LESSON 1
STATISTICAL CONCEPTS
Science, Observations, Statistics
Science is based on the empirical method for making observations - for systematically obtaining
information. It consists of methods for making observations. Observations are the basic empirical "stuff" of
science. Statistics is a set of methods and rules for organizing, summarizing and interpreting information.
Statistics is a set of concepts, rules, and procedures that help us to:
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organize numerical information in the form of tables, graphs, and charts;
understand statistical techniques underlying decisions that affect our lives and well-being;
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make informed decisions.
Populations and Samples
A population is the set of all individuals of interest in a particular study. We will also refer to
populations of scores. A Sample is a set of individuals selected from a population, usually intended to
represent the population in a study. We will also refer to samples of scores.
Parameters and Statistics
A Parameter is a value, usually a numerical value that describes a Population. A parameter may be
obtained from a single measurement, or it may be derived from a set of measurements from the population.
A Statistic is a value, usually a numerical value that describes a sample. A statistic may be obtained from a
single measurement, or it may be derived from a set of measurements from the sample.
Variable and Constant
A variable is any information that differs from one member to another in a population or sample. A random
variable (designated as X) is one whose numerical value is determined by chance. The key elements here
are that the variable assumes a number (sales volume, rate of return, test score, etc.) and that the sample
selection process generates the numbers randomly, i.e., by a “random” selection.
A constant is an information about the population or sample that is true to all members.
Quantitative and Qualitative Variables
Quantitative variable refers to that which exists in different AMOUNTS. When it is measured, the
scores tell something about the amount or degree of the variable. At the very least, a larger score indicates
more of the variable than a smaller score does.
Examples: age, class size
Qualitative variable is one that exists in different KINDS. A number may be assigned to this
variable but the scores or members are simply used as names or labels (dummy). It does not have
quantitative meaning.
Examples: religion, citizenship
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Discrete and Continuous Variables
A discrete variable is obtained by counting indivisible units. It can take specific values only as it is
always a collection of whole numbers and can never be a part of a unit.
Example: enrolment, class size
Continuous variable is one which comes in units which are divisible into an infinite number of
fractional parts. It can take any point in the number line.
Examples: distance traveled, land area
Independent and Dependent Variables
Independent variable is that which is manipulated by the researcher in a study – the treatment
variable in an experiment. It is the presumed cause of the differences in the dependent variable.
Dependent variable is that which is measured and analyzed in an experiment. Its values are tested
to determine whether they are dependent upon values of the independent variable. It is the presumed effect
of the independent variable
Research Title: Mathematics Achievement of Grade VI Pupils Taught Under Three Methods of Teaching
Dependent Variable: Mathematics Achievement (the variable measured after employing the treatment)
Independent Variable: Methods of Teaching (the variable that is manipulated)
Classification of Scales
1. Nominal scale - the lowest level and primitive type of measurement scale. It permits classification
of individuals into two or more categories. It likewise permits the making of statements of equality or
difference. The basic requirement is to assign an item or individual to one and only one category
and specify the criteria for placing individuals into classes.
Example: sex (you are either a male or female; never both)
2. Ordinal scale – specifies the relative position of items/individuals with respect to a given
characteristic, with no indication as to the distance between the positions. It has the same quality
with a nominal scales, plus the characteristic of greater than or less than. One must be able to
determine whether a n item has more, same or less of the attribute than another item or individual
has.
Example: socio-economic status (you are either poor or rich)
3. Interval scale – permits the making of statements of sameness or difference, greater than or less
than, and the added property that the intervals between items are equal. However, it doe not have a
true zero point. Being zero does not mean absence of something or nothing.
Example: test score (one who scored 4 has twice more of the one who got 2, but one who got 0 does
not mean he knows nothing about the lesson discussed)
4. Ratio scale – permits the making of statements of sameness or difference, greater than or less than,
equal rations between items, and the presence of a TRUE zero point, which means absence of the
attribute being measured
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Example: Distance traveled (The measurement starts from point 0 always)
Introduction to Statistical Methods
The more important notion to be got across at this early stage is how the subject of statistical
methods is organized. This diagram may help:
Descriptive Statistics
Example: "The average income of the 104 families in our company is Php 18,673." In descriptive
statistics, our objective is to describe the properties of a group of scores or data that we have "in
hand," i.e., data that are accessible to us in that we can write them down on paper or type them into
a spreadsheet. In descriptive statistics we are not interested in other data that were not gathered but
might have been; that is the subject of inferential statistics.
What properties of the set of scores are we interested in? At least three: their center, their spread,
and their shape. Consider the following set of scores, which might be ages of persons in your
professional club:
28, 38, 45, 47, 51, 56, 58, 60, 63, 63, 65, 66, 66, 67, 68, 70
We could say of these ages that they range from 28 to 70 (spread), and the middle of them is
somewhere around 60 (center). Now their shape is a property of a graph that can be drawn to depict
the scores. If I marked the scores along a number line, like so
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then we can see that the ages tend to bunch at the older ages and trail off very gradually for the
younger ages. Later we will learn that this distribution of data is said to be negatively skewed,
because the "trailing off" is toward the negative end of the number line.
Inferential Statistics
Example: "This sample of 512 families from Barangay Macopa indicates with 95% confidence we
can conclude that the average family income in the county is between Php5,187 and Php9,328."
In inferential statistics, our interest is in large collections of data that are so large that we can not
have all of them "in hand." We can, however, inspect samples of these larger collections and use
what we see there to make inferences to the larger collection. How samples relate to larger
collections of data (called populations) from which they have been drawn is the subject of inferential
statistical methods. Inferential statistics are frequently used by pollsters who ask 1000 persons
whom they prefer in an election and draw conclusions about how the entire municipality or province
will vote on election day. Scientists and researchers also employ inferential statistics to make
conclusions that are more general than the conclusions they could otherwise draw on the basis of
the limited number of data points they have recorded.
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Worksheet 1
STATISTICAL CONCEPTS
Activity 1 - Classify the following variables according to quantitative/qualitative, discrete/continuous.
Variables
Weight
Average grade
Type of residence
Annual salary
Academic rank
Height
Qn/Ql
Dis/Con
Variables
Burnout level
Highest degree earned
Birth order
Job satisfaction score
Length of service
Attitude score
Qn/Ql
Dis/Con
Variables
LGU annual income
Municipality class
Leadership style
Nutritional status
Palay harvest (bags)
Palay harvest (kilos)
Qn/Ql
Dis/Con
Variables
Number of children
Skin color
Population growth rate
Insurance premium
Bacteria growth (cm)
Genetic traits
Qn/Ql
Dis/Con
Activity 2 - Classify the above variables into scale type (nominal, ordinal, interval, ratio). If you believe a
variable can be classified in more than one scale type, justify.
Activity 3 – Identify the dependent and independent variables in the following research titles:
1. Effect of Training on the Managerial Capabilities of Newly-Elected Barangay Officials
2. Bacteria Inhibition as Influenced by Plant Part and Concentration Levels of Bangbangsit
3. Enhancing Concept Development and Retention Through the Use of Graphic Organizers
ADDITIONAL>>>>>>>>>>
Scales of Measurement
One of the most influential distinctions made in the field of measurement was Stevens' (1946, 1957)
classification of scales of measurement. He described nominal, ordinal, interval, and ratio scales of
measurement, which are briefly defined below. A more detailed discussion of these scales can be
found in Chapter 4 of the text.
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Nominal: Nominal scales are naming scales. They represent categories where there is no
basis for ordering the categories.
Ordinal: Ordinal scales involve categories that can be ordered along a dimension. However,
we have no way of knowing how different the categories are from one another. We state the
latter property by saying that we do not have equal intervals between the items. Rankings
also represent ordinal scales, because we know the order but do not know how different each
person is from the next person.
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Interval: Interval scales are very similar to standard numbering scales, except that they do
not have a true zero. That means that the distance between successive numbers is equal, but
that the number zero does NOT mean that there is none of the property being measured.
Many measures that involve psychological scales, especially those that use a form of normal
standardization (e.g., IQ), are assumed to be interval scales of measurement.
Ratio: Ratio scales are the easiest to understand, because they are numbers as we usually
think of them. The distance between adjacent numbers are equal on a ratio scale and the
score of zero on the ratio scale means that there is none of whatever is being measured. Most
ratio scales are counts of things.
The most important reason for making the distinction between these scales of measurement is that it
affects the statistical procedures that you will use in describing and analyzing your data.
In this unit, we will be presenting dozens of examples of measures at each of these levels of
measurement, along with some exercises to help you to refine your understanding of these
distinctions. We recommend that you complete the exercises since the best way to learn anything is
to actively process the information by using it to solve real-life problems.
Examples of Each Scale of Measurement
Listed below are several examples of each scale of measurement. We have focused on frequently
used psychological variables to help illustrate what each of the scales represent. We have tried to
provide a wide variety of examples to help make these distinctions clear for you.
Nominal Scale Examples
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diagnostic categories
sex of the participant
classification based on discrete characteristics (e.g., hair color)
group affiliation (e.g., Republican, Democrat, Boy Scout, etc.)
the town people live in
a person's name
an arbitrary identification, including identification numbers that are arbitrary
menu items selected
any yes/no distinctions
most forms of classification (species of animals or type of tree)
location of damage in the brain
Ordinal Scale Examples
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any rank ordering
class ranks
social class categories
order of finish in a race
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Interval Scale Examples
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scores on scales that are standardized (i.e., with an arbitrary mean and standard deviation,
usually designed to always give a positive score)
scores on scales that are known to not have a true zero (e.g., most temperature scales except
for the Kelvin Scale)
scores on measures in which it is not clear that zero means none of the trait (e.g., a math
test)
scores on most personality scales based on counting the number of endorsed items
Ratio Scale Examples
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time to complete a task
number of responses given in a specified time period
weight of an object
size of an object
number of objects detected
number of errors made in a specified time period
proportion of responses in a specified category
Exercises
Listed below are a number of exercises designed to familiarize students with the classification of
measures using Stevens' classification system. For each of the measures listed, determine what scale
of measurement most closely approximates the measure as described. Some of the examples are
deliberately ambiguous. To find out the correct answer, click on the word answer at the end of the
description of the item.
1. the number of questions asked by a customer during a simulated encounter with a
salesperson answer RATIO
2. the religious group that one affiliates with answer NOMINAL
3. the time it takes to complete a checking task answer RATIO
4. the score on a 35-item scale of ambivalence answer INTERVAL
5. the rank of a person's salary within the company answer ORDINAL
6. rank order based on IQ score in the sample answer ORDINAL This is a tricky one, because
you are using an IQ measure, which is assumed to produce an interval scale of
measurement, to rank order people. Once you start using the rank orderings, you have
essentially converted an interval measure into an ordinal measure.
7. the square footage of each participant's house or apartment answer RATIO
8. the size of the cerebellum expressed as a volume answer RATIO
9. the number of frustrated comments made during a laboratory negotiation task answer
RATIO
10. the time it takes for a couple to resolve a custody issue during court ordered mediation
answer RATIO
11. score on the Beck Depression Inventory (a pencil and paper depression scale) answer
INTERVAL It is generally assumed that self-report scales represent an interval scale of
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measurement. One could make reasonable arguments for anything from ordinal to ratio
scales of measurement. For example, the argument for a ratio scale is the the self-report
scale represents the number of items endorsed that represent whatever is being measured.
So endorsing 10 items is endorsing twice as many items as endorsing five items. The
argument for ordinal scales is that the underlying dimension (in this example, it is
depression) has equal intervals, but there is no guarantee that the items of the scale tap
depression precisely enough that the intervals between scores on the scale are equal. Faced
with such ambiguity, the norm is to ASSUME that self-report measures are interval scales
unless there is clear evidence that the scale is less than ordinal (i.e., clear evidence that the
difference between scores on different points of the scale are not equal).
12. ratings of anger shown by those involved in courtroom testimony answer
INTERVAL/ORDINAL
13. the number of pound lost during a six-week diet answer RATIO
14. the proportion of weight lost during a six-week diet answer RATIO The proportion of
weight lost is a ratio scale, since zero represents on weight lost and each score represents
an equal amount of weight lost. However, this is a good example of how the measure may
not map well onto other psychological concepts. For example, it is relatively easy to lose 5%
of your body weight, but usually takes more than four times the effort to lose 20% of your
body weight. Furthermore, the amount of effort needed to lose a given proportion of your
body weight will usually depend on your initial weight. Nevertheless, this is still a ratio
scale.
15. the heart rate of the participant answer RATIO
16. the percent shift in heart rate over baseline during an emotionally demanding task answer
RATIO
17. the percent of errors made on a classification task answer RATIO
18. the number of false alarm responses in a monitoring task answer RATIO
19. the types of grammatical errors made in a writing sample answer NOMINAL One could
convert this to an ordinal scale or better by ranking the errors on their severity in distorting
the meaning of the message. However, as stated, this is simply a classification, and thus a
nominal measure.
20. one's ice cream preference answer NOMINAL
21. how quickly a person gives up on an impossible task that looks like it should be possible
answer RATIO
22. a student's SAT score answer INTERVAL Again, the general assumption is that any
standardized test that has been normed with an arbitrary mean and standard deviation
represents an interval scale of measurement. The SAT subtests scores are normed to have an
arbitrary mean of 500 and standard deviation of 100.
23. the percentile rank from an achievement test answer ORDINAL
24. the type of categorization errors in a sorting task answer NOMINAL
25. the pattern of scores on the MMPI personality inventory answer NOMINAL Such patterns
are referred to as two-point or three-point codes, which means that they refer to the two or
three highest scales. These patterns are often associated with diagnostic categories.
26. the age at which one went on his or her first date answer RATIO
27. the number of children in your family answer RATIO
28. the score on an anxiety sensitivity scale answer INTERVAL
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29. whether one has a pet answer NOMINAL This is an interesting variable, because one could
make an argument for it representing a higher level of measurement. For example, if you
coded the number of pets, that would clearly be ratio data. If you used a code of 1 for at
least one pet and 0 for no pets, that would appear to be at least ordinal. With just a yes/no
distinction as presented here, the data are nominal. Psychologists usually prefer to use the
highest level of measurement possible. Therefore, they would probably specify the variable
here differently--as the number of pets, which produces a ratio scale of measurement.
30. the teacher's rankings of cooperativeness in the classroom answer ORDINAL
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