the sizes of the housing units needed by
month, Community Housing received requests
from 25 such families in need of subsidized housing. Before creating an array, June
lists the names of each of these women along with the number of children they
have (Table 3.1).
tant
provides
these families. In the
an
most recent
Frequency Distributions and Graphs
Table 3.1 Raw Data
Name
-
27
Clients’ Names and Number of Children
Number of Children
Cindy
1
Helen
2
Kathy
1
Chelsea
3
Taylor
2
Nancy
1
Rebecca
4
Ann
5
Paula
2
Zaina
1
Jenny
1
Mary
2
Maria
4
Anna
1
Julia
2
Elizabeth
1
Irene
3
Gisele
2
Sophie
1
Colleen
1
Sarah
2
Claudia
3
Helene
1
Pauline
1
Stephanie
2
Looking at the raw data, June can already see that most of the families have
only one or two children. In order to display her observation more clearly, she
creates an array (Table 3.2) to organize her data from the lowest to the highest
number of children. This array shows her that 11 of the 25 families have 1 child
and 8 families have 2.
28
Statistics for Social Justice
Table 3.2 Array
-
Clients’ Names and Number of Children
Name
Number of Children
Cindy
1
Kathy
1
Nancy
1
Zaina
1
Jenny
1
Anna
1
Elizabeth
1
Sophie
1
Colleen
1
Helene
1
Pauline
1
Helen
2
Taylor
2
Paula
2
Mary
2
Julia
2
Gisele
2
Sarah
2
Stephanie
2
Chelsea
3
Irene
3
Claudia
3
Rebecca
4
Maria
4
Ann
5
Absolute
Frequency Distribution
The array already provides June with
helpful information with respect to the size of
by these families. In order to make these data more useful, June
creates an absolute frequency distribution, which displays the data on the variable
the units needed
she is interested in
along with the different values. The variable is the number of
children and the values
Having already created
each value
the actual number of children in each of the families.
an array, June is able to quickly count how many times
places this number in the “Absolute Frequency” column
(Table 3.3).
occurs
for each value
are
and
Frequency Distributions and Graphs
Table 3.3 Absolute
29
Frequency Distribution - Clients’ Names and Number of Children
Number of
Absolute
Children
Frequency
Cindy, Kathy, Nancy, Zaina, Jenny, Anna,
Elizabeth, Sophie, Colleen, Helene, Pauline
1
11
Helen, Taylor, Paula, Mary, Julia, Gisele,
Sarah, Stephanie
2
8
Chelsea, Irene, Claudia
3
3
Rebecca, Maria
4
2
Ann
5
1
Names
Cumulative
Frequency Distribution
June also creates a table to show the cumulative frequency distribution (Table
3.4), which displays the cumulative total of the values. For instance, June found
that 11 of the 25 clients have 1 child and 8 of the clients have two children. So the
cumulative total for families with 1
Table 3.4 Cumulative
or
2 children is 19.
Frequency Distribution
-
Clients’ Names and Total Number of Children
Number of
Cumulative
Children
Frequency
Cindy, Kathy, Nancy, Zaina, Jenny, Anna,
Elizabeth, Sophie, Colleen, Helene, Pauline
1
11
Helen, Taylor, Paula, Mary, Julia, Gisele,
Sarah, Stephanie
2
19
Chelsea, Irene, Claudia
3
22
Rebecca, Maria
4
24
Ann
5
25
Names
Percentage Distribution
Because
for
our
example includes only 25
cases, a
relatively small number, it is
understand the breakdown without the need for any
easy
additional
larger? It may become necessary
to use different tables to make the data more readable. In this case, it may be useful
to create tables to show percentage distributions.
June creates a percentage distribution table in order to display these data in
the form of percentages. She is aware of the fact that as the waiting list for housing
someone to
tables. But what if the number of cases was much
30
Statistics for Social Justice
grows,
ful and
she will have to
to
use percentage
allow the reader
to
distributions to make her data
understand the distribution of families
more use¬
waiting for
subsidized
housing. With the current 25 cases, the 11 families who have 1 child
represent 44% of the total number of families. The 8 families with 2 children rep¬
resent 32% of the total number of families (Table 3.5).
Table 3.5
Percentage Distributions
-
Clients’ Names and Number of Children
Number of
Names
Children
Percentage
Cindy, Kathy, Nancy, Zaina, Jenny, Anna,
Elizabeth, Sophie, Colleen, Helene, Pauline
1
44%
Helen, Taylor, Paula, Mary, Julia, Gisele,
Sarah, Stephanie
2
32%
Chelsea, Irene, Claudia
3
12%
Rebecca, Maria
4
8%
Ann
5
4%
did for the absolute and cumulative frequency distributions, she
a cumulative
percentage distribution. June simply totals the per¬
centage for each value by adding it to the percentage ofthe lower value (Table 3.6).
For example, the cumulative percentage for the value “2” is calculated by adding
32% to 44%. The advantage ofthis table is that it shows the cumulative percentage
of families with certain numbers of children. June could show that 76% offamilies
on the wait list have 1 or 2 children. Down the road, this information could
help
with program planning.
Just
as June
could also
create
Table 3.6 Cumulative
Percentage Distributions -
Clients’ Names and Number of Children
Number of
Cumulative
Children
Percentage
1
44%
Helen, Taylor, Paula, Mary, Julia, Gisele,
Sarah, Stephanie
2
76%
Chelsea, Irene, Claudia
3
88%
Rebecca, Maria
4
96%
Ann
5
100%
Names
Cindy, Kathy, Nancy, Zaina, Jenny, Anna,
Elizabeth, Sophie, Colleen, Helene, Pauline
Frequency Distributions and Graphs
31
It is often useful to show both the absolute
distributions in
one
table. In Table 3.7,
frequency and the percentage
June shows the absolute frequency and
percentage distributions. In Table 3.8, June displays the cumulative frequency and
percentage
distributions.
Table 3.7 Absolute
Frequency and Percentage Distributions
-
Clients’ Names and Number of Children
Number of
Absolute
Absolute
Children
Frequency
Percentage
1
11
44%
Helen, Taylor, Paula, Mary, Julia,
Gisele, Sarah, Stephanie
2
8
32%
Chelsea, Irene, Claudia
3
3
12%
Rebecca, Maria
4
2
8%
Ann
5
1
4%
Names of Clients
Cindy, Kathy, Nancy, Zaina, Jenny,
Sophie, Colleen,
Anna, Elizabeth,
Helene, Pauline
Table 3.8 Cumulative
Frequency and Percentage Distributions
-
Clients’ Names and Number of Children
Number of
Cumulative
Cumulative
Children
Frequency
Percentage
1
11
44%
2
19
76%
Chelsea, Irene, Claudia
3
22
88%
Rebecca, Maria
4
24
96%
Ann
5
25
100%
Names of Clients
Cindy, Kathy, Nancy, Zaina, Jenny,
Colleen,
Anna, Elizabeth, Sophie,
Helene, Pauline
Helen, Taylor, Paula, Mary, Julia,
Gisele, Sarah, Stephanie
Grouped Frequency Distribution
When
working with large data sets with many possible values, it is sometimes
comprehend the meaning of the data if they are grouped.
For instance, June is interested in creating a table that displays the various income
levels of all of the families on her waiting list. Rather than including every annual
income amount, it makes more sense to group the incomes of the clients. June
creates groups of income, such as $0 to $4,999, $5,000 to $5,999, $6,000 to $9,999
and so on (Table 3.9).
easier to visualize and
Statistics for Social Justice
32
Table 3.9
Grouped Frequency Distribution
Income Levels and Number of Families
Number of Families
Income Level
Absolute
Frequency
$0 to $4,999
5
$5,000 to $5,999
16
$6,000 to $9,999
28
$10,000 to $14,999
14
$15,000 to $19,999
6
$20,000 and
-
2
over
Determining which frequency distributions to use depends on what information
most useful and have the greatest impact. As social workers, we are naturally
concerned about the plight of individual families and their children, but providing
the numbers in a manner that is easily understandable to agency administrators,
will be
government departments and the general public will have the best results in terms
of reaching our
social justice goals. (See Box 3.1 for an example.)
Box 3.1 Shelter Use
The
by Homeless Families
following summary describes a study using descriptive statistics.
Homeless families
diverse in structure, with some
including two parents, and
headed by a single parent (usually female). Family homelessness is largely
underpinned by structural factors, including inadequate income, lack of affordable
housing and family violence. Following the withdrawal of government housing
programs and decreased supports, more families are turning to emergency shelters.
are
many
A
significant finding from the Segaert study was that the sharpest increase in shelter
has been among families (in most cases headed by women) and therefore
children. For instance, the number of children staying in shelters increased by over
50% between 2005 (6,205) and 2009 (9,459). Segaert identifies that the average
length ofshelter stay for families was 50.2 days, an increase of 50% over five years, and
more than
triple the average stay for the total population of people who experienced
use
homelessness. This
that while families accounted for just 4%
of all shelter
they used 14% of total bed nights. This puts incredible pressure on the family
shelter system, which has not had the capacity to deal with this increase. It is worth
noting, once again, that these figures do not include female-headed families using
shelters providing accommodation for women fleeing violent partners.
means
stays,
Source:
Sagaert 2012
Frequency Distributions and Graphs
33
GRAPHS
While
frequency tables are useful for presenting data in a readable manner, with
larger data sets, graphs often display these data in a more easily understandable
way. The advantage of graphs is that they can quickly convey, in the form of a
picture, large volumes of data. The drawback is that they often sacrifice detail, but
they are nevertheless effective in conveying meaning to the audience. This may
be especially important, for example, when making a presentation to a grassroots
association whose members do not have the academic training to comprehend
tables that convey large data sets.
Graphs displaying one or two variables generally have two perpendicular lines.
The horizontal line is called the x axis, the vertical line is they axis, and the point
where they join is the point of origin.
Bar
A
Graph
of graph is the bar graph, also called a bar chart. In this graph,
equal widths and they do not touch; this is to acknowledge
that the data are qualitative in nature with values at the nominal or ordinal level
of measurement. As stated in Chapter 1, nominal level measurement involves
classifying observations into mutually exclusive categories, with no inherent order
or rank. If we use the
example of June’s study, looking at families with children,
versus families with no children, versus
single adults, we could create a frequency
distribution table with the three categories of clients. Let’s assume that there are a
total of 26 families with children on the waiting list, 15 families with no children
and 38 single individuals. Table 3.10 shows the frequency distribution for these
data. Category one includes families with children, category two shows families
with no children and category three covers the single individuals. Figure 3.1 is a
bar graph showing these same data. The bar graph clearly shows that the majority
of clients are single individuals with no children and the next largest category is
common
the bars
are
type
drawn in
families with children.
Table 3.10
Categories of Clients
Cumulative
Clients
Categories
Frequency
Percent
Families with Children
1
26
32.9%
Families,
2
15
19%
51%
3
38
48.1%
100%
79
100%
no
Individuals
Totals
Children
Percent
32.9%
34
Statistics for Social Justice
Figure 3.1 Bar Graph - Three Categories of Clients
Client Categories
Pie Charts
Pie charts
generally used for nominal level data. When the values add up to a
pie chart can be used. Each piece of the pie can reflect a segment of the
whole. If we again turn to the example ofthe data from June s study shown in Table
3.10, which displays the three categories of clients, the pie chart for these data is
illustrated in Figure 3.2.
whole,
are
a
Figure 3.2 Pie Chart - Three Categories of Clients
clients
■ roc
□ :oc
□ 3.00
Frequency Distributions and Graphs
Pie charts
effective if there
are
are
few
35
categories. If there are too many
categories the chart can become confusing and the data will not be understood.
Pie charts
frequently used to show the breakdown of a budget of an
organization — hence the phrase “bigger slice of the pie.”
are
Histograms
Another type
of graph that is commonly used is the histogram. This is similar to
that the bars do touch; this is to reflect the fact that the data is
rank ordered and to reflect differences in quantity. The data must be at the interval
or ratio level. As stated in
Chapter 1, the interval/ratio level involves observations
that are mutually exclusive, have an inherent order and have equal spacing between
categories. The height of the bars varies to reflect the difference in frequency. The
bars can be of equal width, reflecting equal category intervals; however, the width
can
vary if the category or “bin” sizes vary. In some cases, the bars can be drawn on
both sides of they axis, reflecting positive values on the right, and negative values
on the left, but this
may be confusing for the reader.
Let’s say that Community Housing’s intake form for clients requesting subsi¬
dized housing includes information on income level and years of education. Both
of these variables are at the ratio level. Looking first at income levels of all clients
from the previous month requesting subsidized housing, June creates a frequency
table (Table 3.11).
the bar graph except
Table 3.11 Absolute
Frequency and Percentage Distributions -
Income Levels of Clients
Income Level
Category
Frequency
Percent
Cumulative Percent
$0 to $4,999
1
2
6.7
6.7
$5,000 to $5,999
2
5
16.7
23.3
$6,000 to $9,999
3
11
36.7
60.0
$10,000
$14,999
4
7
23.3
83.3
$15,000 to $19,999
5
3
10.0
93.3
$20,000 and
6
2
6.7
100.0
30
100.0
to
over
Total
Next she used these data to
develop a histogram (Figure 3.3). This histogram
of the clients fit in categories 2, 3 and 4, with income
levels ranging from $5,000 to $15,000. The distribution of income levels in this
histogram resembles a bell curve, also called a normal distribution, which we
discuss in Chapter 5.
shows that the majority
36
Statistics for Social Justice
Figure 3.3 Histogram
-
Income Levels of Clients
June then does the same with the data on education levels. She creates a frequency
table and
a
histogram (Table 3.12 and Figure 3.4).
Table 3.12 Absolute
Frequency and Percentage Distributions - Years of Education
Years of Education
Frequency
Percent
Cumulative Percent
6
2
6.7
6.7
7
7
23.3
30.0
8
9
30.0
60.0
9
We
6
20.0
80.0
10
3
10.0
90.0
11
2
6.7
96.7
100.0
12
1
3.3
Total
30
100.0
clearly see that the majority of clients had education levels of seven,
eight and nine years and that the distribution of education levels is beginning to
look like a normal curve. This information is useful because, as we explain in later
chapters, certain statistical tests can only be done for interval/ratio level data that
are normally distributed.
can
Frequency Distributions and Graphs
37
Figure 3.4 Histogram - Years of Education
education
Frequency Polygons
Frequency polygons are similar to histograms, except that dots are used instead of
bars. If we place a dot at the top middle of each bar, then join the dots, we have a
frequency polygon. Polygons are normally used to display data at the ratio/interval
level. Let’s consider, for example, a frequency polygon for grade levels reached. If
the sample is large enough, the polygon will begin to resemble a normal curve.
Scatterplots
Scatterplots are useful for displaying the relationship between two variables which
both at either an interval or ratio level. For example, we could use a scatterplot
to show the relationship between the numbers of treatment sessions and the score
on a related
psychological test. The two variables would each be shown on either
the x axis (the independent or predictor variable, discussed in Chapter 10), or the
y axis (the dependent or outcome variables, also discussed in Chapter 10). Create
a
scatterplot by plotting the points on the graph for each subject, such that each
subject is represented by one point of the graph. If there is a relationship between
the two variables, the dots should begin to resemble a line going up diagonally, in
the case of a positive relationship, or down in the case of a negative relationship.
In June’s study, the data on income and education levels are both at the ratio
level. June believes that there is likely a relationship between years ofeducation and
income level, such that clients with higher education levels also have higher levels
of income. She decides to plot these data on a scatterplot with each point on the
chart showing where the education level and income level intersect (Figure 3.5).
are
38
Statistics for Social Justice
Figure 3.5 Scatterplot - Education and Income Levels
The
scatterplot shows data from fourteen subjects. Each point on the graph
represents one subject in the study. One subject, with 6 years of education, has
an
income of $2,000.
The graph shows a definite pattern, with the higher the
education
level, the higher the income level. June therefore concludes that there
positive relationship between education and income. This positive type of
relationship is discussed further in Chapter 10 when we look at the Pearson r
is
a
correlation.
COMMON MISTAKES
The purpose
of displaying data in the form of tables and graphs is to provide the
to support an argument in a way that words alone
cannot do. While this method of describing data is useful, it is also important to
make sure that we are not misrepresenting the data. For instance, ifJune receives
5 new applications, each from families with children, she would be correct in stat¬
ing that there was a 20% increase in the families with children on her waiting list.
Five families does not seem like a lot, but 20% could send a misleading message.
Another common mistake made by researchers is to include too much informa¬
tion in the table or graph. If a table is too complex, it will be hard for the reader to
interpret the information. It is easy with the aid of computers to create complex
tables with large amounts of data. As a general rule, it is much better to develop,
simple easy-to-understand tables and charts.
reader with visual evidence
Frequency Distributions and Graphs
39
EXAMPLES OF REAL STUDIES
The
following example, from Christine Marlow (2005: 240), provides a good
frequency distributions can be used. In this data
there are twenty-five university students with preschool-aged children. Each
illustration of how the various
set
student represents one
observation and the study is interested in their number of
children, their ethnicity, their expressed need for daycare and the number of miles
they live from campus. The different ethnic groups are coded as follows: 1 = white,
non-Hispanic; 2= Hispanic and 3= African American. On the expressed need for
daycare, a score of 1 is for the least need, while 4 is for the greatest need. The number
of children and the miles from campus are represented by the actual numbers as
reported by the students. In terms of level of measurement, the variable ethnicity
is at the nominal level, the need of daycare is at the ordinal level, and the number
of children and distance from campus are at the ratio level. Table 3.1 shows the
frequency and percentage distributions for all the variables.
Table 3.13
-
Frequency and Percentage Distributions
Four Variables
Observation #
by Each Observation
Number of
Children
Ethnicity
Need for
Miles from
Daycare
Campus
1
2
1
3
2
2
1
1
4
1
3
1
3
4
10
4
1
1
4
23
5
1
2
3
4
6
1
1
3
2
7
2
2
4
1
8
2
1
3
1
9
1
2
2
6
10
3
1
4
40
11
2
1
3
2
12
1
2
1
1
And
so on
Source: Marlow 2005: 240
40
Statistics for Social Justice
Tables 3.14 to 3.17 describe the
Table 3.14
frequency distributions for the four variables.
Frequency and Percentage Distribution
Label
-
Ethnicity
Value
Frequency
Percent
Non-Hispanic white
1
17
68
Hispanic
2
6
24
African American
3
Total
Table 3.15
2
8
25
100
Frequency and Percentage Distributions - Need for Daycare
Label
Value
Frequency
Percent
No need
1
3
12
A little need
2
2
8
Some need
3
7
28
Great need
4
Total
Table 3.16
13
52
25
100
Frequency and Percentage Distributions - Number of Children
Value
Frequency
Percent
1
13
52
2
11
44
3
Total
Table 3.17
1
4
25
100
Frequency and Percentage Distributions - Miles from Campus
Value
Frequency
Percent
Less than 5
14
56
5-9 miles
5
20
10
-
14 miles
15 miles and
Total
The
over
2
8
4
16
25
100
following example of a study on homelessness also provides good illustra¬
use of
frequency tables and graphs to inform governments, policy
analysts and the general public about this important social justice issue. In 2009,
research was conducted by Bri Trypuc and Jeffrey Robinson whose goal was “to
help people across Canada have a better understanding of our homeless situation
based on evidence rather than myths, and bring to the public s attention programs
tions of the
Frequency Distributions and Graphs
41
that work in
helping the homeless. With better information, we hope Canadians
(2009: 1).
This research found staggering numbers. There are an estimated 157,000 home¬
less people in Canada. Of the 20% who remain homeless for more than three
months (and are therefore considered chronically homeless), life on the streets
can lead to addiction, abuse and suicide. An estimated 1,350 homeless
people die
each year; the average life expectancy of a homeless person is 39 years. Not only
is there a large personal cost to homelessness, but there is also a societal cost; it is
estimated that Canada spends 1.1 billion, or $35,000 per person, per year to keep
homeless people in shelters, jails or hospital emergency wards. As stated by Trypuc
and Robinson, homelessness occurs in every region of Canada. Figure 3.6 is a
bar graph showing the distribution of shelter use in the provinces and territories.
can
make informed decisions that will create results for those in need”
Figure 3.6 Distribution of Shelter Use by Province
§ 1200
Provinces
Source:
Trypuc and Robinson 2009: 4
The
graph reveals that the prevalence of shelter use is highest in Alberta and
shows the prevalence ofshelter use,
the authors of the study also felt it necessary to put a human face to the statistics:
lowest in Atlantic Canada. While the bar graph
Andrew
boy scout and a good student in school.
began. His schizophrenia was difficult to
control, he rebelled against medications which left him feeling numb. His
family could no longer cope alone with Andrew’s erratic behaviour and
he went to live in a group facility. When this did not address his needs,
Andrew struck off on his own. Living alone was too great a challenge,
was a
When he
was
happy child,
a
17 the voices
42
Statistics for Social Justice
and without
steady wages, Andrew was evicted from his apartment and
became homeless.
Carrie has
long blond hair and beautiful blue eyes and loves to read
Dostoevsky. At age 8 her step-father began raping her. Living with ongoing
sexual abuse, Carrie escaped from ‘home’ at age 16. Carrie lives on the
streets with Patches, her part-Rottweiler dog. Patches is her only source
of unconditional love and companionship, offering protection, trust and
body heat. Dogs are not allowed in the emergency shelters, so for four
years Carrie has lived in a make-shift shanty camp.
Rob drank his first beer with his dad when he
he
was an
was
11. Within 2 years
alcoholic, hiding his daily drinking from his parents. To pay
for booze, Rob
began stealing, starting first with petty theft escalating
By 20, Rob was in federal jail. For the next 22 years,
he was either in jail or drinking, moving from job to job. Rob would be
dry for sometime, holding a job, but his alcoholism was always lurking,
leading Rob to homelessness for years.
Homelessness is the result we see
people lining up at shelters,
sleeping on park benches, and squatting in doorways. But the causes
to
bank robberies.
-
of homelessness
abuse
are
are
varied.
Addiction,
severe
mental illness, and child
primary causes of years living on the streets. In most every case
homelessness is
triggered by a single crisis beyond a persons control
which cascades. Without effective early intervention and family or com¬
munity support, people fall through the gaps leading to a desperate life
on
the
of God we
or
Homelessness
streets.
too
can
happen to
anyone.
But for the Grace
could be homeless.
Sadly we may view the homeless through the distorting lens of morality
character, judging those living on the streets as lazy, undeserving or
less worthy than
ourselves. Worse is the attitude that people choose to be
in their right mind would choose to be homeless with
its violence, stress and degradation. Sometimes sleeping on the streets is
safer than being in a crowded emergency shelter. Homelessness reflects
a failure in us and
organizations to provide appropriate and responsive
care. It doesn’t have to be this
way. There are effective programs and ser¬
homeless. No
one
vices that work to intervene with those who
of these programs
constrained
In
are
homeless. The waitlists
and their capacity to work with
by the lack of funding. (2)
more
people, is only
study published in 2005, Steve Pomeroy examined the question of the
addressing homelessness through institutional settings such as
hospitals, treatment centres, prisons, emergency shelters and hostel programs,
a
relative cost of
Frequency Distributions and Graphs
43
compared to community-based and affordable housing. Figure 3.7 provides a
comparison of the approximate per diem costs averaged over four major cities —
as
Halifax, Montreal, Toronto and Vancouver.
Figure 3.7 Comparative Costs of Responses to Homelessness
Prison/Jail
Psychiatric Hospitals
Emergency Shelters
Emergency Shelters Families
Emergency Shelters Family Violence
Psych/Detox Treatment Centres
Group transitional Supportive
Group Longterm Supportive
Board/Room House - Community
Supports
Independent Apart. Single
Independent Apart. Family
0
■ per
Source:
50
100
150
200 250 BOO 350 400
diem cost
Pomeroy 2005: iv
What the bar
graph by Pomeroy clearly reveals is that it is much cheaper, and
arguably more effective, to deal with the problem of homelessness by providing
supportive and affordable housing than it is to keep people in shelters, hospitals
and
prisons.
SUMMARY
In this
chapter, we introduced descriptive statistics. We showed how to display
quantitative data in a variety of frequency distribution tables, including absolute,
cumulative and percentage distributions. We described a few of the main type of
tables, charts and graphs, such as bar graphs, histograms, pie charts and scatterplots.
These illustrated how complex numerical data can be displayed in ways that are
easily understood by academics, government officials and by the general public.
We added a word of caution that including too much complex data may reduce
the effectiveness of the table
or
chart.
44
Statistics for Social Justice
REVIEW QUESTIONS
1.
Describe
a
situation where you
lute, percentage or
would want to use a distribution table (abso¬
cumulative) and why you would choose
to present your
data in this way.
Discuss each
3.
type of graph (bar graphs, pie charts, histograms, frequency pol¬
and scatterplots), indicating their strengths, weaknesses and the level of
data each is appropriate for.
Provide an example of how data could be manipulated to present an inaccu¬
4.
Discuss how tables and
2.
ygons
rate
picture.
graphs
work researcher compared to a
cal framework.
may be used differently by a structural social
researcher working from a mainstream empiri¬
4
Central Tendency and
Variability
In this chapter we look at measures of central tendency and examine ways in
which we can describe a typical case. We show how to calculate mode, median
and mean and examine the impact of outliers. We also discuss the degree
of variability that exists in a population, using range, variance and standard
deviation. Together, Chapters 3 and 4 provide enough information to allow
researchers to use descriptive statistics to argue for social change.
Social workresearchers
want to descri
e things lihow
ke onetheirof
typical
the and practitioners
of
of may
the clients
theirbcaseload,
their
cases,
range
scores
on
clients compare to
others and how their clients compare to the general population.
following hypothetical example.
Anne Matthews is a social worker for a municipal social assistance department. She
has a large caseload of over a hundred people, the majority of whom are single mothers.
She is familiar with the literature that states that the rates of depression among lowincome single mothers is proportionally much higher than itisin the general population.
She decides to find out ifthis is true ofthe clients on her caseload. Recognizing that she is
not be able to conduct the research herselfbecause of her conflict of interest in this situa¬
tion, she approaches the School of Social Work at her local university and asks a social
work research professor ifsome of her students would be interested in conducting the study.
After getting the necessary approvals and clearance from the university’s research
ethics board, the student researchers ask the single mothers on Anne’s caseload if
they would be willing to participate in their study.
They decide to use the Depression Scale, a well-known standardized selfadministered (paper and pencil) measure of depression. The scale reveals scores
that range from 1 (lowest possible depression score) to 25 (highest possible
depression score), with scores of 1-5 indicating no depression, and scores greater
than 20 indicating extreme depression.
Of the 75 single mothers on Anne’s caseload, 55 agree to participate, so the
sample size is 55. After collecting the data, the students find that the mean (average)
score of the
single mothers is 8, well above what is considered within the normal
Let’s look at the
45
46
Statistics for Social Justice
and many of Anne’s clients score over 10. This confirms for Anne what the
indicates, that the single mothers on her caseload are far more likely to
be depressed than people in the general population.
What she is able to show as a result of the research is that a typical client on her
caseload is a single mother in her early twenties with one preschool-age child, no
permanent address, has been staying temporarily with friends and relatives and is
clinically depressed. This description of a typical case provides the management
of Anne’s department with a real sense of the problem of homelessness involving
female-headed, single-parent families and the level of depression that these clients
are
facing. It also raises concerns about the lack of support available for families
with very young children.
range,
literature
MEASURES OF CENTRAL TENDENCY
In
descriptive statistics, it is often useful to provide an idea ofwhat a typical case in
the population or sample being investigated looks like. Measures ofcentral tendency
describe
is
typical cases in research. There are three accepted ways of describing what
typical: mode, median and mean.
Mode
The mode is the value that
at
occurs
most
often with the data. It
all four of the levels of measurement:
can
be used with data
nominal, ordinal, interval and ratio. For
example, the following data set displays test scores out of 10: 5,3,8,5,4,7,4,5,6,6.
As a first step to determining the mode (or median), we display the data set in
an
array (discussed in Chapter 3). An array organizes the data from the lowest
value to the highest or the other way around. The array for the data above looks
like this: 3,4,4, 5, 5, 5, 6, 6, 7, 8. With the data arranged from smallest to largest,
it is easy to see which values occur more than once and which occur most often
(the mode). In this example, the value 5 appears most often, and therefore, the
mode is 5. Data sets can be bimodal, with two values occurring most frequently.
Consider the following array: 3,4,4,5,5,5,6,6,7,7,7,8. The values 5 and 7 each
appear three times. Therefore, this data set is bimodal, and we would report the
modes as both 5 and 7. If we were to draw a histogram (as described in Chapter
3), it would have two distinct peaks (Figure 4.1). Data can also be multimodal,
that is,
have
more
than two modes.
Central
Tendency and Variability
47
Figure 4.1 Histogram - Bimodal Distribution
Histogram
scores
Of the three
for its
used
use.
measures
While it
often
can
other
of central
tendency, the mode has the fewest restrictions
be used with all levels of measurement, the mode is not
The
problem is that, while it does identify the
often, this value may not be the most accurate portrayal
of a typical value. When dealing with data that is at the ordinal, interval or ratio
level, a more accurate description of central tendency can be obtained by using
as
as
measures.
value that appears most
either the median
or
the
mean.
Median
Unlike the mode, the median has the restriction that it
that
are
at
the ordinal, interval
level where
or
can
only be used for data
ratio level. This is because the data must be at
a
they can be placed in a meaningful rank order, from lowest to highest
around. The median is the halfway point in the list of values. It
divides a set of values into two equal halves. If the values were arranged in an array,
we would simply count the number of values and divide by two. For example,
consider the following data set: 5, 3, 8, 5, 4, 7, 4, 5, 6. Displayed as an array this
data set is as follows: 3,4,4, 5, 5, 5, 6, 7, 8. The value at the halfway point is 5, and
therefore 5 is the median. If there is an odd number of values, it is easy to pick the
middle number. In our example there are four values below 5 and four values above
5. If there is an even number of values, the halfway point will fall between two
values, as with the following data set: 3, 4, 4, 5, 6, 7, 7, 8, 8, 9. The halfway point
is between the 6 and the 7. Therefore, the median is 6.5, or the average of 6 and 7
(6 plus 7 divided by 2). Unlike the mode, the median may not correspond to an
actual value in the distribution; the halfway point may lie in between two values.
or
the other way
Statistics for Social Justice
48
The main
advantage ofthe median is that it is not influenced by extreme values.
(discussed in Chapter 5) and thus is com¬
monly used to describe income levels because this type of data tends to be skewed.
The main disadvantages are that it cannot be used for variables at the nominal level
and it can be a bit more difficult to compute than the mode.
It is often used for skewed distributions
Mean
The mean,
commonly referred to as the average, is the most common method of
describing central tendency. It is easy to compute: add all the values together and
divide by the number ofvalues. It is viewed as the most accurate measure of central
tendency because it uses all ofthe values in the data set in its computation; it is not
simply the one in the centre of an array, as is the case ofthe median, or the one that
appears most frequently, as is the case of the mode. Consider, for example, the fol¬
lowing data set: 5, 3, 8, 5,4,7,4, 5,6,6. If we add all the values, we arrive at a total
of 53. Since there are 10 values, we divide 53 by 10, and we obtain a mean of 5.30.
Outliers
It may
happen that one value is atypical. Consider the following data set: 3, 4, 4,
5,5, 5,6, 6,7, 8, 15. We
is considered
side of the
an
area
see
that the value 15 is unlike the other values. This value
outlier. If we
were
to
where the other values
draw
are
a
histogram, this value would lie out¬
found. Of all of the
tendency, the median is the least affected by outliers, but the
Figure 4.2 Histogram with Outlier
Histogram
= 618
Std. De v - 3 .25
Mean
N-11
scores
measures
mean
of central
certainly is.
Central Tendency and Variability
49
Figure 4.2 illustrates the main disadvantage of the mean, which is that it is
by outliers. It is also the most restrictive of the three measures of central
tendency in the sense that it can only be used with interval or ratio level data.
affected
DECIDING WHICH TO USE
The decision of which
measure
what describes the data
most
of central
tendency to use is usually based on
accurately. With nominal data, the mode is the only
appropriate measure of central tendency. With ordinal data, it may be helpful to use
both the mode and median, but we cannot use the mean in this case. With interval
and ratio data that have skewed distributions, such as income values, the median
is used most often. With interval and ratio level data that
normally distributed
mean is used most commonly and provides the most accurate
measure of central
tendency. In a perfect normal distribution, the mode, median
and mean are all the same value (discussed in Chapter 5). Consider the following
hypothetical example.
are
(bell shaped), the
Assume that you are a
social work student doing a placement at a community agency
working with new immigrants, conceptualized by the agency as those who have been in
the country five years or less. This agency has recently developed a Head Start Program
for the preschool children in the community. The agency director is concerned that both
recent immigrants and families with the lowest income levels are not accessing the pro¬
gram. She asks you to do a review of case files offamilies in the community to determine
who is and who is not accessing the program.
You and the director have come up with the following two research questions:
1.
Who is
2.
What is the income level of the
You
accessing the Head Start Program?
people using the Head Start Program?
randomly select 25 active case files to answer
you construct Table 4.1 to describe your data.
your
research questions, and
50
Statistics for Social Justice
Table 4.1 Five Variables
by Each Observation
Number of
Times
Number of
Families
Preschool
Ethnicity
Years in
Head Start
Canada
Program in
Children
1
2
Using
Income
(x 1000)
Rounded
to
the
the Previous
Nearest
Month
1,000
3
4
5
6
1
2
1
5
7
35
2
4
2
1
1
8
3
3
3
3
4
10
4
1
1
5
6
23
5
2
2
3
3
24
6
2
1
2
3
15
7
4
2
1
1
8
8
2
1
2
3
16
9
1
2
4
4
26
10
3
1
5
7
40
11
2
1
2
3
20
12
3
2
2
1
7
13
1
1
1
2
23
14
2
3
2
2
7
15
3
3
2
4
8
16
1
1
3
4
19
17
1
1
4
6
15
18
2
1
3
4
25
19
1
2
2
1
9
20
2
4
2
1
8
21
2
1
3
4
22
22
1
2
4
7
36
23
2
1
4
5
31
24
1
1
3
4
28
25
2
1
2
3
18
Total 50,
Mean 2.00
Note that columns 2, 5
Mode 1
Total 70
Total 90
Mean 2.80
Mean 3.6
and 6 contain the actual numbers
Median 19
reported by the par¬
ticipants. For the data on the ethnicity of recent immigrants (column 3), 1 stands
as
Central Tendency and Variability
for
51
European, 2 is for South Asian, 3 is for Arabic, 4 is for Chinese, and 5 is for
ethnicity is at the nominal level of measurement, and the other
other. The variable
variables
are
at
the ratio level.
Because the variable for the number of preschool
children is at the ratio level,
any of the three measures of central tendency. However, the one that
is used most often for ratio level data is the mean, which is calculated by adding
up the total number of children (50) and dividing by the number of families (25).
Therefore, the mean number of children is 2.00 (50 4- 25 = 2.00).
we can use
Table 4.2
Frequency and Percentage Distribution
-
Ethnicity
Ethnicity
Value
Frequency
Percentage
European
1
14
56
Arabic
2
7
28
Chinese
3
3
12
Other
4
1
4
25
100
Total
For the variable
ethnicity, since we are dealing with nominal level data, the only
of central tendency that is appropriate is the mode. Table 4.2 clearly shows
that the value 1 occurs most frequently. Therefore, the mode for the ethnicity of
measure
immigrants is 1, which represents immigrants who identify as European.
in Canada is at the ratio level of measurement; therefore, any
of the three measures of central tendency can be used. We chose to use the mean,
recent
The variable years
number of years in Canada, as it is the most often used and the one
typical client, given that there are no outliers within
this variable. The mean for this variable is 2.8 years in Canada (70 4 25 = 2.80).
The variable for the number of times new immigrant families used the Head
Start Program is also at the ratio level and any of the measures of central tendency
or
average
which will best represent a
can
be used. We chose
the Head Start
to
use
the
mean
number of times families have attended
Program which is 3.6 (90 4 25
The variable income is also
=
3.6).
the ratio level,
at
but instead of reporting the
chose to report the median. Remember that the
mean, we
income is to
use
the median
as
the
measure
of central
common
practice with
tendency as it is often
more
representative of the sample or population. This is because income distribution in
the
general population has a positively skewed distribution, making the median a
accurate measure of a typical case. In our sample of 25 families, the median
more
income level is
to
also combine the values into four income
be able to quickly examine those in the lowest income bracket (income
$10,000).
groups, to
less than
$19,000. We decide
52
Statistics for Social Justice
Let’s
original research questions: Who is accessing the Head
Program? and What is the income level of the people using the Head Start
Program? We can now describe the typical family in our sample of agency clients
that use the Head Start Program. For the most part, they come from either Europe
or Arabic
speaking countries, have been in Canada between 1 and 3 years, and have
return to
our
Start
a
median income of $ 19,000.
With respect to
the director s concern that those with the lowest income (con¬
ceptualized as those with an annual income of less than $ 10,000) are not using the
Head Start Program as often, we can see that those families only used the Head Start
Program once as compared to an overall mean of almost 4 times (see Table 4.3).
Table 4.3
Frequency - Use of Head Start Program by Each Low-Income Family
Income
Frequency of Use
7,000
1
7,000
2
8,000
1
8,000
1
8,000
1
8,000
4
9,000
1
Mean
1.6
This confirms the
held
by the director that the lowest income families
making much use of the program. It would now be important to find out
why. Is it because of the transportation costs or are there other reasons?
are
concern
not
VARIABILITY
Variability, also called dispersion, tells us how the values are distributed, whether
all the values are close to the mean or spread apart. Why is it important to describe
variability? If we provide only the measure of central tendency it may not give us
a
complete picture of the cases in our sample or population. For instance, in the
previous example of single-female-headed families, we indicated that the mean
score on the
Depression Scale was 8, but how many clients scored close to this
mean? If most clients scored between 5 and 10,
somewhat
this would indicate that most
are
clinically depressed. But what if many clients scored within the normal
range (below 5) and many others scored at the more severe range of depression
(above 20)? This would give us a different picture of the clients in our population
and may require a different approach.
Central Tendency and Variability
There
five
53
of variability: range,
interquartile range, mean deviation,
chapter, we focus on those that are used
range, variance and standard deviation.
are
measures
variance and standard deviation. In this
most
often:
Range
The range
is the difference between the lowest and highest values. It is simply cal¬
culated
by subtracting the minimum value from the maximum value and adding
are included. Consider the following data set: 5,
3, 8,5,4,7,4, 5,6, 6. We start by displaying this data set as an array: 3,4,4, 5, 5,5,
6, 6, 7, 8. Next we calculate the range: 8 - 3 + 1 = 6. For this data set, the range is
6. The main disadvantage of the range is that it is influenced by outliers.
Consider the array from the previous example, except with an outlier: 3, 4, 4,
5, 5, 5, 6, 6, 7, 12. Now the range would be 10 (12 - 3 + 1 = 10). This data set has
only one different value (changing an 8 to a 12), and the range increases from 6 to
10, indicating that there is more variability within the data.
1; 1 is added so that all the values
Variance
be defined
Variance
can
from the
mean.
the number of
It is the
scores
deviation and is
a
as
the average
sum
of the
of the dispersion of the individual scores
squared deviations from the mean divided by
minus 1. It is the basis of the calculation for the standard
fundamental way
of describing the variability of a normally
distributed variable.
It is calculated
using the following formula.
s2
Zfx-y)2
n-i
=
where:
s2
=
Variance of the
£
=
Sum of
x
=
Individual
x
=
Mean of the
n
=
Number of participants
Consider the
data
set is
sample
raw score
sample
following array: 2,3,3,4,4,4,5,5,6. The total of the values in this
are 9 values, we divide 36 by 9, which gives us a mean
36. Because there
of 4. Next, we calculate the variance as
subtract the
from it, and square
follows:
we
take each value in the data set,
this value. This gives us what is called the
squared deviation from the mean. (This makes sense as it is simply how much
the values are deviating from the mean, squared.) The reason we have to square
the difference is because if we added up the differences without squaring them,
we would come up with 0 every time. Next, we total the
squared deviations and
mean
54
Statistics for Social Justice
divide this value
by the number of participants (in this case 9) minus 1. This value
is what is called the variance.
x
-
x
x-x
(x-x)2
2
-
4
-2
4
3
-
4
-1
1
3
-
4
-1
1
4
-
4
0
0
4
-
4
0
0
4
-
4
0
0
5
-
4
1
1
1
5
-
4
1
6
-
4
2
4
Total
12
12 h-
Box 4.1 Note
on
(9-l)
=
1.50 (variance)
Calculating Formulas
This is the first time in this text that
we
introduce
a
formula. In order
to
use
this
formula
properly, you need to ensure that you are following the order of operations,
which states the order in which different parts of the equation must be calculated.
A common acronym used to remember the order is bedmas: brackets, exponents,
division, multiplication, addition, subtraction. There are a number of YouTube
videos that do a good job of explaining this, if you need a refresher. A second
practice you may wish to review is the proper application of rounding principles.
Keep in mind that we usually want to keep two decimal places. When we round,
we
typically use the rule that a 5 or more causes the number to the left to round up,
whereas a 4 or less causes it to round down. For example, 1.546 would round up to
1.55, whereas 1.544 would round down to 1.54.
THE STANDARD DEVIATION
Standard deviation is
by far the most commonly used measure of variability in
by getting the square root of the variance. The formula
statistics. It is calculated
is
as
follows:
EC*
n
-
*)2
1
Notice that this formula is very
ference is that you
similar to the formula for variance; the only dif¬
take the square root of the calculated number as the last step
Central Tendency and Variability
(therefore, the
square root
variance
score
above of 1.50, we
Table 4.4
Depression Scores
55
of the variance is the standard deviation). Using the
obtain a standard deviation of 1.22, by simply
taking the square root of the variance (Vl-50 = 1.22).
How does the standard deviation help us to understand what a sample or a
population looks like? Let’s use the example of the research on depression levels
of single mothers, described at the beginning of this chapter. To make the calcula¬
tion of the standard deviation a bit easier, let’s take a sub-sample of 10 participants
from the 55 participants who originally agreed to be part in the study and display
their score in a table (Table 4.4).
Depression
Client
Client
Depression
Score
Score
Mary
6
Jane
7
June
8
Shelley
7
Susan
10
Tania
8
Julie
8
Sonia
8
Martha
9
Tina
9
If we add up
these
sample), we obtain
obtain
a
scores
a mean
and divide by 10 (the number of participants in our
of 8. If we then calculate the standard deviation, we
standard deviation of 1.15.
x-x
(x-x)x
6
8
-2
4
8
8=0
0
10
8=2
4
=
8
8=0
0
9
8=1
1
7
8=1
1
7
8=1
1
8
8=0
0
8
8=0
0
9
8=1
1
Total
12
-r
(10 - l)
=
12
1.33 (variance)
V 1.33= 1.15 therefore the standard deviation is 1.15
Statistics for Social Justice
56
This suggests
that the variability within the data is fairly narrow. This becomes
display these data in a frequency table (Table 4.5) and a his¬
togram (Figure 4.3). The histogram is particularly effective at demonstrating that
there is little variability within the data, as it is easy to see that the data are all close
more
to
the
evident if we
mean
Table 4.5
and the mode.
Frequency and Percentage Distribution
-
Depression Scores
Cumulative
Frequency
Percentage
6.00
1
10.0
10.0
7.00
2
20.0
30.0
8.00
4
40.0
70.0
Percentage
9.00
2
20.0
90.0
10.00
1
10.0
100.0
Total
10
100.0
Figure 4.3 Histogram
-
Depression Scores
Histogram
score
Central Tendency and Variability
57
SUMMARY
In this
chapter, we covered two important ways of describing data: measures of
tendency and variability. We explained that there are three measures of
central tendency: mode, median and mean. We showed how to calculate each
and examined the impact ofoutliers. We also looked at the measures of variability,
including range, variance and standard deviation. The information covered in this
chapter and Chapter 3 provide what is needed to make use of descriptive statistics
in order to advocate for social change.
central
REVIEW QUESTIONS
1.
Discuss the
2.
tendency.
Explain how different
tures of a typical case.
3.
Describe the effect that outliers have
4.
5.
strengths and weaknesses of each of the three measures of central
measures
of central tendency could paint different pic¬
data
they relate to describing a
typical case. How might this work in the researcher’s favour at times and to
their disadvantage in other cases?
Describe a situation where a social work researcher might use a measure of
variability to advocate for social change.
Self-reflexive exercise: take a few minutes to think about your work to date in
the social work profession. Have you encountered descriptive statistics? If so,
when and where, and how were they being used ? How did you react to this in¬
formation? Were the numbers overwhelming, or did they make sense to you?
How might your past experiences shape when and how you use descriptive
statistics moving forward in your practice?
on
as
5
Probability and the Normal Distribution
In this chapter, we move into the area of inferential statistics, which are
used when we want to generalize our findings to a larger population based
sample. We begin by introducing one of the fundamental aspects of
inferential statistics, that is, the concept of probability, and The rules related
on a
determining the different levels of probability. We explain how the normal
can be used to establish the likelihood or probability of an outcome
occurring based on all possible outcomes. Finally, we show how to calculate z
scores and percentiles and how they can be used to make comparisons.
to
distribution
f the
sample has been selected using a probability approach, meaning that each
population of interest had an equal chance ofbeing selected,
then we can infer the characteristics of the population based on the characteristics
of our sample. Inferential statistics can also be used to tell us if there is a rela¬
tionship between two or more variables, for instance, if the independent variable
(for example, alcohol consumption) is likely to be related to a change (increase or
decrease) to the dependent variable (for example, injection drug use).
individual within the
PROBABILITY AND SOCIAL JUSTICE
In a completely egalitarian society, everyone’s life chances would be the same;
however, we know that this is far from the case. We also know that children grow¬
ing up in poverty, people of colour, First Nations people and other marginalized
people may well not have the same life chances and opportunities as people who
are raised in an affluent, White, mainstream
family. The German sociologist Max
Weber coined the term “life chances”
their chances of achieving
explain how someone’s class influences
their goals in life.
to
It is the most elemental economic fact that the way in
which the disposi¬
is distributed among a plurality of people
meeting competitively in the market for the purpose ofexchange, in itself
creates specific life chances. (1978: 927)
tion
over
material property
58
Probability and the Normal Distribution
Eric
Krieg (2012)
59
the term “social capital” to refer to the number and range
people who have political and financial influence. Those with
greater social capital obviously have greater life chances.
There are many examples of the use of probability figures help to describe
important social justice related issues. For instance, in an article published by Forbes
Magazine online, Daniel Fisher (2012) points out that while 79% of students in
the United States from the top income quartile (incomes over $98,875) obtained
a bachelor’s
degree, only 11 % of students from the lowest quartile (incomes below
$33,000) had the same level of education. In other words, the probability of stu¬
dents from high-income families earning a bachelor’s degree is roughly eight times
higher than students from low-income families.
Another example involves poverty rates among First Nations children. In a
study conducted in 2013 for the Canadian Centre for Policy Alternatives, David
uses
of connections with
Macdonald and Daniel Wilson state that 40% of First Nations children in Canada
live in poverty.
child
The authors also point out that the probability of a First Nations
living in poverty is two and half times that of the general population of
children in Canada.
BASIC LAWS OF PROBABILITY
of the fundamental concepts of statistical analysis. It is based
repeated observations of a phe¬
nomenon, a certain pattern will be evident. For instance, let’s say that the height
of every grown Canadian male is measured and the average height is found to be
175 cm. While there may be large fluctuations in the height of individual males,
with a large enough sample, the laws ofprobability indicate that the average height
of men in the sample will be approximately 175 cm. Another example is the coin
toss. Because there are only two possible outcomes, heads or tails, if you toss a coin
often enough, in the long run, you should get roughly an equal number of heads
and tails. Of course, if we only toss a coin 10 times, it is quite possible that we get
6 or 7 heads and only 4 or 3 tails.
Another basic law is that probability is expressed in a range from 0 (never) to
1 (complete certainty). Since there are only two possible outcomes in a coin toss,
the probability of getting heads is .5. On a multiple choice question with four pos¬
sible outcomes, the probability of getting a correct score is .25. Probability is used
to forecast weather, anticipate economic growth and predict election results. It is
also used by social workers interested in social policy because it can predict the
impact of social policies affecting thousands of people. Calculating probability is
therefore an effective and essential tool for social justice oriented social workers.
Probability is
on
one
certain basic laws. One such law is that within
60
Statistics for Social Justice
Calculating Probability
Probability is represented by the letter p and is expressed as a proportion of 1.0.
Remember that since there are two possible outcomes, the probability of getting
heads in a coin toss is .5 (p = .5). The formula for calculating probability is the
following.
Number of outcomes in the event
v _
*
Number
of all possible outcomes that can occur
example of tne coin toss, since there are two sides to the coin, the
formula is expressed as follows:
Turning to
our
Number of outcomes in the event
Number of all possible outcomes that can occur
P=
_
~
1
2
=
Another
example often used to explain how to calculate probability is rolling
calculate the probability of rolling a die and getting a 3.
Since there are 6 possible outcomes with a die, the probability of getting a 3 is. 167.
dice. Let’s say we want to
Number of outcomes in the event
Number of all possible outcomes that can occur
P=
1
=
6
=
-167
Addition Rule
There
few basic rules
follow when
calculating probabilities. The first is the
probability of flipping a coin and getting heads
is .5, and that the probability of getting tails is also .5. By adding these two prob¬
abilities we obtain a 1.0 (p = .5 + .5 = l). This addition rule applies because ofwhat
is called disjoint events. Disjoint events are those that when one occurs no other
event can occur
the events are mutually exclusive. Flipping a coin and getting
a head
precludes the possibility of getting a tail.
The addition rule can be applied to the example of rolling dice. If we want to
know the probability of obtaining either a 3 or a 4, and we know that each of these
two events has a probability of. 167, we can add these probabilities together to get
our answer.
Adding these two together, the probability of obtaining either a 3 or a
4 is .334 (p = .167 + .167 = .334). As a word of caution, Krieg (2012) points out
that we need to be aware of rounding errors. If we were to add up the fractions of
rolling either a 3 or a 4, as opposed to adding the individual probabilities, it would
are a
to
addition rule. We know that the
—
be
as
follows:
+
2
6
.333
Probability and the Normal Distribution
Ap
be
of .334 as opposed to .333
may not seem
61
like a big difference but it could
important. As Krieg suggests, the best way to avoid rounding errors is to add
frequencies (e.g., 1/6) instead of the individual probabilities (e.g., 1.67).
The
Multiplication Rule
The
next
rule
have
consider in
calculating probabilities is the multiplication
first look at what are called independent
events. These are events where the occurrence of each event has no impact on the
occurrence of
any other event. If we were to calculate the probability of multiple
independent events, we would simply multiply the probability of each event with
the others. Looking at the example of a coin toss, the probability of getting one
heads is .5. Getting one heads on the first toss has no impact on getting a second
heads with the second flip of the coin. The probability of getting two heads then
is .25, or .5 times .5. The same applies to rolling dice. If we want to know the prob¬
ability of getting two 3s by rolling two dice each just one time, the probability of
the first event (rolling a 3) is. 167, times the probability of the second event (rolling
a 3), which is also .167,
equals .028.
The last thing to consider within the multiplication rule is the probability with
or without
replacements. Replacement involves selecting a case for one event and
then replacing this case for the second event. The example used by Krieg is the
alphabet. If we put all 26 letters of the alphabet in a bag and select one, we would be
left with 25 letters. The probability of obtaining any one specific letter, say an A, is
1 -h26. or .038. If we replace this same letter in the bag, the probability of obtaining
a B is
again 1h-26, or .038. However, if we did not replace the A, the probability
of obtaining a B is W25 or .04. This is because there are now only 25 letters left.
we
to
rule. To understand this rule
we
must
What do these calculations have
out, it is a
do with social justice
issues? As Krieg points
small step from calculating probability to analyzing social data. Going
to
back to the example
of First Nations children in the study conducted by MacDonald
(2013), they found that 40% of First Nations children live in poverty.
The probability of a First Nations child living in poverty is 4-MO, or .4.
Assuming that we have a random sample of 25 female-headed families in need
of public housing, as described in Table 3.3 on housing needs and numbers of
and Wilson
children, out of a total of 25 families, the number of families with
The
we
probability of having families with
found that there
were
one
child is 1W25.
or
one
child is 11.
.44. In Table 3.3
8 families with 2 children. If wanted to know what the
probability of having families with either one or two children, we would add the
probability of each, 11/25 + 8/25 = 19/25 = .76. If our sample was large enough,
we could
generalize these results to future service users, suggesting that there is a
.44 (44%) chance that any new service user will have one child and a .76 (76%)
chance that they will have either one or two children.
Statistics for Social Justice
62
THE NORMAL DISTRIBUTION
The normal distribution
be used
establish the likelihood
probability of
occurring based on all possible outcomes. It is thus an essential tool
in statistics. However, it is important to understand that the normal distribution
exists only as a theoretical concept. Although many distributions approximate the
shape of the normal distribution, it is an abstract ideal.
an
can
to
or
outcome
Figure S.l The Normal Distribution
99.74
95.44
♦—
68.26
—*
Standard
Deviations
Source: Lui n.d.
The normal distribution has
a
number of
properties that help in calculating
probabilities.
1.
The mean, median and mode are all at
the
2.
The total
area
under the
and 50% above the
3.
curve
equals 1.0,
or
100%
—
50% below the
mean
mean.
The normal distribution contains six standard
mean
4.
the highest point and in the center of
curve.
deviations, three above the
and three below.
The ends of the
touch the horizontal axis
reflecting the fact that
values (outliers) beyond the three standard deviations.
standard deviation above the mean is equal to the dis¬
curve never
there may be extreme
5.
The distance of one
tance
of one standard deviation below.
As illustrated in
and
one
Figure 5.1, approximately 34% of cases fall within the mean
mean. Similarly, 34% of all cases fall within
standard deviation below the
Probability and the Normal Distribution
the
mean
say
that about 68% of all
we move
look
at
and
one
standard deviation above the
cases
further from the
fall within
mean, more
one
and
mean.
Adding these
up, we can
standard deviation of the
more cases are
three standard deviations from the mean,
63
mean.
As
included. If we were to
above and below,
we
would be
including approximately 100% of all cases.
Kurtosis
Kurtosis is the
degree to which a distribution is peaked as opposed to Hat.
are peaked, that is to say with narrow variability, are described as
leptokurtic (also called positive kurtosis). Distributions that are flat, that is to say
with wide variability, are described as platykurtic (also called negative kurtosis).
Distributions that are neither peaked nor flat, and resemble most closely a normal
Distributions that
distribution,
are
described
as
mesokurtic.
Figure 5.2 Kurtosis
Source:
Signal Trading Group 2014
Skewed Distributions
Some distributions
skewed. When
they are skewed to the right, the slanting
right side and more data than would be expected is on the left of the
mean. Such a distribution is called
positively skewed. When they are skewed to
the left, the slanting tail is on the left side and more data than would be expected is
on the
right of the mean. Such a distribution is called negatively skewed. As stated
in Chapter 4, income is a typical example of a positively skewed distribution, with
most people being grouped in the lower end of the distribution. When a curve is
free of skewness it is said to be symmetrical; a distribution that is symmetrical and
bell shaped is called a normal distribution.
tail is
on
the
are
Statistics for Social Justice
64
Figure 5.3 Skewed Distributions
(b) A negatively skewed distribution
(a) A positively skewed distribution
Source: Dean and
i
Illowsky 2008
SCORES
Another important concept used
score.
Z scores convert scores to
direct
comparisons
in statistics is the z score, also called the standard
standard score, which allows us to make
a common
on measurements
different research studies
taken from either two different populations,
using different measurement scales. Z scores
tell us where the score is in relation to the rest of the population described in a
normal distribution. They are used for calculating the results ofsuch measurement
across
or
the Scholastic Aptitude
Test (sat) and the Graduate Record Exam
(gre). To calculate the z score simply divide the raw score minus the mean by the
instruments
as
standard deviation.
raw score
z =
The
-
mean
standard deviation
following calculation shows the z score for a raw score of 81 when the
mean
is 75 and the standard deviation is 5:
81-75
z=
The
1.2
3
following calculation shows the z score for a
raw score
is 75 and the standard deviation is 5:
z=
70-75
zl
5
5
-1.0
of 65 when the mean
Probability and the Normal Distribution
65
Percentiles
Another way in
which z scores are used is to calculate percentiles. For instance,
56% on a test, you may feel that you have done poorly. However, if
you find out that the percentile of your score was 96, you would probably feel a
whole lot better. Why? Because 95% of the students did more poorly than you
did. Scores on measurement instruments such as sats and gres are normally
expressed as percentiles.
Convert z scores to percentile scores by using a z table, which describes the area
under the normal curve. If the z score is positive you must add 50. If the z score
is negative, you must subtract the number from 50. A little practice is essential.
What is the percentile if the z score is 1.2? Using Table 5.1, which describes
if you score
the
area
under the normal curve, the number indicated
for 1.2 is 38.49. Since the
we add 50 (38.49 + 50 = 88.49). The percentile is 88.49.
percentile if the z score is -1.0? Again using Table 5.1, the number
indicated for 1.0 is 34.13. Since the z score is negative, we subtract 34.13 from 50
(50 - 34.13 = 15.87). The percentile is 15.87.
z score
is
positive,
What is the
Box 5.1
Suggestions for Using a z Table
Table 5.1 shows the
First, find the
area
z score
under the normal
curve.
It is also known
as a z score
table.
that you have calculated with first decimal point within the
first column
(remember columns are vertical, or go up and down). Next, find the
point in the first row on top (remember rows are horizontal, or go
from side to side). Last, find where the column and row intersect and use that score
to complete your calculation.
second decimal
For
calculated a z score of 1.78. Lookup 1.7 in the first column
row and that column intersect and you’ve
found the value that you need to continue your calculations — 46.25. Since this
z score is
positive ( + 1.78, not -1.78), add the number found in the table (46.25)
to 50 to arrive at the percentile score (46.25 + 50 = 96.25). The percentile score is
example,
say you’ve
and .08 in the first
therefore 96.25.
row.
Find where that
66
Statistics for Social Justice
Table 5.1 Area under the Normal Curve
(z score table)
.04
.05
.06
.07
.08
.09
1.20
01.60
01.99
02.39
02.79
03.19
03.59
05.17
05.57
05.96
06.36
06.75
07.14
07.53
09.10
09.48
09.87
10.26
10.64
11.03
11.41
12.55
12.93
13.31
13.68
14.06
14.43
14.80
15.17
16.28
16.64
17.00
17.36
17.72
18.08
18.44
18.79
z
.00
.01
.02
0.0
00.00
00.40
00.80
0.1
03.98
04.38
04.78
0.2
07.93
08.32
08.71
0.3
11.79
12.17
0.4
15.54
15.91
.03
0.5
19.15
19.50
19.85
2019
20.54
20.88
21.23
21.57
21.90
22.24
0.6
22.57
22.91
23.24
23.57
23.98
24.22
24.54
24.86
25.17
25.49
0.7
25.80
26.11
26.42
26.73
27.04
27.34
27.64
27.94
28.23
28.52
0.8
28.81
29.10
29.39
29.67
29.95
30.23
30.51
30.78
31.06
31.33
0.9
31.59
31.86
32.12
32.38
32.64
32.90
33.15
33.40
33.65
33.89
1.0
34.13
34.38
34.61
34.85
35.08
35.31
35.54
35.77
35.99
36.21
37.08
37.29
37.49
37.70
37.90
38.10
38.30
1.1
36.43
36.65
36.86
1.2
38.49
38.69
38.88
39.07
39.25
39.44
39.62
39.80
39.97
40.15
1.3
40.32
40.49
40.66
40.82
40.99
41.15
41.31
41.47
41.62
41.77
1.4
41.92
42.07
42.22
42.36
42.51
42.65
42.79
42.92
43.06
43.19
1.5
43.83
43.94
44.06
44.18
44.29
44.41
43.32
43.45
43.57
43.70
1.6
44.52
44.63
44.74
44.84
44.95
45.05
45.15
45.25
45.35
45.45
1.7
45.54
45.64
45.73
45.82
45.91
45.99
46.08
46.16
46.25
46.33
1.8
46.41
46.49
46.56
46.64
46.71
46.78
46.86
46.93
46.99
47.06
1.9
47.13
47.19
47.26
47.32
47.38
47.44
47.50
47.56
47.61
47.67
2.0
47.72
47.78
47.83
47.88
47.93
47.98
48.03
48.08
48.12
48.17
2.1
48.21
48.26
48.30
48.34
48.38
48.42
48.46
48.50
48.54
48.57
2.2
48.61
48.64
48.68
48.71
48.75
48.78
48.81
48.84
48.87
48.90
2.3
48.93
48.96
48.98
49.01
49.04
49.06
49.09
49.11
49.13
49.16
2.4
49.18
49.20
49.22
49.25
49.27
49.29
49.31
49.32
49.34
49.36
2.5
49.38
49.40
49.41
49.43
49.45
49.46
49.48
49.49
49.51
49.52
2.6
49.53
49.55
49.56
49.57
49.59
49.60
49.61
49.62
49.63
49.64
2.7
49.65
49.66
49.67
49.68
49.69
49.70
49.71
49.72
49.73
49.74
2.8
49.74
49.75
49.76
49.77
49.77
49.78
49.79
49.79
49.80
49.81
2.9
49.81
49.82
49.82
49.83
49.84
49.84
49.85
49.85
49.86
49.86
3.0
49.87
3.5
49.98
4.0
49.997
5.0
49.99997
Source: Fisher and Yates 1963
Probability and the Normal Distribution
67
EXAMPLES OF SOCIAL JUSTICE ISSUES
Normal distributions and standard deviation
be used
identify impor¬
issues. For instance, in a study conducted byjustin Doubleday for
the Chronicle oj Higher Education, percentiles are used to show that students from
low-income families score much lower than students from high-income families
tant
on
scores can
to
social justice
standardized
tests.
Critics of standardized tests
(such as sats) contend that the examinations
backgrounds. Many of those
students, they argue, don’t have the same access to advanced classes and
test-preparation materials as their more-affluent peers do. The College
Board’s report showed that test takers in the lowest income percentile,
whose families make less than $20,000 per year, averaged a score of 1326,
well below the mean. The average score for students from families who
make more than $100,000 was 1619. (2013: 1)
are
unfair
to
students from low-income
A
study conducted by Jane Friesen and Brian Krauth from Simon Fraser
University in British Columbia uses standard deviation z scores to show the gap
between First Nations and non-First Nations students.
First Nations students in
grade 7 score more than 0.6 standard deviations
average below non-First Nations students on... Foundations Skills
Assessment (fsa) exams. The results by quartile are similar: the achieve¬
ment gap ranges from 0.51 to 0.77 standard deviations. Among students
who wrote the fsa numeracy test in both grades, the gap between the
on
mean
by
the
test
scores
of First Nations and non-First Nations students grew
additional 0.05 standard deviations between grades 4 and 7, and
reading test score gap grew by 0.09 standard deviations. (2010: 7)
an
SUMMARY
In this
chapter we covered a number of key aspects of analysis used for inferential
was
probability; establishing the probability of a relationship
existing between variables is the underlying purpose of any inferential statistical
analysis. We looked at the normal distribution, its properties and how it can be used
to help identify the probability of any possible outcome
occurring. We showed how
to calculate z scores and percentiles and how they can be used to make comparisons.
statistics. The first
68
Statistics for Social Justice
REVIEW QUESTIONS
1.
Describe the addition rule and the
2.
Imagine
importance of mutual exclusivity.
casino at a roulette table with a friend. You can see that
“red” has come up the past 8 spins. Knowing that you are taking a statistics
course, your friend asks you for advice placing their bet. They ask, “I should
you are at a
probably bet
black is due
a
lot
black, right? Since red has come up the past 8 times,
up!” What advice would you give this friend and explain
on
to come
reasoning.
Explain the importance of specifying if there was or was not replacement
your
3.
done for
subsequent events when calculating probability. Create an example
justice oriented research.
Discuss the meaning of the number 99.74 as it relates to normal distributions.
Come up with an example of a variable which you think is skewed. Is it posi¬
tively or negatively skewed? Why do you think this is?
Describe the difference between obtaining a 90% as a final mark in a statistics
course, versus being in the 90th percentile in a statistics course.
that relates to social
4.
5.
6.
6
Hypothesis Testing
In this
chapter, we cover the basic concepts involved in hypothesis testing,
including testing for relationships, design flaws, statistical significance,
festing the null hypothesis, the two different types of hypotheses and finally,
the two types of research errors.
Hypothesis testing involvrelationship
es followingbetween
a specific set of statisvariables.
tical procedures
There to
determine if there is
two or more
a
are
kinds of relationships that could be of interest to social workers; some
involve cause and effect, but within the social sciences, many do not. For instance,
a researcher
may determine through a study that there is a relationship between
poverty and crime. However, she cannot say that this is a cause and effect relation¬
ship; she cannot say that poverty causes someone to become involved in crime.
In terms of relationships that do involve cause and effect, one of the most com¬
mon forms of research carried out in social work are
program evaluations. Social
workers are naturally interested to find out if there is a relationship between an
intervention program and the outcomes for service users, in particular whether
the program causes the desired outcomes. Social workers are also interested in the
effects of social problems on citizens. For instance, the causes of poverty remain
in dispute, with some people blaming the poor for their poverty and others blam¬
ing the lack of secure well-paying jobs with benefits. Regardless of the causes, the
effects of poverty on children are clear. For one thing, children who grow up in
poor families experience more severe and persistent health problems (see Box 6.1).
many
CLASSIC EXPERIMENTAL DESIGN
As stated
above, a common type ofresearch carried out in social work is a program
evaluation. If a
foundation
must
a
agree to
summative
measurable
community organization applies for funding from a government or
to create
a new
conduct
identified social need, it typically
evaluation. Many funding bodies will insist on
program to meet an
a program
evaluation, also called
an outcome
evaluation, which relies
on
quantitative evidence to show that the program produces the desired
69
70
Statistics for Social Justice
Box 6.1 The
Impact of Poverty on Health Status of Children
Child poverty in
Canada is a significant public health concern. Because child
development during the early years lays the foundation for later health and
development, children must be given the best possible start in life. Family income
is a key determinant of healthy child development. Children in families with greater
material resources enjoy more secure living conditions and greater access to a range
of opportunities that are often unavailable to children from low-income families.
On average, children living in low-income families or neighbourhoods have poorer
health outcomes. Furthermore, poverty affects children’s health not only when they
are
young, but also later in their lives as adults. The health sector should provide
services to mitigate the health effects of poverty, and articulate the health-related
significance of child poverty, in collaboration with other sectors to advance healthy
public policy.
Source: Paul-Scn
Gupta de Wit, and McKeown 2007
Ideally, this type of program evaluation makes use of what is referred
experimental design. This type of research is considered to be
the gold standard against which other research designs are compared. This is the
only research design that will allow the people responsible for running the program
to state conclusively that the program is effective and has produced the desired
outcomes. If the experimental group scores significantly higher on the measure¬
ment instrument than the control group, this is generally accepted as proof that
the program is effective.
The classic experimental design must include the following three conditions:
outcomes.
to
as
the classic
1.
The selection of participants
2.
There must be two groups
for the study must make use of probability sam¬
pling (random sampling), meaning that every member within the population
of interest has an equal chance of being selected.
of participants: the experimental group, made
of participants who are included in the program, and the control group,
made up of participants who are not included in the program.
The measurement instrument used must have been tested for reliability and
validity.
up
3.
Box 6.2 highlights the use of experimental program evaluation design. To con¬
duct research using classic experimental design, the researcher must be familiar with
the concept
ofhypothesis testing. To illustrate the concepts related to hypothesis
testing, consider the following hypothetical example.
Suppose you are running a Head Start Program in your community to help preschool
children from
chances
low-income families improve their school readiness and ultimately their
of succeeding in school. Suppose also that you are interested in demonstrating
Hypothesis Testing
Box 6.2
71
Evaluating the Effectiveness of a Social Skills Program
for Preadolescents
Peer relations
play
Social workers
who
are
a major role in the social development of children and youth.
well aware of the potential negative consequences for children
socially excluded and they frequently work with these children concerning
truancy, academic performance, delinquency and substance abuse. This
article discussed the outcome of a social skills program designed to improve the
social interaction ot fifth grade children. Results indicated that the treatment group
made significant gains on sociometrics, observational and self-perception measures.
The importance of using school social workers and staff and integrating social skills
programs into elementary school curricula is also discussed.
are
issues of
Source:
Hepler 1994
the
effectiveness of your Head Start Program. Because you want to determine if there
and effect relationship between your Head Start Program and better school
readiness, you decide to use the classic experimental design. The program would be the
cause and the school readiness outcome would be the effect.
To carry out this rigorous form of program evaluation, three basic conditions
must be met in order to explain causality:
is
a cause
1.
The
two
variables
be
empirically linked to one another. In other words,
Program must be linked to the outcome of school readiness.
must precede the effect in time. This means that the program must
must
the Head Start
2.
The
cause
occur
3.
The
before the
outcome
of school readiness is measured.
relationship between the factors cannot be explained by other factors.
are no other
explanations for the improve¬
You need to demonstrate that there
ment in
school readiness.
TESTING YOUR RESEARCH HYPOTHESIS
You
begin by stating your research hypothesis. In this example, your research
hypothesis is that your program is effective in helping preschool children improve
their readiness for school. Next you identify and define your variables. As stated in
Chapter 1, the term conceptualization refers to the process we use in choosing
the variables and clearly defining the variables that are included in our study. In
this case, the program is the independent variable, and school readiness is the
dependent variable. The independent variable is the variable which is believed to
affect the dependent variable, and it is the one manipulated in some way. You need
to clearly describe your
program and how you will define “school readiness,” such
as
preschool literary or math skills. Next you state how you will measure your vari¬
ables. Again turning to the definitions in Chapter 1, the term operationalization
72
Statistics for Social Justice
refers
the method used
the variable. Your
independent variable is
possible levels, either children
are in the
program (they are the experimental group) or they are not in the pro¬
gram (they are the control group). For the dependent variable, school readiness,
you find an instrument to measure the outcomes of your program. Let’s call it the
Head Start Outcome Measure, and it measures preschool literary and math skills.
These measurements give you data at the interval/ratio level of measurement.
You want to establish that there is a relationship between the independent variable
and the dependent variable.
Let’s say that you were able to use a probability sampling method and ran¬
domly select two groups of preschool children from your community. One group
is participating in your Head Start Program. This is your experimental group.
The second group is not involved in the program. This is your control group. By
using a control group, you are able to control for other possible variables which
are not
part of your study but which may have an impact on the outcome, such as
differences in parental support or differences in natural academic ability. You test
both groups at the start ofthe program with your Head Start Outcome Measure to
show that both groups are similar. This is your pre-test. You then test both groups
again at the end of the program, for your post-test. If the experimental group
scores
higher than the control group on the post-test, are you safe in concluding
that there is a relationship between the independent variable and the dependent
variable and that the your program is effective? Maybe, but you need to consider
other possible problems.
At this point you must be open to the fact that there may still be other expla¬
nations which may exist and which may also explain the relationship. Robert
Weinbach and Richard Grinnell (2010) identify two other possible explanations:
rival hypotheses and research design flaws.
at
to
to
measure
the nominal level of measurement and has
Rival
You
two
Hypothesis
testing your hypothesis that your Head Start Program is effective with
preschool children and the results confirm that it is. However, there maybe other
explanations for the improvement in school readiness. The rival or alternative
are
hypothesis suggests that some non-random cause, or some other variable, may
causing the relationship you have found. For example, if the children attend¬
ing your program receive more family support in the form of learning activities
at home, the relationship between the independent and the dependent variable
may be explained by the family support as opposed to your Head Start Program.
Another possibility is that variables co-vary, that is to say they interact to create a
relationship. In our example, perhaps the family support together with the Head
Start Program created the outcome.
be
Hypothesis Testing
Research
73
Design Flaws
There may also be
design flaws in your study. Two possible design flaws are measure¬
sampling bias. Measurement error may be the result of consistent
distortion of the measurement of variables, which can distort the quality of the
data and the subsequent analysis of the results. This could create a measurement
bias that goes undetected. There may also be a systematic error, which occurs
when the researcher is using an invalid measure. This may happen if your Head
Start Outcome Measure is not a valid measure of preschool literary or math skills.
There may also be random error, which could occur because of mood or health
changes on the part of the children.
The second type of design flaw is referred to as sampling bias, which is the
systematic distortion of a research sample. What if the sample you selected is not
typical of the population you are investigating? In our example, what if the children
attending the Head Start Program are not typical of the children from low-income
families in your community? For instance, perhaps because of the lack of a school
bus service, many low-income parents may not be able to bring their child to your
program, and only families who have access to a car are able to take advantage of
ment
error
and
the program.
These
possible design flaws can be dealt with before you begin your study
by using proper research procedures. Because you are using a control group, you
are able to dismiss the
possibility of rival hypotheses. If your research instrument,
two
the Head Start Outcome Measure, is a
reliable and valid
measure
of school readi¬
also dismiss the possibility of measurement error. Finally, if you used
random sampling to select your sample and were able to show that your sample is
representative of the low-income families in your community, you can disregard
the problem of sample bias.
ness, you can
Sampling Error
Even if you
carefully followed the most rigorous design procedures, there is one
possible problem that could still affect the quality of your data and your ability
to show that there is a true relationship between your independent and depend¬
ent variables. This problem has to do with sampling error, which is the concept
that there is a natural tendency for any sample to differ, if only slightly, from the
population from which it was drawn. As you can imagine, it would be a very rare
occurrence for the
sample statistics and the population parameters to be identical.
However, there are two ways ofrefuting sampling error: replication and inferential
statistical analysis. Replication refers to doing a study over and over again. The
more we
repeat it, the more we can be sure that the results are true. If we repeat
the study 100 times and we obtain the predicted outcome 95 times, we are safe in
concluding that our hypothesis is confirmed. A cheaper and more practical method
74
Statistics for Social Justice
inferential statistical analysis.
Ifwe rigorously follow scientific procedures,
significant results after running inferential statistical analyses, we can
arrive at the same conclusion, that our hypothesis is confirmed. This brings us to
the issue of statistical significance.
is to
use
and
we
find
STATISTICAL SIGNIFICANCE
Significance is a common term used in everyday conversation. We might, for
example, say that someone has made a significant contribution to a social work
organization. We may also say that the results of our work are significant. However,
in discussing statistics, significance has a specific meaning.
Weinbach and Grinnell (2010: 99) state: “Statistical significance is the dem¬
onstration, through the use of mathematics and the laws of probability, that the
relationship between variables in a sample is unlikely to have been produced by
sampling error.” Regardless of the direction, if there is a low probability that the
results we obtain are due to chance, we say that the results are statistically signifi¬
cant. It cannot be emphasized
enough that there is always a chance, albeit a slim
one, that the results could be the result of sampling error.
If we are trying to prove that a relationship exists and we accept that because
of sampling error there is always some chance that no relationship exists, at what
point are we safe in accepting that there is a relationship? Within the social sciences,
the accepted level is 95%.
Looking back at the concept ofprobability, we said that probability is expressed
as a
range of 0, meaning that there is no relationship, to 1.0, meaning that there is
a 100% chance that a
relationship exists. Therefore if the accepted level is 95% or
higher, we can then say that there is less than a 5% chance (or a .05 probability)
that no relationship exists. The .05 is our level of statistical significance.
THE NULL HYPOTHESIS AND THE REJECTION LEVEL
We mentioned that
hypothesis is a tentative answer to a research question derived
through a review of the literature. It is a testable statement. Looking at our example
ofthe evaluation of our Head Start Program, our test hypothesis is that our program
is effective in helping children from low-income families improve their readiness
for school. However, in the interest of being scientifically rigorous, we must now
introduce a new concept, the null hypothesis. The null hypothesis is a statement
that no relationship exists between the variables in our study, and, if one does seem
to exist, it is simply occurring by chance.
In scientific studies, we are, therefore, not directly interested in finding evidence
to accept our test hypothesis. Instead, we are looking for reasons to reject the null
hypothesis, which would allow us to say that the relationship that we have found
a
Hypothesis Testing
between the variables has
75
occurred by
chance. This brings us to the concept of
rejection level. The rejection level is the level at which we are safe in rejecting
the null hypothesis and concluding that a real relationship exists; this is the level
of sampling error that we are willing to accept within our research. If there is less
than a 5% chance that the relationship that we found is due to sampling error, then
we are safe in
rejecting the null hypothesis, and we can then say that the relation¬
ship is statistically significant. In social work, as is the case in most social sciences,
a
probability value of .05 is considered acceptable for our purposes. The p = .05
is our rejection level, also called the significance level. This level was chosen
through convention.
not
the
TYPES OF HYPOTHESIS
There
of research hypotheses, and each has a function in rejecting
hypothesis. These two types are called the one-tailed and two-tailed
hypotheses. The term “tailed” refers to the tails at either end of a normal distribu¬
tion formed by the distribution of all possible outcomes
are
two
types
the null
One-Tailed Hypothesis
A one-tailed hypothesis is also known as a directional hypothesis. This is where we
predict the direction of the relationship between the two variables. For instance, if
we
say that children in the Head Start Program will score higher on the Head Start
Outcome Measure than children in our control group, we are implying a direction
(we are predicting they will score higher, not lower, and are therefore specifying
the direction we think the relationship will show).
With a one-tailed hypothesis, to obtain a p of .05 or less, the results of our sta¬
tistical analysis would have to be located at or above the 1.645 standard deviation
above the mean in a distribution of all possible outcomes (Figure 6.1).
Figure 6.1 One-Tailed Hypothesis, Area under the Curve
Source: Institute
Critical Value
=
-1.64
for Digital
Research and Education n.d.
76
Statistics for Social Justice
TWO-TAILED HYPOTHESIS
If we believe, on the other
in what
direction,
hand, that there is a relationship but we do not know
use a two-tailed hypothesis, or a non-directional
would
we
hypothesis. For example, if we say that there is a relationship between participating
in a Head Start Program and outcome on the Head Start Outcome Measure, we
would not be implying a direction. With a two-tailed hypothesis, to obtain ap of
.05 or less, we need to split the .05 level into two, or .025, above the mean and .025
below. The results of our statistical analysis would have to be located at or above
the 1.96 standard deviation either above
or
below the
mean
in
a
distribution of all
possible outcomes (Figure 6.2).
Figure 6.2 Two-Tailed Hypothesis, Area under the Curve
Critical Values
=
-1.96 and +1.96
Source: Institute for Digital Research
and Education n.d.
ERRORS
Once
have
ready to state if we are able to accept
errors that we could make in
drawing our conclusions: type I and type II errors. These are two possible errors
in interpreting the research data.
or
we
completed our study and
are
reject the null hypothesis, there are two possible
Type I Error
This
error
is where
we
reject the null hypothesis and conclude that there is a rela¬
tionship when in fact there is
none. Although we can never totally eliminate the
I error, we can l) use larger samples because larger
samples reduce the possibility of sampling error, and 2) replicate our study to
chance of committing a type
confirm
our
results.
Hypothesis Testing
11
Type II Error
The second type
the null hypothesis and say that there is no
relationship when in fact there is a relationship. Although we would obviously want
to reduce our chances of making either type of error, type II errors can result in us
missing potentially useful relationships. We can reduce the chances of making a
type II error by using a test with a higher level ofstatistical power. Statistical power
refers to the ability of the test to correctly reject the null hypothesis, to detect a
true relationship between variables. Factors that affect statistical power include the
strength of the actual relationship, the amount of variability within the variables,
the rejection level being used, whether a one-tailed or two-tailed hypothesis is
used and the size of the sample.
is where
we accept
SUMMARY
In this
chapter we explored the concept ofhypothesis testing and explained that it
identifying whether a true relationship exists
among variables. We looked at different kinds of relationships, and how to use the
classic experimental research design to determine if a cause and effect relationship
exists. We explained that, even if the results of our research show that there is a
relationship, there may still be other explanations, including errors, as to why we
obtained these results. We introduced the concepts of statistical significance, the
null hypothesis, and one- and two-tailed hypotheses.
is
a
fundamental part ofthe process in
REVIEW QUESTIONS
1.
Describe what is
2.
Discuss the
3.
Explain how the concepts of sampling
nificance
4.
6.
by cause and effect.
importance of the criteria for the classic experimental design.
are
error,
rejection level and statistical sig¬
related.
Describe the importance
of formulating a hypothesis and a null hypothesis
conducting research.
Provide an example of a situation in which a rival hypothesis might explain
the relationship found in a research study.
Discuss why committing a type I error is so dangerous. Explain why commit¬
ting a type II error is also a bad thing.
when
5.
meant
1
Sampling Distributions
In this chapter, we examine the important concept of the sampling distribution
and how we use this concept to make inferences about the population
parameters based on the sampling statistics. A sample never provides a
completely accurate representation of the population. Therefore, we need
to be familiar with concepts such as sampling error, sampling distributions,
central limit theorem and confidence infervals and learn how to use the
information from these concepts to answer the question: at what point are we
safe in concluding that the sample statistics accurately reflect the population
parameters? We attempt to provide that familiarity this in this chapter.
Conservative politicalvalues favour low taxatiin peoples
on rates, lives.
a reliaWith
nce onthetheConservative
free market
and minimal involvement of government
Party of Canada winning three successive elections, forming minority governments
in 2006 and 2008 and then
majority in 2011, one could be forgiven for thinking
public attitudes in Canada during the last decade have shifted towards such
conservative values. And yet, a survey entitled Focus Canada 2011 by the polling
company Environics found that the opposite is true. Michael Adams, founder of
a
that
Environics, had this to say:
Our Focus Canada survey
showed three-quarters of Canadians believe
generally a positive thing, as opposed to one in five (19 per¬
cent) who think taxes are mostly a bad thing.... A strong majority (68
percent) agree that “governments are essential to finding solutions to the
important problems facing the country.”... A large majority (82 percent)
agree either strongly (50 percent) or somewhat (32 percent) that “gov¬
ernments in Canada should actively find ways to reduce the gap between
wealthy people and those less fortunate.”... In short, Canadians tend to
think government is reasonably effective in how it operates (although
many see room for improvement); large majorities think government
has an important role to play in addressing society’s problems, including
inequality and the excesses of the private sector; and three-quarters are
taxes
are
78
Sampling Distributions
79
quite happy to fork over some of their own money to make a functioning
government possible. These are not attitudes one would expect from a
population that is utterly disgusted with public services or interested in
burning government institutions to the ground. (Adams 2013)
For those of us who believe in the role that
government plays in supporting
the most vulnerable in our society, “hard data” such as those above
describing the attitudes of Canadians about the role of government in people’s lives
provide important evidence to dispel the myth promoted by right-wing conserva¬
tives, who claim that they speak for the majority of Canadians. Adams continued:
programs to help
When
Foreign Affairs Minister John Baird revealed his behind-theto oppose anti-gay policies signed into law in Russia in
June [of 2013], the socially conservative lobby group real Women of
Canada condemned him as a “left-wing elitist” who was out of step with
“grassroots Canada.” Unless “grassroots Canada” excludes the majority
scenes
efforts
of Canadians, real Women is mistaken: Minister Baird’s work on this
file fits
quite nicely with public attitudes. Social values research
well
polling indicates that Canadians are becoming more socially liberal;
more at ease with diverse
family models, diverse sexual orientations and
gender identities; and generally more comfortable with sexuality, in real
life and in popular culture.
as
as
we know if the conclusions drawn
by the Focus Canada 2011 are accu¬
they really describe the attitudes of the average Canadian? Is it possible
that the statement made by the real Women of Canada that Minister Baird is “out
of step” with grassroots Canada is true?
How do
rate? Do
SAMPLING ERROR
We take it
self-evident that there is
natural
tendency for any sample to differ
population it is taken from. Put in another way, the sample statistics, for
example, the mean and standard deviation of the sample, are very rarely a perfect
representation ofthe true mean and standard deviation of the population. Sampling
error is the difference between the
sample statistics and the population parameters.
Obviously, the larger the sample, the more likely it is that the sample statistics
will be similar to the population parameters. To illustrate this point, let’s assume
that we have a class of 100 undergraduate social work research students. Say that
the class average (the mean) on an exam is 75 (out of 100), with a standard devia¬
tion of 5; these are the population parameters. If we randomly select the exams
of three students, it is highly likely that the average score on the exams of these
from the
as
a
80
Statistics for Social Justice
three students will differ from the overall class average.
It is possible that the three
of 85. If we randomly select the exams of
another three students, the average score of these three maybe 65. But what if we
select a much larger sample, say 25 students? We could expect that the average
score of the 25 students would be much closer to the
average of the entire class.
The average of the 25 students could be 74 or 76 and the standard deviation of this
sample would also be close to the standard deviation of the total class; it could be
something like 5.2. To put it simply: the larger the sample, the smaller the sample
error, whereas the smaller the sample, the greater the sample error.
students could have
an
average score
SAMPLING DISTRIBUTION OF THE MEAN
A
sampling distribution of the mean is an abstract concept which may be best
introduced through a relatively simple example. Imagine that Susan is an economics
student who is interested in public attitudes about raising minimum wage. Suppose
that Susan asks
random
sample of 25 Canadians if they support increasing the
by $ 1 per hour, and let’s say that 15 of the 25 say yes. To verify her
results, Susan randomly selects a different sample of 25 Canadians and asks them
the same question, and this time only 10 say yes. This range from 10 to 15 out of
25 is a large spread. So Susan decides to do the same thing many more times, and
she obtains results ranging from 9 to 16, with most results from all of her samples
clustering at 13. If she were to draw a histogram with all the different results from
the different samples, it would form a normal distribution with a mean score of
13, which would be very close if not the same as the population mean (if we could
possibly know it). This distribution created by the means of all these samples is
the sampling distribution. The sampling distribution of the mean is the mean
(average) of the means.
An important characteristic of the sampling distribution is that it will form a
a
minimum wage
normal distribution
even
if the actual data of a variable are skewed. For instance, we
know that income is positively skewed
in our population, with most people having
income that is in the low end of the income distribution. But if we
randomly
samples, the distribution created'of all of the average incomes of all
of our samples (the mean of all of the means) would form a normal distribution.
an
select many
Standard Error of the Mean
Another
important characteristic of the sampling distribution is that because of
sampling error, the standard deviation of this distribution will not be the same as
the standard deviation of the population. Instead, it is called the standard error
of the mean, and it is calculated by dividing the standard deviation of the popula¬
tion by the square root of the sample size. Once again, the larger the sample, the
smaller the sampling error will be.
Sampling Distributions
81
Central Limit Theorem
Of course, it is not realistic or necessary to
do what Susan did. We do not need to
repeat our study over and over again to create a sampling distribution in order to
establish the population parameters. Instead, we rely on a statistical theory called
the central limit theorem. This theory states that for any variable, if the sample is
large enough, that is to say at least 30 participants, the sampling distribution of the
mean will form a normal distribution and will
approximate the population param¬
eters. This is true even if the variable itself is skewed within the population. The
concepts of sampling distribution and central limit theorem form a fundamental
basis for statistical analysis. This is because once we have a normal distribution,
we are able to
carry out a variety of statistical procedures, which allows us to make
inferences about the population based on the sample statistics.
CONFIDENCE INTERVALS
But what if we don’t know the
our
parameters of the population? How can we tell if
sample is representative of the population? Public opinion polls such as Focus
Canada 2011 make inferences about the attitudes of Canadians without
ever
knowing the actual population parameters. To answer these questions, we look
at confidence intervals. Simply put, a confidence interval is a range of values in
which the true mean of the population is expected to fall and is calculated based
on
sample statistics. To calculate the confidence interval, we must first choose the
confidence level. Public opinion polls typically choose a 95% confidence level.
They state that the results of their poll are accurate 19 times out of 20, or allow for
a 5%
margin of error, which represents the confidence level of 95%.
So how do
we use
Here we must go
Figure 7.1.
the confidence level
to
calculate the confidence interval?
back to the concept of normal distribution and z scores. Consider
82
Statistics for Social Justice
Figure 7.1 Area under the Normal Curve Lying within 1.96
Standard Deviations of the Mean
Critical Values
Source:
=
-1.96 and +1.96
Psychstatistics n.d.
Table 5.1 in
Chapter 5 demonstrates the area under the normal curve, and
that will give 95% of the all the values within a normal
distribution. We find that the z score is 1.96, assuming we are using a two-tailed
test. This means that, in any normal distribution, 1.96 standard deviations above
and below the mean will always include 95% of all the possible values. In order to
estimate the parameters of the population based on the sample statistics, we can
use this
figure of 1.96. If we want to be 95% confident (confidence level) that
the population mean falls between two levels, we take the sample mean less the
z score (1.96) times the standard error. This is the lower level of the confidence
interval. For the upper level we add the z score to the sample mean and multiply
by the standard error.
To illustrate how this works in practice, let’s assume you are planning to run
an assertiveness
training program for women staying in a shelter and you want
to know their current level of assertiveness. You will be using a standardized test
measuring assertiveness with a known mean of 100 and a standard deviation of
10. (In this case we know the population parameters because the measurement
instrument has been standardized for the population.) Let’s say that you want to
calculate the 95% confidence interval of the population mean for the residents in
the shelter. You randomly select a sample of 25 residents from the shelter and you
we
look for the
obtain
a mean
z scores
of 80 for assertiveness. Remember that the formula for determin¬
ing the standard error is the standard deviation divided by the square root of the
sample size and that the z score for a 95% confidence interval is always 1.96 when
using a two-tailed test.
Sampling Distributions
83
Example 7.1 Calculating the Confidence Interval
Mean of your
80
sample is
The standard deviation is
The
z score
for
a
10
95% confidence
interval
The
(two-tailed) is
standard error is (10
4-
1.96
V2S)
2
Lower limit:
80.00
1.96x2
=
-
3.92
76.08
Upper limit:
80.00
1.96x2
=
+3-92.
83.92
Therefore,
be 95% confident that the true assertiveness score of the
population of residents in the women’s shelter lies between 76.08 and 83.92.
In
our
you can
example,
we were
able to calculate the standard
error
by dividing the
standard deviation ofthe population on a standardized test measuring assertiveness
by the square root of the sample size. The standard deviation on the assertiveness
scale is 10, and we have a sample of 25 women. The square root of 25 is 5, and so
10 divided by 5 gives us a standard error of 2.
STANDARD ERROR OF THE PROPORTION
In the
of public opinion polls,
researchers do not know the standard deviation
population. In this case, instead of using the standard error of the mean for
a
population, we use the standard error of the proportion. Let’s go back to our
example of a public opinion poll at the beginning of this chapter. The poll found
that 82% of Canadians agree that governments should actively find ways to reduce
the gap between wealthy people and those less fortunate. How accurate is this
percentage? These researchers surveyed a sample of 1,500 people. The formula
for calculating the standard error of the proportion is as follows.
case
of the
SE
P(l-P)
=
84
Statistics for Social Justice
We
SE
can now
.82(1
plug in the actual figures.
-
.82)
=
1500
The standard
of the proportion
then is .0099, or 1.0%. We can use this
figure in the same way as we use the standard error of the mean, as in Example 7.1.
Assuming that we are interested in a 95% confidence interval, the calculation is as
follows, shown in Example 7.2.
error
Example 7.2 Calculating the Confidence Interval
The standard
of the proportion
error
1.0
multiplied by the z score is
1.96= 1.96
x
Lower limit:
82.00
1.96
-
80.04
Upper limit:
82.00
+
1.96
83.96
Therefore,
be 95% confident that the true proportion of the population
of
between the rich and those less fortunate lies between 80.04%
you can
of Canadians who believe that the Government of Canada should find ways
reducing the
gap
and 83.96%.
What this
is that the results of the survey
published by Focus Canada
This
example demonstrates the important role that statistics can play to help ensure
that the truth is known
Canadian attitudes are becoming more liberal, despite
what some Conservatives would have us think. So how are these polls conducted?
In Box 7.1 the polling company Gallup offers an explanation. Box 7.2 provides an
example of polls showing public opinion on issues related to poverty.
means
2011 do reflect the real attitudes of Canadians about the role of governments.
-
Sampling Distributions
Box 7.1 How Does
85
Gallup Polling Work?
Gallup polls aim to represent the opinions of a sample of people representing the
same
opinions that would be obtained if it were possible to interview everyone in
a
given country.
The majority of Gallup surveys in the U.S. are based on interviews conducted by
landline and cellular telephones. Generally Gallup refers to the target audience as
“national
adults,” representing all adults, aged 18 and older, living in United States.
The
findings from Gallup’s U.S. surveys are based on the organizations standard
telephone samples, consisting of directory-assisted random-digit-dial
(rdd) telephone samples using a proportionate, stratified sampling design. A
computer randomly generates the phone numbers Gallup calls from all working
phone exchanges (the first three numbers of your local phone number) and notlisted phone numbers; thus, Gallup is as likely to call unlisted phone numbers as
listed phone numbers.
national
Within each contacted household reached via
landline, an interview is sought with
of age or older living in the household who has had the most
recent birthday. (This is a method pollsters commonly use to make a random
selection within households without having to ask the respondent to provide a
complete roster of adults living in the household.) Gallup does not use the same
respondent selection procedure when making calls to cell phones because they are
typically associated with one individual rather than shared among several members
an
adult 18 years
of a household.
When
respondents to be interviewed are selected at random, every adult has an
equal probability ot falling into the sample. The typical sample size for a Gallup poll,
either a traditional stand-alone poll or one night’s interviewing from Gallup’s Daily
tracking, is 1,000 national adults with a margin of error of ±4 percentage points.
Gallup’s Daily Tracking process now allows Gallup analysts to aggregate larger
groups of interviews for more detailed subgroup analysis. But the accuracy of the
estimates derived only marginally improves with larger sample sizes.
After
Gallup collects and processes survey data, each respondent is assigned a
weight so that the demographic characteristics of the total weighted sample of
respondents match the latest estimates of the demographic characteristics of the
adult population available from the U.S. Census Bureau. Gallup weights data to
census estimates for
gender, race, age, educational attainment, and region.
Source:
Gallup 2010
86
Statistics for Social Justice
Box 7.2 Is It
Really Possible to Have a Poverty-Free Canada?
For many
experts, the answer is a clear yes, and the best way to reach that goal is
through a guaranteed annual income. It’s a radical idea that to date has been largely
dismissed by government leaders as too costly, too difficult to implement and
lacking in public support. Because of the perception that voters don’t generally like
the idea, few politicians bother even to think about such a program, let alone come
out in support of it. However a major new poll conducted this fall may provide
the evidence that some risk-averse politicians need before giving their support
to a
guaranteed annual income. The survey showed more Canadians like the idea
than oppose it. The findings are important because it’s the first time a national poll
has ever asked Canadians what they think of the idea of providing everyone with a
guaranteed income.
Such a program “is often dismissed as giving free money to people who won’t
work,” said Keith Neuman, Executive Director of the Environics Institute for Survey
Research, which conducted the poll earlier this fall for the Montreal-based Trudeau
Foundation. Neuman said the results suggest there’s a “potential foundation for
building public support for it (guaranteed income) by some bold government,”
especially if it was accompanied by the elimination of other programs. A guaranteed
annual income is a single, cash payment that would replace all current social
programs, such as welfare and employment insurance. It would create a minimum
income below which
income line” at about
no
Canadian would fall. Statistics Canada
$22,200 for
now
sets
a
“low-
single person and $47,000 for a family with
three children. Proponents, such as Conservative Senator Hugh Segal, argue such
a
plan wouldn’t cost Ottawa more money because it would get the needed dollars
from other programs that would be killed. Also, they contend the idea would actually
encourage people to work because it would eliminate provisions in the current
welfare system that penalize the poor who take very low-paying or part-time jobs.
The poll found 46 per cent of Canadians strongly (19 per cent) or somewhat
(27 per cent) favour such a policy. Another 42 per cent said they strongly (25 per
cent) or somewhat (17 per cent) oppose the idea. About 10 per cent said it would
depend on how such a program was actually implemented or had no opinion.
Support was highest in Quebec at 55 per cent and lowest in Alberta at 38 per cent.
A majority of Canadians with household incomes under $100,000 and those with
no
post-secondary education also backed the idea while support was lowest (38 per
cent) amongst Canadians earning more than $100,000. To date, no major national
political party has embraced the idea of a guaranteed income, although all talk
vaguely about the need to study it more closely just like any other policy option.
Clearly we are in an era when our politicians are more wary than brave, afraid to
champion new programs for fear of upsetting voters whose minds are focused only
on
cutting taxes. As Keith Neuman says, it would require a bold government to
make the guaranteed income a reality. Given the poll results though, the idea could
be a winner for the political party with the courage to make it a serious part of the
debate on tackling poverty in our country.
Source:
Hepburn 2013
a
Sampling Distributions
87
SUMMARY
In this
a
chapter, we introduced the concept of sampling distributions. We covered
topics, including sampling error, central limit theorem, confi¬
number of related
dence intervals and how we
can use
the information from these concepts to answer
the
question: “At what point are we safe in concluding that the sample statistics
accurately reflect the population parameters?” We explained how sampling distri¬
butions
be used
bring attention to important social justice issues such as
The examples of public opinion polls included in this
chapter show how confidence intervals based on the standard error of the propor¬
tion can be powerful tools in convincing governments at various levels about the
political advantages of pursuing a social justice agenda.
can
to
the elimination of poverty.
REVIEW QUESTIONS
1.
Explain what is meant by “sampling error.” Why is it important to understand
this concept? What level of sampling error is generally accepted within the
social sciences?
2.
Describe the
relationship between sampling statistics and population
param¬
eters.
3.
The
4.
What is the
sampling distribution of the mean is the mean of the means. Elaborate.
relationship between the central limit theorem and the concept of
normal distribution?
5.
Provide
to
6.
an
calculate
example of a scenario where a social work researcher would want
a
confidence interval.
Using a structural and aop perspective, critique the research methodology of
Gallop polls.
8
Chi-Square
In this chapter, we
introduce the chi-square test of association, a nonparametric
test that social workers may use if their data is at the nominal or ordinal level.
We illustrate how to create a cross-tabulation table and how to use observed
frequencies to calculate
a
chi-square. We explain that this does not prove
us about the strength of the relationship among
and effect but will tell
variables.
cause
Social workers are often faced with having to conduct studies where
the only
If, for
data available
are
at
the nominal
or
ordinal level of measurement.
exam¬
ple, they want to show that their intervention program is related to a positive
outcome, they would use a well-known non-parametric test called the chi-square
test of association, or simply chi-square. Consider, for instance, the following
hypothetical example.
Julie Chow is working in a shelterfor women leaving abusive relationships. She runs
a
counselling program for the residents designed to help these women plan for a future
without their former abusive relationships. She recognizes that there are many factors
related to why women return to abusive relationships, certainly not the least of which
is income security. In order to applyfor funding to keep the program going, she needs
to demonstrate to a local private foundation dedicated to addressing violence against
women that her
program is effective. She has designed a study to test the efficacy of the
counseling program.
In Julie s study, the independent variable is simply whether the women attended
the program or not. The dependent variable is whether the women who attended
the program return to their former abusive relationship or not. Both of these
variables are at the nominal level of measurement. The research hypothesis in this
case could be stated in the
following manner: Women who attend a counselling
program designed to help women refrain from returning to an abusive will be less
likely to return than women who do not attend the program. This is a one-tailed
directional hypothesis because June is predicting a positive outcome. The null
hypothesis would state: there is no relationship between attending the program
88
Chi-Square
89
and returning to an abuse
relationship. To comply with the rigorous research design,
testing the research hypothesis, Julie needs to test the null hypothesis
that there is no relationship. If she obtains a significant result, she will be able to
reject the null hypothesis.
Julie decides to conduct a pilot project with a small group of women in the
hopes of being able to reject the null hypothesis. She randomly selects a sample of
15 women from the shelter and, after obtaining the necessary informed consents,
invites them to participate in an intensive five-session group counselling program.
She plans to compare the outcome for this group with the outcome of a similar
sized group of women who did not attend the program. Upon completing the
program and at the point when the women in her counselling program are ready
to leave the shelter, there are four possible scenarios for the women in both groups:
rather than
attended the program
2. attended the program
1.
3.
4.
and did not return to an abuse relationship.
and did return to an abusive relationship.
did not attend the program and did not return to an abusive relationship.
did not attend the program and did return to an abuse relationship.
Let’s say that
were as
the number of women who fit into each of the four categories above
follows:
1.
attended the program
2.
attended the program
3.
4.
and did not return to an abuse relationship. (10)
and did return to an abusive relationship. (5)
did not attend the program and did not return to an abusive relationship. (5)
did not attend the program and did return to an abuse relationship. (10)
CROSS-TABULATION TABLE
For the
chi-square test of association, the above results would be displayed in what
a cross-tabulation table (see Table 8.1). It would be set
up for the above
example with four cells, a, b, c and d, which display the observed frequencies.
These are the actual results of the study. This cross-tabulation table is an example
of a two-by-two table because for each of the two variables, there are only two pos¬
is called
sible values
If there
—
in this case,
were more
than
two
returned/not returned and attended/did not attend.
values for each variable, we would
have
more
cells.
90
Statistics for Social Justice
Table 8.1 Cross-Tabulation Table
Returned
Did Not Return
Totals
Attended
(a) observed
frequencies
(b) observed
frequencies
marginal total
Did Not Attend
(c) observed
frequencies
(d) observed
frequencies
marginal total
marginal total
marginal total
overall total
Totals
Table 8.2 Cross-Tabulation Table
-
Observed
Frequencies for Women Surviving
Abuse
Returned
Did Not Return
Totals
Attended
a
5
b
10
15
Did Not Attend
c
10
d
5
15
15
30
Totals
15
The far right
column of the table displays the totals for each row. For instance, if
the observed frequencies recorded in the row that included cells a and
b, we have a total of 15. Similarly, if we add up the observed frequencies for cells
we
c
add up
and
d,
we
also have
a
total of 15. If we do the same for each of the two columns,
end up
with totals shown in the bottom row of 15 and 15. These are called the
marginal totals. The overall total of 30 is recorded in the bottom right hand box.
Another important point is that the minimum number of cases for each cell should
not be less than 5. As you can see, cells a and d each have 5 cases.
we
CALCULATING THE CHI-SQUARE
To
the
answer
the
question ofwhether a true relationship exists, we must first calculate
expected frequencies. These are the frequencies that are most likely to occur
hypothesis is correct. The expected frequencies are calculated by using
if the null
the
following formula:
E
=
(RXC)
(N)
Where:
E
=
Expected frequency of a particular cell
R
=
Marginal total for the row in which the cell appears
C
=
Marginal total for the column in which the cell appears
N
=
Total number of cases
Chi-Square
These
91
expected frequencies are then compared with the observed or actual frequen¬
cies
using the chi-square formula.
Note that, normally, one would need to calculate the expected frequencies
separately for each of the four cells (a, b, c, d), but because the marginal totals for
each row and column are the same, the expected frequencies for each of the four
cells are the same as well. The expected frequency for cell a is 15 times 15, divided
by 30. The expected frequency for cell b is 15 times 15, divided by 30. The expected
frequency for cell c is 15 times 15, divided by 30. The expected frequency for cell
d is 15 times 15, divided by 30.
DEGREES OF FREEDOM
The
larger the table and the more cells, the more there is a chance for a large dif¬
expected frequencies. The number of cells is
expressed in terms of degrees of freedom. Once we have obtained the chi-square
results, we must calculate the degrees of freedom using the following formula:
ference between the observed and
df= (R -1)(C - l)
The R refers
to
the number of rows,
and C stands for the number of columns.
Chi-Square Formula
,
4f= (2-l)(2-l) =1
(O-E)2
^
Cell
fS
Cell b
a
7.501 2
-
+
.83
=
x2
=
3.32,
df=l,p =
+
7.5
7.5
x2
(10-7.512
Cell
+
<
.83
(TO 7.51 2
-
Cell d
c
+
7.5
+
.83
(5-7.5)2
7.5
+
.83
.05
Where:
O
E
=
=
observed
frequency
expected frequency
The critical value for x2 with 1
degree of freedom (see Table 8.3) at the .05 level
the chi-square scorejulie obtained was 3.32 and
is greater than the x2 critical value, she is safe in rejecting the null hypothesis and
accepting the research hypothesis that there is a significant relationship between
attending the counselling program and not returning to an abusive relationship.
for
a
one-tailed
test is 2.71. Since
Statistics for Social Justice
92
Table 8.3 Critical Values of Chi-square
Level of Significance
.10
.05
for a One-Tailed Test
.025
Level of Significance
.01
.005
.0005
for a Two-Tailed Test
Df
.20
.10
.05
.02
.01
.001
1
1.64
2.71
3.84
5.41
6.64
8.83
2
3.22
4.60
5.99
7.82
9.21
13.82
3
4.64
6.25
7.82
9.84
11.34
16.27
4
5.99
7.78
9.49
11.67
13.28
18.46
5
7.29
9.24
11.07
13.39
15.09
20.52
6
8.56
10.64
12.59
15.03
16.81
22.46
7
9.80
12.02
14.07
16.62
18.48
24.32
8
11.03
13.36
15.51
18.17
20.09
26.12
9
12.24
14.68
16.92
19.68
21.67
27.88
10
13.44
15.99
18.31
21.16
23.21
29.59
11
14.63
17.28
19.68
22.62
24.72
31.26
12
15.81
18.55
21.03
24.05
26.22
32.91
13
16.98
19.81
22.36
25.47
27.69
34.53
14
18.15
21.06
23.68
26.87
29.14
36.12
IS
19.31
22.31
25.00
28.26
30.58
37.70
16
20.46
23.54
26.30
29.63
32.00
39.29
17
21.62
24.77
27.59
31.00
33.41
40.75
18
22.76
25.99
28.87
32.35
34.80
42.31
19
23.90
27.20
30.14
33.69
36.19
43.82
20
25.04
28.41
31.41
35.02
37.57
45.32
21
26.17
29.62
32.67
36.34
38.93
46.80
22
27.30
30.81
33.92
37.66
40.29
48.27
23
28.43
32.01
35.17
38.97
41.64
49.73
24
29.55
33.20
36.42
40.27
42.98
51.18
25
30.68
34.38
37.65
41.57
44.31
52.62
26
31.80
35.56
38.88
42.86
45.64
54.05
27
32.91
36.74
40.11
44.14
46.94
55.48
28
34.03
37.92
41.34
45.42
48.28
56.89
29
35.14
39.09
42.69
46.69
49.59
58.30
30
36.25
40.26
43.77
47.96
50.89
59.70
32
38.47
42.59
46.19
50.49
53.49
62.49
34
40.68
44.90
48.60
53.00
56.06
65.25
36
42.88
47.21
51.00
55.49
58.62
67.99
38
45.08
49.51
53.38
57.97
61.16
70.70
40
47.27
51.81
55.76
60.44
63.69
73.40
Source: Fisher and Yates 1963
Chi-Square
She thus showed that women who attended the program were
abuse relationship.
93
less likely to return
to an
It is
important to note that the chi-square test will not demonstrate that there is
and effect
relationship between variables even if there is a strong relation¬
ship. In other words, in the case ofJulie’s study, she cannot state if the counselling
session caused the women to not return to their former abusive relationship. What
the chi-square test can show is that the relationship between the independent
variable (the counselling program) and dependent variable (women refraining
from returning to an abusive relationship) is so strong that sampling error alone
is unlikely to explain the relationship.
The above example was based on a study with a relatively small sample of indi¬
viduals. It is also possible to use the chi-square test for much larger macro-level
studies. Let’s look at another hypothetical example.
Karen Weinstein is a social worker with Municipal Child Care Services who is
interested in demonstrating the social and economic value of increasing the number of
subsidized child care spaces in her city. She is aware that child poverty rates in Canada
remain stubbornly high and that one of the most effective ways of reducing the rate of
child poverty is through offering families quality regulated child care at an affordable
price. Currently her department has 600 subsidized spaces where parents pay $10 per
day, which is just under what someone would make per hour at minimum wage. She has
400families on the waiting list, and most of these parents are single, female, unemployed
and surviving on social assistance. She is aware that the child poverty rate for singlefemale-headedfamilies is proportionally much higher than forfamilies headed by single
males or couples. Karen is therefore particularly interested in looking at the impact of
having access to subsidized child care on single-female-headedfamilies. She would like
to demonstrate that there is a strong relationship between having access to subsidized
child care and being employed full-time. Karen is hoping that a snapshot look at the
impact ofsubsidized child care on employment status willprovide the evidence she needs
to argue for more funding for child care.
In this example, there are again two variables, both at the nominal level; the
independent variable is access to subsidized care with two scores: yes or no (on
the waiting list), and the dependent variable is employment status with two scores:
employed full-time or unemployed/ employed part-time.
The one-tailed research hypothesis for this study is: Women who have access
to subsidized child care are more likely to have full-time employment. The null
hypothesis for this study is: There is no relationship between having access to
subsidized child care and being employed full-time.
The observed frequencies of a random sample chosen from families accessing
child care and from the waiting list are shown in Table 8.4.
a cause
94
Statistics for Social Justice
Table 8.4 Cross-Tabulation Table
-
Observed
Frequencies for Single-Female-Headed
Families
Access to
Employed
Subsidized Care
Full-Time
Unemployed or
Employed Part-Time
Totals
Yes
a
80
b
20
100
No
c
30
d
60
90
80
190
Totals
110
The first step
is to calculate the expected frequencies for each of the four cells.
Using the following formula, the results
c
E
are as
follows:
(r)xca
=
N
The
expected frequency for cell a is 100 times 110=11,000, divided by 190 = 57.9
expected frequency for cell b is 100 times 80 = 8,000, divided by 190 = 42.1
The expected frequency for cell c is 90 times 110 = 9,900, divided by 190 = 52.1
The expected frequency for cell d is 90 times 80 = 7,200, divided by 190 = 37.9
Next Karen must calculate the chi-square score for the above data.
The
2
(O-E)2
v
Cell
df= (R-1) (C -1) = 1
Cell b
a
(20-42.1) 2
(80-57.9) 2
=
x2
=
42.30,
Here
Cell
Cell d
c
(30-52.1) 2
(60-37.9)2
57.9
+
42.1
+
52.1
+
37.9
8.44
+
11.6
+
9.37
+
12.89
df= l,p = < .05
again, the critical value for x2 at the .05 level for a one-tailed test is 2.71.
score that Karen obtained was 42.30, much
higher than 2.71, she is
Because the
definitely safe in rejecting the null hypothesis and accepting the research hypoth¬
esis that women who have access to subsidized child care are much more likely to
obtain full-time employment. These results provide strong evidence in support of
the argument that investing in subsidized child care is related to important social
and economic benefits.
Chi-Square
95
CROSS-TABULATION TABLES FOR VARIABLES
WITH MORE THAN TWO LEVELS
Both of our
examples are relatively simple in that they each involve a two-by-two
cross-tabulation table. It is also possible to use the chi-square test for larger tables.
Consider this third
hypothetical example.
counsellor working for a government employment centre who has
been providing employment counselling services to people who have been laid offfrom
their jobs in the high tech sector as a result ofdownsizing and who need help locating and
securing a position in emerging high tech companies. He offers help in resume writing,
preparingfor interviews and connecting people with possible employers. Up to now, he
has been seeing people on an individual basis but because of the recent increase in the
number of layoffs and the growing number ofpeople on the waiting list, he has decided
to offer workshops to people in groups. He wants to compare the outcomes of the group
workshops with the outcomes of the individual training. Given the number of people on
the waiting list at the start of the study, he wants to also compare the results of people
who took his training with the results of a group ofpeople on the waiting list, who may
find work on their own before entering either of the services he provides.
In Jim’s study, there are two variables; the independent variable is the type of
employment service provided (individual, group, waiting list), and the dependent
variable is success in obtaining employment (yes, no). Since there are three values
for the delivery of service and two values for success in obtaining employment, a
two-by-three table is required. The cross-tabulation table for this study is shown
Jim Morris is
a
in Table 8.5.
Table 8.5 Cross-Tabulation Table
-
Observed
Frequencies for Employment Training
Program
Type of Training
Employed
Unemployed
Totals
Individual
a
20
b
10
30
Group
c
10
d
5
15
Wait List
e
10
f
30
40
45
95
Totals
c
E
40
=
(R) (C)
N
The
expected frequency for cell a is 30 times 40 = 1,200, divided by 95 = 12.63
expected frequency for cell b is 30 times 45 = 1,350, divided by 95 = 14.21
The expected frequency for cell c is 15 times 40 = 600, divided by 95 = 6.32
The
96
Statistics for Social Justice
The
The
The
expected frequency for cell d is IS times 45 = 675, divided by 95 = 7.11
expected frequency for cell e is 40 times 40 = 1,600, divided by 95 = 16.84
expected frequency for cell/is 40 times 45 = 1,800, divided by 95 = 18.95
Next Jim must
calculate the chi-square score for the above data.
(O-El2
Cell
df =
Cell
Cell b
a
(20- 12.62)2
f 10
-
Cell d
c
(10 6.3212
14.213 2
+
4.3
x2
=
+
17.54,
1.25
df= 2,p =
<
+
2.14
Cell
+
=
2
Cell f
e
flO- 16.841 2
(5-7.nl2
-
6.32
14.21
12.63
(R- l) (C- 1)
(30- 18.951 2
7.11
+
16.84
+
18.95
.63
+
2.78
+
6.44
.05
Looking at the Table 8.3, showing the critical values for chi-square for a one.05 level with 2 degrees of freedom, the value we are looking
for is 4.60. Because Jim obtained a result of 17.55, he is confident in concluding
that he can reject the null hypothesis that there is no relationship and accept the
tailed hypothesis at the
Box 8.1 Link Between Sexual Abuse in Childhood
and in Adolescence
The aim of this
study was to examine the link between childhood experiences of
subsequent revictimization in adolescence.
A sample of 281 female adolescents between 17-20 years of age, who
participated in a prevalence survey of unwanted sexual contacts, completed the
Sexual Experiences Survey as a measure of unwanted sexual contacts in adolescence
and indicated whether or not they had experienced childhood sexual abuse.
Childhood experiences of sexual abuse were reported by 8.9% of the
respondents, a further 8.5% indicated they were not sure if they had been sexually
sexual abuse and
abused
as
children. Both abused
victimization status
contacts as
were
women
significantly
adolescents than
women
and
more
who did
women
uncertain about their
likely to report unwanted sexual
not state
abuse. The link between
childhood abuse and
subsequent victimization was mediated by a higher level of
activity among the abuse victims.
The results support existing evidence on the impact of childhood sexual
abuse on sexual relationships in subsequent developmental stages and underline
sexual
the need
to
consider childhood sexual abuse
as a
risk factor of adolescent sexual
victimization.
Source:
Krahe, Scheinberger-Olwig, IVaizenhofer and Koplin
(1999)
Chi-Square
97
research
hypothesis that there is a strong relationship between his services to laidand helping them obtain employment. Remember, he cannot say
if there is a cause and effect relationship, but can say that there is a relationship.
Furthermore, proportionately as many people obtained employment who received
individual counselling as the ones in the group workshops, which suggests that the
individual counselling and the group workshops may be equally effective. However,
significantly more people who received Jim’s services were able to obtain employ¬
ment than those who were on the waiting list, so being in either of his programs
is better than nothing at all.
off employees
SUMMARY
In this
chapter, we introduced the chi-square test of association. We explained that
conducting research may only have access to nominal or ordinal level
data. We showed how to create a cross-tabulation table, where we input the actual
observed frequencies and compare these to the expected frequencies, which are
the frequencies that would occur if there was no relationship between independ¬
ent and dependent variables. We explained how to calculate a chi-square test of
association and what can be inferred from the results. Finally, we explained that
we cannot state if there is a cause and effect
relationship, but we can say that the
relationship is so strong that sampling error is not likely to explain the relationship.
social workers
REVIEW QUESTIONS
1.
Explain how the number of rows and columns in
a
cross-tabulation table is
determined.
2.
Discuss the
relationship between observed and expected frequencies as well
relationship with the null hypothesis.
Describe the steps from start to finish of calculating a chi-square test of asso¬
as
3.
their
ciation.
4.
Create
a
hypothetical example of a research study for which a chi-square test
most appropriate inferential statistical test to use.
of association would be the
9
/ Tests and ANOVA
This
chapter examines three types of inferential statistical tests called t tests:
one-sample /tests, paired /tests and independent /tests. We also briefly look
at the test used to compare three or more groups called the anova. which
stands for analysis of variance. We will begin by describing / tests in general,
and will continue to explain the theory behind the one-sample /test. Next, we
will explain paired and independent / tests and demonstrate how to calculate
the latter. We will then explain how to report findings of / tests. Lastly, we will
conclude our description of the various / tests by introducing the anova.
In Chapter 1, we noted that social workers
oftenfunding
calledbodies,
upon tosuchdemonstrate
explainedarethat
the effectiveness of their work. We
ments
and private foundations,
as govern¬
continue to require empirical evidence as proof that
the programs
they are funding are effective. In Chapter 6, we stated that the gold
of program evaluations is the classic experimental design, where
the experimental group, the group receiving the program, is compared to a control
group. To test whether the experimental group scores higher on a given variable
than the control group, we generally use an inferential statistical test called the t test.
standard in
terms
WHEN TO USE/TESTS
of inferential statistical test that use means as their primary way
comparisons. These t tests can be used when you wish to determine if
there is a significant relationship between l) a sample and its population (onesample t test) or 2) two variables (independent and paired t tests). To run any t
test, the independent variable must be at the nominal level of measurement, and
the dependent variable at the interval or ratio level of measurement. In addition,
the dependent variable must be normally distributed, as this is a parametric test.
An additional criterion for the independent t test is that participants/cases must
be randomly selected. We use the following hypothetical example throughout the
chapter to help us better understand the different types of t tests and their uses.
Kiran Tran is a social worker at a non-profit organization, Spectrum Group, whose
t tests
are a
type
of making
98
t Tests and
ANOVA
99
mandate is to provide services to
children on the autism spectrum, using the principles oj
applied behaviour analysis (aba), an evidenced-based approach to working with children
on the
spectrum. Spectrum Group uses the Verbal Behaviour Milestones Assessment
(vb-mapp), an assessment tool for children with autism, before placing children into a
program. The assessment is re-administered at six-month intervals to check on the child’s
progress. Recently the non-prof t organization was given a small amount offundingfrom
the provincial government to evaluate the effectiveness of three of their programs offered
to children prior to their entering the public school system: one-on-one aba therapy in the
centre, one-on-one aba therapy in the child's home and a school-readiness preparation
program. Kiran is tasked with carrying out the evaluation.
ONE-SAMPLE (TEST
The
one-sample t test is most often used to determine it a research sample is
may first want to ensure that the children
in the sample from Spectrum Group do not somehow differ significantly from
other children in Ontario on the autism spectrum. She may want to ensure that
the severity of their autism and their functional status are representative, in order
to be able to confidently generalize the results. When using a one-sample t test to
determine if a sample is representative, we hope we are not able to reject the null
hypothesis, that there is no difference between the sample and the population.
This is the opposite of what we are usually doing, which is collecting evidence
which we hope will allow us to reject the null hypothesis and conclude that there
is a significant difference between our sample and the control group. In the case
of a one-sample t test, we want to prove that there is no difference between the
sample and the population, and that our sample is therefore representative of the
population. Remember that in failing to reject the null hypothesis, we are stating
that any difference between the mean of the sample and the mean of the population
is likely not due to sampling error, but rather, is a true difference.
Using their scores from the Verbal Behaviour Milestones Assessment (vbmapp), Kiran can compare the children at Spectrum Group to other children in
Ontario (assuming she has the data of other children’s scores on this assessment).
Let’s say she knows the mean score for children in Ontario on this assessment and
the mean score for all of the children in the three programs at Spectrum Group.
She could use the one-sample t test to compare these means. Ifthe resulting p value
is greater than .05 (p > .05), she would fail to reject the null hypothesis, and she
would conclude that her sample is not significantly different from the population
in regard to their scores on the vb-mapp. She could then confidently proceed to
run other statistical tests with the data and
generalize the results to the population
of children on the autism spectrum in Ontario. If the results of her one-sample t
representative of its population. Kiran
100
Statistics for Social Justice
resulted in
value of less than .05 (p < .05), however, she would reject the
hypothesis and conclude that her sample is significantly different from the
population in regard to their scores on the vb-mapp. She would, therefore, not be
able to generalize her findings, making them much less valuable. Kiran would likely
present this information to Spectrum Group and its funder prior to continuing
with any further statistical analyses.
test
a
p
null
NON-PARAMETRIC ALTERNATIVE: CHI-SQUARE GOODNESS OF FIT TEST
When
designing a research study, it is usually best to design it in such a way that
robust statistical tests, parametric tests, can be used to evaluate the data.
This, however, may not always be possible, for instance, if we are given data that
have already been collected that do not meet the criteria for a parametric test. When
this is the case, non-parametric alternatives must be used. For every parametric
test, there is a non-parametric alternative. For the one-sample t test, the alternative
test is called the chi-square goodness of fit test. It fulfills the same purposes as
the one-sample t test, but is used when the dependent variable is at the nominal
level of measurement (remember that for the one-sample t test it must be at the
interval/ratio level of measurement). It is similar to the chi-square test of associa¬
tion, described in Chapter 8. A less robust test, the chi-square goodness of fit test
does not compare means, but instead compares the percentage of cases in each
category of the nominal variable. This less precise test produces a chi-square value
(instead of a t value), represented by a2.
the
most
PAIRED-SAMPLES f TEST
There are many names for a paired-samples t test:
dependent or dependent-groups
paired or paired-groups t test, or matched-groups t test. This type of t test
does not compare a sample mean with a population mean, as did the one-sample t
test. Instead, it compares two sample means that are somehow related. The sample
may be the same sample measured twice, often a one-group pre-test/post-test
research design, or the test may involve two related samples, each measured once.
t test,
Let s look
a
bit closer at these
two scenarios.
Same Sample, Measured Twice
Using Kiran’s task of evaluating the effectiveness of the program at Spectrum
Group, let’s say she wants to look at the effectiveness of one program in particular:
the one-on-one aba therapy in-home. As mentioned, the Spectrum Group uses
the vb-mapp assessment on all children prior to starting the program and then
again after six months. Here we have the same sample of children being measured
twice, often called a one-group pre-test/post-test research design. We could run a
t Tests and ANOVA
101
paired-samples t test to determine ifthe childrens scores had changed significantly
from one time point (pre-intervention) to another (mid-intervention). The research
hypothesis could be: Children who participate in the one-on-one aba therapy inhome program will have improved scores on the vb-mapp assessment.
Two Related
Samples, Each Measured Once
Here the two
samples may naturally be similar in some way, such as two of the
being siblings, one of whom is put in a therapy group, while the other
receives no treatment. Or, the samples may be matched based on some other
characteristic, for example, based on their vb-mapp scores (low and high), with
half of the low scores and half of the high scores each assigned to the one-on-one
aba
therapy in-home program and the one-on-one aba therapy in-centre program.
This would ensure that both groups (in-home and in-centre) have an equal number
of low and high scoring children on the vb-mapp.
children
NON-PARAMETRIC ALTERNATIVE: WILC0X0N SIGN TEST
The
non-parametric alternative to the paired-samples t test is the Wilcoxon Sign
This test is used when the
dependent variable is not at the interval/ratio level of
(through their natural match (e.g.,
sibling) or calculated match [e.g., based on assessment scores]), and then assigns
each case a positive or negative, based on whether their score on the dependent
variable is higher or lower than its matched counterpart. If there are many more
positives or negatives in one group over the other, this suggests that a significant
difference may be found.
test.
measurement.
The
test
matches the individuals
INDEPENDENT SAMPLES t TEST
Like the
paired-samples t test, the independent samples t test compares the means
samples. It is commonly used in experimental or quasi-experimental designs
with two intervention conditions (intervention A/intervention B, or intervention/
control group). With an independent samples t test, in addition to the criteria
mentioned before (nominal level independent variable, interval/ratio dependent
variable and a normally distributed dependent variable), the samples must also
be randomly selected. That is, all cases have an equal chance of being selected
for participation in the study. Importantly, the two samples do not have to be the
same size. The
degrees of freedom for an independent t test are calculated using
the following formula, dj= ni+n2-2. The independent samples t test compares the
means of the two
samples, and a t value and p value are produced, which tell us if
the two means are significantly different from each other. We can then either reject
or fail to
reject the null hypothesis.
of two
102
Statistics for Social Justice
In Kiran’s task of looking
at the effectiveness of Spectrum Group’s three pro¬
she may choose to compare the effectiveness of one program over another.
She could choose to determine which one-on-one aba therapy location is most
effective: in-home or in-centre. Assuming the children were randomly selected to
participate in the in-home or in-centre program, and that those who that partici¬
pated in one program did not also participate in the other, the two groups would
be independent of each other. Therefore, an independent t test would be used to
test the following research hypothesis: Among children receiving one-on-one aba
therapy, those who receive their therapy in-centre will have higher scores on the
vb-mapp assessment than those who receive their
therapy in-home.
grams,
First, we need to check to
•
•
ensure
that the criteria for the test have been met:
Independent variable at the nominal level: therapy location (at home, in-cen¬
tre).
Dependent variable at the interval/ratio level: score on the vb-mapp assess¬
ment
•
•
•
Dependent variable normally distributed: assumed
Groups are independent of each other
Random samples
Next, we calculate the independent samples t test.
Formula for the
t
Independent t Test
*1 “ *2
=
if ss.+ssrvf
yj\N1 + /V2 — 2/ V
Where
Xi =
The
mean
of group 1
x2 =
The
mean
of group 2
5S,=
The
sum
of squares of group 1
of squares of group 2
ss2 =
The
N,
=
The number of participants in group 1
N2
=
The number of participants in group 2
sum
Below are the results from the vb-mapp assessments after six months of therapy.
Each group
(in-home and in-centre) had 10 children.
t
Tests and ANOVA
103
Table 9.1 Raw Data
Group 1: In-Home
Group 2: In-Centre
40
40
30
60
40
60
20
80
50
30
70
90
80
60
40
80
50
70
60
70
OO
^1
.5?
II
SO
Step 1: Calculate 3ci and Jc2.
Step 2: Calculate ssi and 55: by doing the following for each case: subtract the
mean, square the given value, sum the squared values for all cases.
Table 9.2 Sums of Squares
Calculations
SSi
ss2
40
-
48
=
-8
=
64
40
-
64
=
-24
30
-
48
=
-18
=
324
60
-
64
=
=
64
60
-
64
-
64
=
576
-4
=
16
=
-4
=
16
=
16
=
256
1156
40
-
48
=
-8
20
-
48
=
-28
=
784
80
50
-
48
=
-2
=
4
30
-
64
=
-34
=
70
-
48
=
22
=
484
90
-
64
=
26
=
676
80
-
48
=
32
=
1024
60
-
64
=
-4
=
16
40
-
48
=
-8
=
64
80
-
64
=
16
=
256
50
-
48
=
2
=
4
70
-
64
=
6
=
36
-
48
=
12
=
144
70
-
64
=
6
=
36
60
SSi
=
2960
ssi =
3040
104
Statistics for Social Justice
Step 3: Plug the calculated values in the formula and calculate.
X1
t
X2
~
ss, + ss2
N,+N2-2
48
t
-
64
=
2960 + 3040
010
10 + 10
—
2
J
)
(± ++ ±)
10/
vlO
-16
V(333.33)(0.2)
1
_
~
-16
816
t
=
-1.96
Degrees of freedom df= Afi + Ni -2
f = -1.96, df= 18, p > .05
=
18
Step 4: Use the t distribution table (Table 9.3) to look
up the t score for the
degrees of freedom and the p value you are using (.05). In this case, we
are
using a two-tailed test as we do not have sufficient evidence to show that either
in-home or in-centre therapy is superior, so the t critical value is 2.101.
calculated
(-1.96) falls within the rejection region (±2.101), Kiran fails
reject the null hypothesis that there is no difference between therapy location,
and she concludes that therapy location is not significantly related to scores on
As the (calculated
to
the
vb-mapp.
t Tests and
ANOVA
105
Table 9.3 Critical Values of T
Level of Significance
.10
.05
for a One-Tailed Test
.025
Level of Significance
.01
.005
.0005
for a Two-Tailed Test
df
.20
.10
.05
.02
.01
.001
1
3.078
6.314
12.710
31.821
63.657
636.619
2
1.886
2.920
4.303
6.965
9.925
31.598
3
1.638
2.353
3.182
4.541
5.841
12.941
4
1.533
2.132
2.776
3.747
4.604
8.610
5
1.476
1.015
2.571
3.365
4.032
6.859
6
1.440
1.943
2.447
3.143
3.707
5.959
7
1.415
1.895
2.365
2.998
3.499
5.405
8
1.397
1.860
2.306
2.896
3.355
5.041
9
1.383
1.833
2.262
2.821
3.250
4.781
10
1.372
1.812
2.228
2.746
3.169
4.587
11
1.363
1.796
2.201
2.718
3.106
4.437
12
1.356
1.782
2.179
2.681
3.055
4.318
13
1.350
1.771
2.160
2.650
3.012
4.221
14
1.345
1.761
2.145
2.624
2.977
4.140
15
1.341
1.753
2.131
2.602
2.947
4.073
16
1.337
1.746
2.120
2.583
2.921
3.015
17
1.333
1.740
2.110
2.567
2.898
3.965
18
1.330
1.734
2.101
2.552
2.878
3.922
19
1.328
1.729
2.093
2.539
2.861
3.883
20
1.325
1.725
2.086
2.528
2.845
3.850
21
1.323
1.721
2.080
2.518
2.831
3.819
22
1.321
1.717
2.074
2.508
2.819
3.792
23
1.319
1.714
2.069
2.500
2.807
3.767
24
1.318
1.711
2.064
2.492
2.797
3.745
25
1.316
1.708
2.060
2.485
2.787
3.725
26
1.315
1.706
2.056
2.479
2.779
3.707
27
1.314
1.703
2.052
2.473
2.771
3.690
28
1.313
1.701
2.048
2.467
2.763
3.647
29
1.311
1.699
2.045
2.462
2.756
3.659
30
1.310
1.697
2.042
2.457
2.750
3.646
40
1.303
1.684
2.021
2.423
2.704
3.551
60
1.296
1.671
2.000
2.390
2.660
3.460
120
1.289
1.658
1.980
2.358
2.617
3.373
Source: Fisher and Yates 1963
106
Statistics for Social Justice
NON-PARAMETRIC ALTERNATIVE: MANN-WHITNEY U TEST
One of the most
common
non-parametric alternatives to the independent t test is
the
Mann-Whitney U test. This alternative test can be used when the dependent
1) not at the interval/ratio level or 2) not normally distributed. Like its
parametric test, the samples do not have to be of equal size. The test compares the
ranked cases in one group which fall above/below the ranked cases in the second
group. For the purposes ofthis textbook, we will not demonstrate how to calculate
this test, but rather only explain the theory behind it.
variable is
PRESENTING FINDINGS OF f TESTS
The proper way to report t test
For
a
For
a
significant test: t = _.
non-significant test: t =
findings is the following:
,
df=
,
,
df-
p
/>
,
.
p =
.
Note that the t,
df and p are always italicized, and there is a space both before
equal sign. Then summarize the numerical findings using a few
concluding sentences for the reader. For example: An independent samples t test
was conducted to evaluate the
hypothesis that:
The test was/was not
significant (insert test results here: t(df) = _. ,p=/<
). This supports/does
not support the research hypothesis.
Participants in the group (M =
, SD =
) on average scored higher/
lower than participants in the group (M = _.
, SD =
).
and after the
.
_.
_
_
.
HOW TO USE A (TEST IN A PROGRAM EVALUATION
As stated
previously, the gold standard for program evaluations is the classic
experimental design, which compares an experimental group with a control group.
Let’s look at how the t test can be used in this type of program evaluation design.
A social work agency has been providing counselling to women survivors of abuse
on an individual basis
for several years with the goal of raising their self-esteem. Their
service user list is over a hundred women and growing. Recently, to become more cost
effective and treat more women at any one time, they have decided to begin offering the
same service in
groups. They are concerned that the group program may not be as effec¬
tive as the individual one-to-one sessions. Justine and two colleagues at this agency have
decided to run a pilot project to evaluate the results.
As a first step, Justine reviews the literature on the internet to find journal articles
that describe research projects carried out with similar populations. She also needs
to
find out about the kinds of instruments that are used to measure the effectiveness
of programs
designed to raise the self-esteem of women survivors of abuse. She
t Tests and
finds standardized
to
fit the
is
one
tests
ANOVA
commonly used in this type of research. One that
107
seems
requirements, the Rosenberg Self-Esteem Scale (ses) (Rosenburg 1989),
of the
widely used measures of self-esteem. While this test was not
specifically designed for women survivors of abuse, after a careful review of the
items in the scale, the social work researchers agreed that it would measure what
they were interested in.
As a next step, Justine designs the study to include two groups: the experimental
group is made up of 10 women randomly selected from the population of service
users to
participate in the group treatment program, and the control group includes
randomly selected women to receive the usual one-to-one counselling. Both
groups are asked to complete the ses at the start of the counselling program. This
is to ensure that the two groups are similar in terms of their level on the ses. The
experimental group participates in the group treatment program, with one group
session per week for ten weeks, and the control group continues receiving one-toone
counselling. Once the programs finishes, both groups again complete the SES.
The research hypothesis is that the scores of the experimental group will be
different than the scores of the control group. The null hypothesis is that there will
be no difference between the two groups. Justine carries out the analysis using a t
test for independent samples. Below are the results of her analysis.
most
Table 9.41 Test Results
Output
Mean Scores
n
t
df
Probability
Experimental
23.6
10
2.287
18
.034
Control
20.6
10
Group
Presentation of results t
=
2.29,
df= 18, p
<
.05
Therefore, Justine rejected her null hypothesis and concluded that that there
is
difference between the
of experimental group
and the control group,
experimental group scoring significantly higher, lending support to the
proposal to offer more group counselling. Not only will it be a cost saving for the
organization, but it will also provide a more effective treatment for the women.
a
scores
with the
ANOVA
anova
stands for
used
compare means.
at
to
the nominal
interval
we are
or
analysis of variance. It is similar to t tests in the sense that it is
It is also similar in that there is an independent variable
or ordinal level of measurement and a
dependent variable at the
ratio level. One difference is that the
anova
is used in situations where
comparing three or more groups instead of two. That is, the independent
108
Statistics for Social Justice
Box 9.1
The
Indigenous Parenting Practices
following
summary
describes
a
study carried out
on
Indigenous parenting
practices using the t test for independent samples.
Indigenous parenting practices are generally viewed as different than mainstream
(Euro-Canadian) parenting. For instance, raising children is seen by Indigenous
parents as a community responsibility whereas the mainstream views parenting
primarily the responsibility of individual families (Northwest Indian Child
1986). Furthermore, it has been recognized that the Canadian
governments policy on assimilation has had a devastating effect on Indigenous
communities (Shkilnyk 1985). So what has been the impact of assimilation
on
Indigenous parenting practices? This article presents the results of a study
comparing Indigenous and mainstream parenting practices. The study attempted
to answer the question: Are there significant differences between Indigenous and
as
Welfare Institute
mainstream
To
parenting?
the difference between
Indigenous and mainstream practices, the
principal author developed a fiffy-five-item paper-and-pencil questionnaire based
on an
Indigenous parenting program called Cherish the Children (Minnesota
Indian Women’s Resources Center 1988). A sub-group of items in this questionnaire
called Family Life Skills looked at the role played by the extended family and elders
in the care and supervision of children. A convenience sample of 102 Indigenous
parents for this study was drawn from the First Nations communities situated in a
semi-rural region of Northern Ontario west of Sudbury. For comparison purposes,
a convenience
sample of mainstream parents consisted of 60 parents who live in the
same
geographic region. Both groups were asked to complete the questionnaire.
A t test for independent samples was used to compare the difference between
the two samples’ results on the Family Life Skills questions. The table below shows
the results of the analysis.
measure
Subscale
Family Life Skills
Group
N
Mean
t score
df
Prob.
Indigenous parents
102
40.42
2.13
160
.036
60
37.88
Non-Indigenous parents
= 2.13,
df= 160, p < .05
Presentation of the results: t
These results indicate that, even after years
of assimilation, Indigenous parenting
significantly different than those of mainstream parents and that
Indigenous parents continue to see parenting as a community responsibility.
practices
are
Note: The term used in the
original article was Native parenting. An acceptable
today is Indigenous. Because the samples were not randomly selected the
authors are limited in their ability to generalize the restults to a wider population
term
of Indigenous parents.
Source:
van
de Sande and Menzies 2003
t Tests and ANOVA
variable has
more
than
two
109
levels. Another difference is that instead of the t value,
the anova produces an F ratio.
If there is only one dependent variable, we refer to
simple or one-way anova. Where there is more than one dependent
variable, we use the term manova, which stands for a multivariate analysis of
variance. With advanced statistical computer programs like SPSS, even the more
complex anovas are relatively simple to calculate.
The simple anova compares the means of each group with each other as well as
with the overall mean of all the groups combined. This is referred to as the between
group variance. It also looks at the amount of variability within each group. This
this
test as
a
is referred to
as
the within group
variance.
To see how the
simple anova is used, let us consider the example of the program
evaluation carried out byjustine. She compared the experimental group of service
users who
participated in group counselling with a control group of users who
received one-to-one counselling. What ifjustine wants to add a third group? Let’s
say that she adds a group of service users who are on the waiting list for counselling.
Instead of doing multiple t tests comparing various combinations of two groups,
she could instead carry out one test using the simple anova. The results of the
test would tell Justine if the differences between the three groups are the result of
random error, in which case she would have to accept the null hypothesis, or if the
differences are great enough, she can reject the null hypothesis and conclude that
there is a real, statistically significant difference.
SUMMARY
In this
chapter, we presented a well-known parametric test called the t test. We
explained that, since the t test is a parametric test, certain conditions must be met.
While the independent variable should be at the nominal level, the dependent
variable
must
be
at
the interval
or
ratio level of measurement and the results must
be
normally distributed. We stated that there are three variations of the t test: the
one-sample t test; the paired t test; and the independent samples t test, and we
learned how to calculate the independent samples t test. We also introduced the
anova, which is used when we have three or more samples.
REVIEW QUESTIONS
1.
2.
Explain why the classic experimental design is considered the gold standard
in research methodology. How might this be related back to what was covered
in Chapter 2 on the history of empiricism?
Describe when you might use a non-parametric alternative to any of the three
types of t tests discussed in this chapter.
Statistics for Social Justice
3.
After
comparing a control group with an experimental group, the researcher
.07. What does this mean in terms of statistical significance? What is
finds p =
the usefulness of these
4.
findings to the research?
hypothetical example of when a social worker research would use the
anova. Explain why they would use the anova in this case instead of a t test.
Create
a
Correlation Analysis
In this chapter, we examine studies where the data for both the independent and
the dependent variables are at the interval or ratio level of measurement. The
parametric test commonly used for this type of study is called the correlation
coefficient, and it tests whether there is
a
linear relationship between the
independent and the dependent variables. We explain that correlation analysis
determine whether the relationship is significant and illustrate how to use
it to further social justice.
can
There is a vast amount of literature comparing data on twojustice
or moreissues,
variasuch
bles at
the ratio level that identifies
a
correlation between social
as
poverty, and a host of social issues, such as health, including mental health, low
levels of educational attainment and crime. For instance, an
article by Gerald
Ogbuja (2012), Correlation between Poverty and Mental Health: Towards a Psychiatric
Evaluation, reviews a number of epidemiological studies which identify poverty
and socio-economic problems as some of the most important factors related to
mental health issues. In a study by Silvernail, Sloan, Paul, etal. (2014: i) entitled The
Relationship between School, Poverty and Student Achievement in Maine, the authors
state that, as the level of poverty increases, the performance of students decline.
While they admit that other factors affect school performance, the authors insist
that the level of poverty is the “single best predictor of average student performance.”
There are also cross-national studies which look at income inequality and
health. In a report published by Inequality.org, an online journal that focuses on
social justice issues, a comparison is made between countries with various levels
of income inequality and infant mortality, both of which are ratio level variables.
Figure 10.1, published by a U.K. organization called Equality Trust and cited in
Inequality.org, plots the 22 richest countries and shows the ratio between the top
20% of income and the lowest 20%. It reveals that countries such as Japan, Finland
and Norway have the lowest ratio (thus least income inequality) while countries
such as Singapore, the United States and Portugal have the highest ratio (thus
most income inequity).
Statistics for Social Justice
112
Figure 10.1 Ratio of Top Income to Bottom Income
(Average for years 2003-06)
9.7
Source:
lnequality.org 2011
In
Figure 10.2, these same countries are compared based on the rates of infant
mortality, defined as the death of children during the first year oflife. It shows that
some of the countries, like the U.S. and
Portugal, also have the highest level ofinfant
mortality. An exception is Singapore, which has the highest income inequality but
the lowest infant mortality. Figure 10.3 displays the same data on a scatterplot
that shows the levels of income inequality with infant mortality. Figure 10.4 is
a
scatterplot showing the same developed countries’ life expectancy compared
to income inequality. Here again, we can see that the countries with the
highest
income inequality have the lowest life expectancy.
Figure 10.2 Infant Mortality, 2005 (Deaths in first year of life per 1000 live births)
6.9
Source:
Inequality.org 2011
Correlation
Analysis
113
Figure 10.3 Scatterplot on Infant Mortality
•
1
USA
6.5
£ 6.0
• New Zealand
Ireland •
■2 5.5
Denmark •
-1
°
5.0
t
•'Netherlands
—-'-"Germany •
• France
Italy
•
Spain •
il 4.0
c
w
•&
Swit2erland.
AuStfia
45
Portugal
• Canada
Belgium •
co
•
* UK
Israel •
Finland
•* NorwaV
3.5
• Sweden
•
2 3.0
Q
Japan
Singapore •
2.5
3.00
4.00
5.00
Income
Source:
7.00
6.00
Inequality (Top 20%
:
8.00
9.00
10.00
Bottom 20% Ratio)
lnequality.org 2011
Figure 10.4 Scatterplot on Life Expectancy
82
•
Japan
81
2
CO
>-
80
c
>.
o
2
o
„
Israel
„
.
Spain M Canada
79
•
Norway t
*
•TBeJgiura^ France
Oprmanu
V
78
m
• lta|y
Austria <
d»
^
Switzerland
Greece <
Netherlands
Singapore •
New Zealand
.2
_i
77
Denmark •
•
76
3.00
4.00
5.00
Income
Source:
6.00
7.00
Portugal
8.00
9.00
10.00
Inequality (Top 20% : Bottom 20% Ratio)
Inequality.org 2011
What
Figures 10.1 to 10.4 reveal is that quantitative data at the ratio level
can be
displayed graphically to illustrate the impact of income
inequality. Reports such as the one published by Inequality.org can be used by
those who wish to argue for the development of progressive social policy changes.
An article published in the Toronto Star newspaper by Conservative Senator Hugh
Segal (2011) states:
of measurement
While all those Canadians who live beneath the poverty
means
prisons
line are by no
almost all those in Canada’s
from beneath the poverty line. Less than 10 per cent of
associated with criminal activity,
come
114
Statistics for Social Justice
Canadians live beneath the poverty
line but almost 100 per cent of our
There is no political ideol¬
ogy, on the right or left that would make the case that people living in
poverty belong in jail.
prison inmates
come
from that 10
per cent.
OTHER USES OF THE CORRELATION COEFFICIENT
In addition to their
use
in
social
justice related studies, correlations are also used
authors developing standardized tests,
for a wide variety of purposes. For instance,
like the
Rosenberg Self-Esteem Scale (ses), mentioned in Chapter 9, can use it to
mentioned in Chapter 1,
reliability is defined as the degree to which the measurement instrument provides
consistent results over time, and validity refers to the degree that an instrument
will truly measure what it is supposed to measure, and not something else. The
test-retest reliability scores range from r = .82, to r = .85 whereas the criterion
validity score is r = .55. This shows that the reliability scores for the ses is quite
good while the validity score is acceptable (Rosenberg 1965).
establish the reliability and validity oftheir instrument. As
SCATTERPLOTS
Earlier we stated that correlations
are
used to
test whether
there is
a
linear relation¬
ship between the independent and dependent variables. To refresh your memory
on what is meant
by a linear relationship, we suggest going back to Chapter 3,
where we provide a description of scatterplots. Examples such as Figures 10.3 and
10.4 portray case values for two variables simultaneously. Each dot on a scatterplot
represents the intersection between the two variables. In this way we can see if a
relationship exists. Ifthe dots form something close to a line, a relationship exists.
If there is no pattern, then no relationship exists.
Ferderich Huebler (2005) provides the scatterplot in Figure 10.5 of the correla¬
tions showing the relationship between poverty and education. Using data from
the 2004, American Community Survey, Huebler states:
In the United
States, the states with the highest poverty rates are also
those with the lowest share of high
school graduates. The graph below
[Figure 10.5] plots the percent of the population living below the poverty
level
against the percent of the population above 25 years of age without
complete high school education.
In other
words, the higher the percentage of people who live below the poverty
line, the higher the percentage of people who have not completed high school.
Correlation Analysis
115
Figure 10.5 Poverty Line and Educational Achievement
25.0%
20.0%
15.0%
10.0%
5.0%
5%
7%
9%
11%
Percent of
13%
15%
17%
19%
21%
23%
population below poverty level
Source: Huebler 2005
CORRELATION COEFFICIENT
The correlation coefficient is
expressed as continuum from -1 to +1, where -1 is a
perfect negative relationship, +1 is a perfect positive relationship, and a score of 0, or
close to 0, means that there is no relationship. The correlation coefficient describes
two things: the strength and the direction of the linear relationship between two
variables. The closer the score is to 1.0 (or-1.0), the stronger the relationship. The
positive/negative sign indicates the direction of the relationship. A line going up,
from the lower left side to the upper right side of the scatterplot, indicates a posi¬
tive direction. A line going down from the upper left side to the lower right side
indicates a negative direction (Figure 10.6). If there is no pattern in the dots, it
indicates a very weak relationship or none at all (Figure 10.7). Correlation coef¬
ficients equal to or greater than .80 are considered strong, those between .50 and
.79
are
considered moderate, and less than .49 are considered weak. Remember:
a
significant relationship does not predict causation. It will not tell us if the independ¬
ent variable caused a
change in the dependent viable. The statistic for correlation
is Pearson’s Product Moment Correlation, or simply Pearson’s r.
116
Statistics for Social Justice
Figure 10.6 Positive and Negative Correlation Examples
Figure 10.7 Weak or No Correlation Example
PREDICTOR AND OUTCOME VARIABLES
So far in this text, we
have been using the terms independent and dependent
variables, where the independent variable is the
that causes or contributes
causing the change in the outcome, which is called the dependent variable.
We manipulate the independent variable to find out what effect this will have on
the dependent variable. For instance, we may want to compare a new program or
intervention approach to a control group or an existing approach to see if it will
have the desired positive outcome on service users. The program or approach is
the independent variable and the desired outcome is the dependent variable.
With correlations, the independent variable is referred to as the predictor vari¬
able and the dependent variable is called the outcome variable. These terms more
accurately describe the role that these variables play. In the case of correlations,
we are not
looking at cause and effect relationships; instead, we are interested in
finding out if the variables co-vary, that is, if the two variables are related in some
way. If there is a strong relationship, either positive or negative, we may be able
to look at the predictor variable and actually predict the effect it will have on the
to
outcome
variable.
one
Correlation
Analysis
117
MULTIPLE r's
used to test the relationship between the outcome variable and
predictor variables. For instance, we may want to know about the
relationship between results on a statistics exam and l) the number of statistics
course taken, 2) the
age of the student, 3) the grade point average and 4) scores
Multiple r’s
three
are
or more
the math section of the
on
sat.
Formula for Pearson’s rTest:
nl*y- Q»(Ey)
sJWZx* -Q»2][nly2-G]y)2]
Where:
r
=
Correlation coefficient
n
=
Number of cases
=
Sum of xy
=
Sum of x column
=
Sum
=
Sum of x2 column
=
Sum
I xy
lx
zy
Lx2
Ly2
column
ofy column
ofy2 column
Admittedly, this formula can appear to be a bit intimidating, but approached in
systematic, step-by-step manner, it is straightforward. Let’s look at the following
hypothetical example.
Sheila is a social worker offering assertiveness trainingfor women. She was interested
in finding out if there a relationship between the number of sessions service users par¬
ticipate in on assertiveness training programs and their scores on a standardized scale
measuring assertiveness. She believed that the more sessions service users participate in,
the higher they will score on the assertiveness scale.
Sheila’s test hypothesis is: There is a positive relationship between the number
of sessions in assertiveness training that service users attend and their score on a
a
standardized
test
measuring assertiveness.
The
corresponding null hypothesis is: There is no relationship between the
training that service users attend and their score
standardized test measuring assertiveness. She carries out the research and
number of session in assertive
on a
her
raw
data is shown in Table 10.1. Table 10.2 contains the work necessary to
obtain the values needed
these numbers in.
to
be
plugged into the formula, followed by plugging
118
Statistics for Social Justice
Table 10.1 Raw Data
Service Users
Sessions
(x)
Score
1
June
2
30
Lynn
3
40
Shelly
3
50
Liz
4
60
Carol
5
60
Debbie
5
60
Anne
7
90
Sue
8
80
Helen
8
90
Table 10.2
Plugging in the Numbers
5*
lx2
X/
1
30
1
900
30
2
30
4
900
60
3
40
9
1600
120
3
50
9
2500
150
4
60
16
3600
240
5
60
25
3600
300
5
60
25
3600
300
7
90
49
8100
630
8
80
64
6400
640
8
90
64
8100
720
46
590
266
39300
3190
X.r
Totals
(y)
30
Mary
I xy
Correlation
Analysis
119
Solving the Formula for r Score
7
n^xy
-
(Xx)Q]y)
~~
V["S>2-(I»2][nEy2-Q:y)2]
10(3190)
-v/[10(266)
-
(46)(590)
(46)2][10(39300)
-
31900
V[2660
-
-
-
(590)2]
27140
2116][393000
-
348100]
4760
V[540][44900]
4760
~
V24425600
4760
r
~
4942.23
r =
.96
Sheilas calculations result in
an r score
of .96, which is very
close to
a
perfect
positive score of 1.0. Therefore, Sheila is confident in concluding that there is a
strong positive
a
service
The
user
next
relationship between the number of sessions in assertive training
attends and
a
standardized
question is whether this
measuring assertiveness.
is significant. Looking at Table 10.2,
score
score
showing the critical values of r, with an N (number of participants) of 10, and at
the .05 level of significance for a one-tailed test, Sheila needs an r score of .5494 to
reject the null hypothesis. Since an r score of .96 is well above the required .5494,
she is certainly safe in accepting her research hypothesis that there is a strong posi¬
tive relationship between the number of sessions in assertive training that service
users
attend and their
score on a
standardized
test
measuring assertiveness.
120
Statistics for Social Justice
Table 10.2 Critical Values of r
Level of Significance
N
.05
.025
Level of Significance
for a One-Tailed Test
.01
.005
.0005
for a Two-Tailed Test
.10
.05
.02
.01
.001
.8054
.8783
.9343
.9587
.9912
6
.7293
.8114
.8822
.9172
.9741
7
.6694
.7545
.8329
.8745
.9507
8
.6215
.7067
.7887
.8343
.9249
9
.5822
.6664
.7498
.7977
.8982
10
.5494
.6319
.7155
.7646
.8721
11
.5214
.6021
.6851
.7348
.8471
12
.4973
.5760
.6581
.7079
.8233
13
.4762
.5529
.6339
.6835
.8010
14
.4575
.5324
.6120
.6614
.7800
15
.4409
.5139
.5923
.6411
.7603
.7420
5
16
.4259
.4973
.5742
.6226
17
.4124
.4821
.5577
.6055
.7246
18
.4000
.4683
.5425
.5897
.7084
19
.3887
.4555
.5285
.5751
.6932
20
.3783
.4438
.5155
.5614
.6787
21
.3687
.4329
.5034
.5487
.6652
22
.3598
.4227
.4921
.5368
.6524
27
.3233
.3809
.4451
.4869
.5974
32
.2960
.3494
.4093
.4487
.5541
37
.2746
.3246
.3810
.4182
.5189
42
.2573
.3044
.3578
.3932
.4896
47
.2428
.2875
.3384
.3721
.4648
52
.2306
.2732
.3218
.3541
.4433
62
.2108
.2500
.2948
.3248
.4078
72
.1954
.2319
.2737
.3017
.3799
82
.1829
.2172
.2565
.2830
.3568
92
.1726
.2050
.2422
.2673
.3375
102
.1638
.1946
.2301
.2540
.3211
Source: Fisher and Yates 1963
Correlation Analysis
Box 10.1 Use of Multiple r s
between
121
Study on the Relationship
Poverty and Childhood Depression
in
a
(ses) and depression has been wellpopulations. A number of studies suggest that family ses may
be associated with depression among children and adolescents as well, although the
evidence is mixed. We assessed the relation between family income and depressive
symptoms among 457 children aged 11-13 years old and examined pathways that
may explain this relation. In-person interviews of children and their caregivers
were conducted,
including assessment of family income and administration of the
Computer-based Diagnostic Interview Schedule for Children (c-disc). Family
income was significantly associated with depressive symptoms, with children in the
lowest income group (<$35,000) reporting a mean of 8.12 symptoms compared
to 6.27 symptoms in the middle income group ($35,000-$74,999) and 5.13
symptoms in the highest income group (>$75,000; p<0.00l). Controlling for the
number of stressful life events experienced in the past six months attenuated the
effect of low family income on depressive symptoms by 28%. Indicators of the
family environment explained 45% and neighbourhood median household income
and aggravated assault rate explained 12% of the relation. The family environment,
including parental divorce or separation and perceived parental support, appears to
explain most of the relation between low family income and childhood depressive
symptoms. Further exploration of the pathways between family ses and depression
may suggest potential interventions to reduce the occurrence and persistence of
depressive symptoms in children.
The relation between low socio-economic status
documented in adult
Source:
Tracy, Zimmerman, Stoep, et al. 2008
SUMMARY
In this
chapter, we described another well-known parametric test, the correlation
explained that correlations are used to compare two variables at
the interval/ratio level of measurement. We showed that a scatterplot can be used
to visually portray whether these two variables co-vary, in the sense that they
form a linear relationship. We explained how to use a correlation analysis to test
if this linear relationship is significant and that correlations are often used in large
population studies regarding important social justice issues.
coefficient. We
REVIEW QUESTIONS
1.
Explain the benefits of using a scatterplot to represent data in which both the
predictor and outcome variables are at the interval/ratio level of measure¬
ment
2.
Describe how
scatterplots can be used to advocate for social change.
122
Statistics for Social Justice
Create
3.
hypothetical example of a research study for which a correlation
analysis would be appropriate. How might this study be tweaked in order to
accommodate another predictor variable?
4.
A researcher calculates
a
a
Pearsons
r
correlation and finds
an
rvalue of +.91.
What does this tell us?
5.
What if a researcher calculates
is -.92?
an
rvalue of-.80 but the
What does this tell us? Can this still be useful?
r
critical value needed
Simple Regression Analysis
In this chapter we show that, if there is a strong and significant relationship
between two variables, we can use a simple regression to predict the score
of the outcome variable based on the score of the predictor variable.
introduce the concepts of the regression line and least-squares criterion
show how to calculate the least-squares regression equation.
We
and
Prediction
ng) what a casebased
value, or score,
likely refers
be forto knowing (without
variable. measuri
Predicting
is
to
an outcome
an outcome
on one or
variables is used extensively in several fields
of science. Two familiar examples
are
predicting the weather and predicting the rate of climate change. This method
is also used for predicting economic growth and predicting consumer demand for
commodities such as gasoline or new housing.
In social work, we are interested in demographic changes, so we might want to
predict the proportion of elderly people in the population at some future date or
the increase in the proportion of Aboriginal people. We are regularly called upon
to predict the success of intervention
programs or the likelihood of child abuse
and neglect based on certain risk factors (see Box 1 l.l).
Child psychologists have been able to make predictions of the occurrence of
depression among adolescents based on their self-image:
more
We
investigated the ability of a measure of self-image, two measures of
depression, and demographic characteristics to predict the outcome of
depressive symptoms. Subjects were 47 adolescents who were referred to
outpatient treatment for depression. Subjects were assessed for depressive
symptoms at three time periods. (Fine, Haley, Gilbert and Forth 1993)
Being able to predict the impact of social policy decisions also has important impli¬
cations for policy analysts and government officials. Let’s consider the following
hypothetical example of a study of the relationship between the number of child
protection cases and the level of income support from social assistance programs.
Estelle Young is a social worker working with an ngo called Poverty Ends Now. She
123
124
Statistics for Social Justice
is interested in the
relationship between poverty and child abuse and neglect. She looks
of studies in the literature thatfind a positive relationship. She examined
the number of child protection cases in her province, along with the changes in social
assistance over a twenty-year period. While there was not a perfect correlation, given
the number offactors related to child protection cases, she nevertheless finds that there
is a negative correlation between social assistance levels and child protection cases: as
social assistance amounts available to families drop, the number of child protection
cases increases. She carries out the statistical
analysis and is able to identify a simple
regression model that will predict the number of child protection cases based on a
specific level of social assistance. Acknowledging that governments have been cutting
costs by reducing social assistance rates forfamilies, she believes that it is important for
governments, and the population in general, to understand that there are consequences
to some of these cost-cutting measures. The savings realized by cutting social assistance
will be reduced by the amount of money needed by child protection authorities to deal
at
a
number
with the increased workload.
HOW DOES SIMPLE LINEAR REGRESSION WORK
As
explained in Chapters 3 and 10, scatterplots are graphs that visually display the
relationship between two variables at the interval or ratio level of measurement.
The points along the x axis represent the scores on the predictor variable and the
points along the y axis represent the scores on the outcome variable, also called
criterion variable. Each point on the graph represents the location where these
variables converge for each case (or participant). If the points form something
close to a line, we can say that there is a relationship. If there is a perfect relation¬
ship, a straight line can be drawn that would touch every point. If there is less than
a
perfect relationship, we could still draw a straight line through the points but it
would not touch every point. The line that comes closest to all the points is called
the regression line.
Figure 11.1 shows a scatterplot comparing experience and income, with expe¬
rience being the predictor variable, or variable x, and income being the outcome
variable, or variabley. The dots on the graph form a rough line, indicating that there
is a strong relationship between these two variables. The line in the graph is the
regression line. If we want to predict a level of income and we only have the score
on years of
experience, we would choose a score on income that sits along this line.
The term “regression” is also referred to as “regression toward the mean,” and it
suggests that for each score on experience, the income score will likely fall along
this regression line. If we were to measure the distance from each dot to the line,
we would be
calculating the deviations. If we squared these deviations and totalled
them, this total would be different depending on where we placed the line amongst
Simple Regression Analysis
125
the dots. The line with the lowest total squared
deviations, called the least-squares
line, or the line of best fit. Fortunately, we
do not need to calculate all of the deviations for each possible line. Instead by using
the least-squares equation, we can easily identify the best regression line.
criterion, would be the best regression
Figure 11.1 Scatterplot Showing the Relationship between Experience and Income
0
10
20
Years of
30
40
Experience
Stating the Research Question
The research
question for simple regressions is not phrased in the form of a hypoth¬
esis but rather is
question worded as follows: “How does knowing a value of the
predictor variable improve the prediction on the outcome variable?”
Limits ot
There
a
Simple Regression
are some
terms of what we can say
using simple regressions.
make predictions using values for the predictor that are
limitations in
For instance, we cannot
larger than the largest value or smaller than the smallest value used in the compu¬
tation of the
equation.
Let’s refer back to the
lished that there is
a
example from Chapter 10, on correlations. Having estab¬
strong correlation between weeks in an assertiveness training
and scores on the assertiveness scale, Sheila is interested in being able to
predict scores on assertiveness based on number of weeks in the program. Using
her data set (Table 1 l.i), she is able to create a model that allows her to make this
prediction. The equation that Sheila uses to create her model is called the least-
program
squares
regression equation.
126
Statistics for Social Justice
Table 11.1 Raw Data and
Client
Preliminary Calculations
Weeks
(x)
Mary
1
30
June
2
30
Lynn
3
40
Shelly
3
50
Liz
4
60
Carol
5
60
Debbie
5
60
Anne
7
90
Sue
8
80
Helen
8
90
Ex
46
590
(y)
Score
Ex2
ly2
Exy
266
39300
3190
Least-Squares Regression Equation
y‘ = a + b(x)
y‘ = Predicted y value from a particular x value
a =
The
point where the regression line
b
The
slope of the line, where the amount of change iny is directly related to
of change in x.
=
would intersect they axis,
amount
x =
A selected value of the
predictor variable used to predict the value of the outcome
variable
Box 11.1 Abstract of a
Study on Child Abuse Risk Factors
Secondary analyses of the 1998 Canadian Incidence Study of Reported Child
were carried out to
investigate the effect of caregiver vulnerabilities
on the substantiation of child abuse and
neglect. Analyses were done of (l)
demographic factors, socio-economic disadvantage, and caregivers history of
abuse; (2) caregiver vulnerability factors; (3) involvement in partner violence; and
(4) the interaction between caregiver vulnerability and partner violence. Results
showed that the total number of caregiver vulnerabilities was the best predictor
of the substantiation of child abuse and neglect. Caregiver substance abuse was
the single most important caregiver vulnerability in predicting maltreatment
substantiation. High caregiver vulnerability and high partner violence increased the
Maltreatment
likelihood that maltreatment would be substantiated.
Source:
Wekerle, Wall, Leung and Troane 2007
Simple Regression Analysis
127
THE REGRESSION COEFFICIENT
The regression coefficient is shown as (b) in the least-squares regression equation
and represents the slope of the regression. As explained by Weinbach and Grinnell
(2010), the slope could be conceived in the same ways as the slope of a hill and
change in the outcome (y) variable for the cor¬
responding horizontal change in the predictor (x) variable.
represents the amount of vertical
Regression Coefficient or Slope Formula
b
NT.xv-(7..v)(T.v)
NEx2-(£x)2
Where
b
=
N
=
I-vy
Zx2
I*
b
Slope
Number of cases
=
Sum of xy
=
Sum ofx2 column
=
Sum
column
ofy2 column
=
Sum of x column
=
Sum
=
ofy column
10(3190}-1461(5901
10(266)-(46)^
31900
-
27140
2660-2116
4760
544
b
_
8.75
128
Statistics for Social Justice
THE y
INTERCEPT FORMULA
After
calculating the slope (b), the next step for Sheila is the calculation of the)/
intercept, which is the (a) in the regression formula. They intercept is the point at
which the regression line crosses they axis. A word of caution: Sheila will not be
able to predict the outcome for a client whose score falls below the lowest score
in her original sample, which was 1 week. Obviously, the score for someone who
attended for 0 weeks would be 0. However, if the lowest score in Sheila’s sample had
been 5 weeks, Sheila will not be able to make predictions for clients who attended
for only 4 weeks or less.
Intercept Formula
a=y-b x
Where:
a
y
-
intercept
y
Mean
bx
Slope times the mean ofy column
ofy column
a
=
59-8.75(4.6)
a
=
18.75
Now that Sheila has calculated the
slope and the y intercept, the least-squares
regression equation for this example isy1 = 18.75 + 8.75(x). This formula is our
prediction model. We can use this model to predict any score on the criterion or
outcome variable based on the score on the predictor variable. For instance, if
have
we
a score
of 6
on
the
predictor variable,
we
could
carry out
the following
calculation:
y'
=
18.75 + 8.75(6)
The criterion
or outcome score
tiveness scale that
we
would
would be 71.25. This is the
score on
the
asser¬
predict an individual to achieve after 6 weeks in the
program.
STANDARD ERROR
Obviously, if there was a perfect correlation, that is, an r-value ofeither + 1.0 or -1.0,
predictor variable would be a perfect predictor of the outcome variable. In this
case, there would be no error. Since there is rarely a perfect correlation between
the predictor and outcome variables, it would be helpful to have an indication of
how well the regression equation will predict the outcome variable based on the
predictor variable. If the r-value is 0.0, which suggests that there is no relationship,
then the regression equation would be of no use. The standard error provides this
the
Simple Regression Analysis
indication. The closer that the r value is
ard
error
to
either + 1.0
or
129
-1.0, the closer the stand¬
(p value) is to 0.0, and the more confident we can be that the regression
equation will accurately predict the value of the outcome variable. However, the
closer the standard error is to 1.0, the less confident we can be in our prediction.
EXCHANGING THE
In
x
AND y
VARIABLES
example of the correlation between experience and income, we were
interested in predicting income (outcome variable) based on level of experience
(predictor variable). However, if instead we were interested in predicting the years
of experience based on the level on income, we could conceivably exchange the
variables with income becoming the predictor variable and experience becoming
the outcome variable. If there is a strong r-value, indicating a strong relationship,
it doesn’t really matter which variable is identified as the predictor variable and
which is identified as the outcome variable. The only difference is that we would
have a slightly different regression equation and so we must be careful to enter the
data to reflect this change.
our
Table 11.2 Raw Data and
Preliminary Calculations
Years of Service
Name of Clients
(x)
Job Satisfaction Scores (y)
Sue
1
8
Jane
6
2
Jim
8
2
Helen
3
4
Sarah
4
5
Ed
2
9
Simone
5
3
Mary
7
2
Pauline
9
2
John
4
4
lx
49
41
I*2
1/
Ixy
301
111
149
Statistics for Social Justice
130
Slope
b
N
Where:
b
=
Ex2
(Ex)2
-
Slope
Number of cases
N
E xy
Sum of xy
E*2
Sum of x2 column
1/
Sum
lx
Sum of x column
ly
Sum
column
ofy2 column
of_y column
10(149)-(49)(41)
b
10(301)-(49)2
1490
-
2009
3010-2401
-519
609
b
-.852
bx
(-.852)(4.9)
a = 4.1 -(-4.17)
a =
y-
a =
4.1
a =
8.27
-
Now that we have calculated the
sion
equation is;y‘
=
slope and the y intercept, the least squares regres¬
8.27 + (-.852)(x).
Simple Regression Analysis
131
SUMMARY
As
a
follow-up to Chapter 10, on correlations, this chapter explained that, if there
significant relationship between two variables at the interval/ratio level of
measurement, we can use a simple regression to predict what the score of the
is
a
on the score of predictor variable. We introduced
of regression line, regression coefficient and least-squares criterion.
We showed that being able to predict the impact of social policy decisions can
be very useful to social work researchers and advocates and also has important
implications for policy analysts and government officials.
outcome
variable will be based
the concepts
REVIEW QUESTIONS
1.
Provide
a
hypothetical example of how
a
structural researcher might
use re¬
gression analysis in order to advocate for social change.
2.
Discuss the
dangers of using regression analyses to make predictions. Why
must we consider before making any predictions?
Explain the relationship between correlation analysis, discussed in Chapter
10, and simple regression analysis. What can the latter do that the former can¬
must
3.
we
be cautious? What
not?
4.
Explain why it is important to first ensure that there is indeed a correlation
two variables before using a simple regression analysis to make pre¬
between
dictions.
5.
What does it
mean
if the “b” within the
regression equation is negative? Do
a
quick sketch of a scatterplot to demonstrate what this would look like graphi¬
cally.
12
Writing a Research Report
In this last
chapter, we focus on the preparation of the final research report
explain fhe importance of following the more mainstream writing style
using the third person passive. We provide an outline of what should be
included in a traditional academic research report and describe in some detail
what each section of the research report should include.
We
n
the
previous chapters, we covered the basic principles of descriptive and infer¬
ential statistics
as
well
as
the
more common
statistical
tests to
determine if a
true
relationship exists among variables. Whether it is to identify and describe important
social justice issues, such as poverty and homelessness, or to evaluate a program
using an experimental or quasi-experimental design, this book provides enough
basic knowledge to carry out a variety of quantitative research. As budgets for social
programs continue to shrink and as smaller non-governmental organizations strug¬
gle to provide essential services to the most vulnerable people in our society, social
workers are increasingly called upon to use their research skills to demonstrate
the effectiveness of these programs and services. While qualitative methods are
gaining more widespread acceptance, they will not replace quantitative methods
as the
approach required by many government departments and funding bodies.
Once the data collection and analysis have been completed, the final step
involves writing the research report. Traditional scientific writing continues to fol¬
low specific guidelines; this was the case in social work writing as well, though it is
less so today. Those who took social work research courses before the 1980s were
told that research reports had to be written in the third person passive. In other
words, they were required to use phrases such as “It was found that...” rather than
“We found that...” This was to maintain the appearance of objectivity.
As structural social work researchers, our main concern is that the report be
accessible to those people we serve. While we want to ensure that our research is
useful to service users, we are often obliged to write the final report following a tra¬
ditional academic format. Regardless of the style, we need to keep in mind that the
ultimate goal of our research and research report is structural change. Increasingly,
132
Writing a Research Report
133
social work researchers are finding that they have to write two reports, a more
formal
audience, and another more accessible
one for service users. Our
graduate students conducting research for community
organizations are often asked to prepare a PowerPoint presentation in plain nonacademic language for community members in addition to the formal report for
the agency requesting the research.
one
for
an
academic and/or government
THE WRITING PROCESS
(2005) offers the following useful suggestions to make the
writing process as straightforward as possible. These steps may seem like common
sense, but following these steps can save much time later on.
Christine Marlow
1.
2.
Keep a log for ideas and decisions taken throughout the research process.
Prepare an outline; the more detail it has the easier the final report will be to
write.
3. Write
a
first draft. This is often the hardest part.
Revise several times if neces¬
sary.
4.
Ask
5.
Have
a
colleague to proofread the draft and don’t be afraid of criticism.
proofread the final copy. There are usually mistakes that the
someone
writer misses.
SECTIONS IN THE ACADEMIC FORMAT
The traditional scientific/academic format
normally includes the following
sections:
•
•
Title page
Abstract
•
Introduction
•
Literature Review
•
•
•
Methodology
Findings
Discussion
•
Conclusion
•
List of References
•
(also called the bibliography)
Appendices
The
following summary of what each section should
(2011).
de Sande and Schwartz
cover
is provided by van
134
Statistics for Social Justice
Title
Page
The title page
should provide the reader with enough information without being
overly wordy (Rubin and Babbie 2008). In addition to the usual details such as
the date and the list of authors with their degrees, try to think of a title that will
catch the reader s interest. However, as Rubin and Babbie point out, with a formal
academic report, it is important to maintain credibility and not come across as
unscholarly. Achieving the right balance can, at times, be a challenge and it is often
helpful to seek feedback from colleagues.
Abstract
The abstract is
brief summary
of the study. Most abstracts are between 150 and
although some journals ask for shorter ones. The abstract should begin
with very basic information on the purpose of the study, the research question
and the central thesis. It should have a sentence or two on the research design or
methodology, and conclude with a couple of sentences on the major findings. If
the plan is to present the research at an academic or professional conference, the
a
200 words
abstract is what is submitted
to
the conference
organizers.
Introduction
The introduction is the first section of the main
body of the report. It should
background information concerning the study. It
should also describe the issue being investigated, the goals of the research and the
specific research question or questions. Next, it should provide a description of the
theory that has informed the research. If the study follows a traditional empirical
design, include the hypothesis and the conceptualization and operationaliza¬
tion of the variables. The conceptualization of the variables provides the reader
with clear definitions of each of the variables, including the independent and the
dependent variables. The operationalization tells the reader how these variables
provide the reader with
some
will be measured.
Literature Review
The literature review provides
the reader with a summary ofwhat has been written
topic under investigation. It is a systematic examination and assessment of
the publications available on the topic ofthe study. It looks at what is known about
the topic and what gaps still exist. If there are studies mentioned in the literature
which give contradictory views, provide a brief overview of the various sides of
the question. Ifyou disagree with what the literature says, provide an argument for
your position. The literature review should conclude with a statement of the gaps
in knowledge and how your research attempts to fill these gaps.
on
the
Writing a Research Report
135
Methodology
The
methodology section of a traditional academic report should begin with a
description ofthe selection of participants, also called the sample selection. Explain
how the participants were contacted and whether a probability sampling method or
a
non-probability sampling method was used. The next section in the methodology
section deals with administration of the measurement instruments used in the study
and the method of analysis. If a standardized measurement instrument was used,
include the reliability and validity scores of the instrument. Finally, identify which
statistical
test
was
used and whether it was
parametric test or non-parametric test.
Findings
The
findings section in a traditional research report provides the reader with a clear
description of the results of the study. With a quantitative design, this sec¬
tion would normally include tables and graphs that visually describe the results.
State if the results obtained were statistically significant and whether or not they
supported the research hypothesis.
concise
Discussion and Conclusion
In the discussion and conclusion section,
the results of the study are related back to
the literature review. Were the results
supported or contradicted by the literature?
implications of the research for social work practice and/or social
policy? In what way have the results added to our knowledge base? The discussion
should also include a subsection on limitations. Describe any problems with the
study. For example, perhaps the response rate from participants was low, result¬
ing in some sample bias, or there were difficulties with the administration of the
instrument, resulting in some design error. The discussion and conclusion section
should end with suggestions for further research.
What
are
the
List of References
The in-text citations and the list of references for
social science reports
fol¬
(American Psychological Association) method of referencing. The
list of references must provide complete and accurate information on all sources
low the
most
apa
cited in the report.
Appendices
The
the
appendices section of the report should include, as a minimum, a copy of
used and copies of ethical consent letters. Many
measurement instrument
researchers include additional tables
findings section but
may
on
the results that
were
not
included in the
still be useful to the readers. Each appendix should be
clearly labeled using letters (A, B, C, etc.).
136
Statistics for Social Justice
Example of a Study Written Using a Traditional Format
Appendix B is an example of a research report, which is based on actual research
carried out byJulie Shaw, one of our graduate students.
SUMMARY
In this last
chapter, we discussed the need to write the research report along tradi¬
lines, using the third person passive, which is still the expectation of most
academic disciplines. We offered some suggestions on how to make the writing
tional
straightforward as possible. The rest of the chapter focused on the
traditional format of an academic research report. While many government and
process as
funding bodies still require an academic format, our first concern is meeting the
needs of the people we serve and using our research to promote social justice. This
may involve writing more than one report, a more traditional one for an audience
made up ofacademic, government or other funding bodies, and a non-traditional,
user-friendly report for a lay audience of service users.
REVIEW QUESTIONS
1.
What
are some
of the
challenges in writing an academic research report for a
lay audience?
2.
How
important is it to maintain academic rigour in a report prepared for
a
mainstream audience?
3.
How do you
versus
feel about writing
active?
the first person
a
research report in the third
person
passive