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statistical analysis with software application
Statistical Analysis (University of San Carlos)
Studocu is not sponsored or endorsed by any college or university
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MODULE 1:
DEFINITION OF STATISTICS
INTRODUCTION TO THE
STATISTICAL
CONCEPTS
Statistics plays a major role in many aspects of our
lives. It is used in sports, for example, to help a
general manager decide which player might be the
best fit for a team. It is used in politics to help
candidates understand how the public feels about
various policies. And statistics is used in medicine to
help determine the effectiveness of new drugs. Used
a p p r o p r i a t e l y, s t a t i s t i c s c a n e n h a n c e o u r
understanding of the world around us. Used
inappropriately, it can lend support to inaccurate
beliefs. Understanding statistical methods will
provide you with the ability to analyze and critique
studies and the opportunity to become an informed
consumer of information. Understanding statistical
methods will also enable you to distinguish solid
analysis from bogus “facts.”
Objectives:
After successful completion of this
module, you should be able to:
• Define statistics.
• Enumerate the importance and
limitations of statistics
• Explain the process of statistics
• Know the difference between
descriptive and inferential
statistics.
• Distinguish between qualitative
and quantitative variables.
• Distinguish between discrete and
continuous variables.
• Determine the level of
measurement of a variable.
Many people say that statistics is numbers. After all,
we are bombarded by numbers that supposedly
represent how we feel and who we are. Certainly,
statistics has a lot to do with numbers, but this
definition is only partially correct. Statistics is also
about where the numbers come from (that is, how
they were obtained) and how closely the numbers
reflect reality.
Statistics is the science of collecting, organizing,
summarizing, and analyzing information to draw
conclusions or answer questions. In addition,
statistics is about providing a measure of confidence
in any conclusions.
Let’s break this definition into four parts. The first
part states that statistics involves the collection of
information. The second refers to the organization
and summarization of information. The third
states that the information is analyzed to draw
conclusions or answer specific questions. The
fourth part states that results should be reported
using some measure that represents how
convinced we are that our conclusions reflect
reality.
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• Statistics is important because it enables
people to make decisions based on empirical
evidence.
• Statistics provides us with tools needed to
convert massive data into pertinent
information that can be used in decision
making.
• Statistics can provide us information that we
can use to make sensible decisions.
What information is referred to in the
definition?
The information referred to the definition is the
data. According to the Merriam Webster
dictionary, data are “factual information used
as a basis for reasoning, discussion, or
calculation”.
Data can be numerical, as in height, or
nonnumerical, as in gender. In either case,
data describe characteristics of an individual.
Field of Statistics
A. Mathematical Statistics- The study and
development of statistical theory and methods
in the abstract.
B. Applied Statistics- The application of
statistical methods to solve real problems
involving randomly generated data and the
development of new statistical methodology
motivated by real problems. Example branches
of Applied Statistics: psychometric,
econometrics, and biostatistics.
Limitation of Statistics
Statistics is not suitable to the study of
qualitative phenomenon.
2. Statistics does not study individuals.
4. Statistics table may be misused.
5. Statistics is only, one of the methods of
studying a problem.
Definitions:
• Universe is the set of all entities under
study.
• A Population is the total or entire group of
individuals or observations from which
information is desired by a researcher. Apart
from persons, a population may consist of
mosquitoes, villages, institution, etc.
• An individual is a person or object that is a
member of the population being studied.
• A statistic is a numerical summary of a
sample.
• Sample is the subset of the population.
• Descriptive statistics consist of organizing
and summarizing data. Descriptive statistics
describe data through numerical summaries,
tables, and graphs.
• Inferential statistics uses methods that
take a result from a sample, extend it to the
population, and measure the reliability of the
result.
• A parameter is a numerical summary of a
population
Example: Consider the Scenario.
You are walking down the street and notice
that a person walking in front of you drops
PHP100. Nobody seems to notice the PHP100
except you. Since you could keep the money
without anyone knowing, would you keep the
money or return it to the owner?
3. Statistical laws are not exact.
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Suppose you wanted to use this scenario as a
gauge of the morality of students at your
school by determining the percent of students
who would return the money. How might you
do this? You could attempt to present the
scenario to every student at the school, but
this would be difficult or impossible if the
student body is large. A second possibility is to
present the scenario to 50 students and use
the results to make a statement about all the
students at the school.
account for the variability in our results. One
goal of inferential statistics is to use statistics
to estimate parameters.
In the PHP100 study presented, the population
is all the students at the school. Each student
is an individual. The sample is the 50 students
selected to participate in the study.
2. Collect the information needed to answer
the questions.
Suppose 39 of the 50 students stated that they
would return the money to the owner. We could
present this result by saying that the percent of
students in the survey who would return the
money to the owner is 78%. This is an
example of a descriptive statistic because it
describes the results of the sample without
making any general conclusions about the
population. So 78% is a statistic because it is a
numerical summary based on a sample.
Descriptive statistics make it easier to get an
overview of what the data are telling us.
If we extend the results of our sample to the
population, we are performing inferential
statistics. The generalization contains
uncertainty because a sample cannot tell us
everything about a population. Therefore,
inferential statistics includes a level of
confidence in the results. So rather than saying
that 78% of all students would return the
money, we might say that we are 95%
confident that between 74% and 82% of all
students would return the money. Notice how
this inferential statement includes a level of
confidence (measure of reliability) in our
results. It also includes a range of values to
PROCESS OF STATISTICS
1. Identify the research objective.
A researcher must determine the question(s)
he or she wants answered. The question(s)
must clearly identify the population that is to be
studied. Identify the research objective.
Conducting research on an entire population is
often difficult and expensive, so we typically
look at a sample. This step is vital to the
statistical process, because if the data are not
collected correctly, the conclusions drawn are
meaningless. Do not overlook the importance
of appropriate data collection.
Example:
A research objective is presented. For each
research objective, identify the population and
sample in the study.
1. The Philippine Mental Health Associations
contacts 1,028 teenagers who are 13 to 17
years of age and live in Antipolo City and
asked whether or not they had been
prescribed medications for any mental
disorders, such as depression or anxiety.
Population: Teenagers 13 to 17 years of age
who live in Antipolo City
Sample: 1,028 teenagers 13 to 17 years of
age who live in Antipolo City
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1. A farmer wanted to learn about the weight
of his soybean crop. He randomly sampled
100 plants and weighted the soybeans on
each plant.
Population: Entire soybean crop
Sample: 100 selected soybean crop
3. Organize and summarize the information.
Descriptive statistics allow the researcher to
obtain an overview of the data and can help
determine the type of statistical methods the
researcher should use.
4. Draw conclusion from the information.
In this step the information collected from the
sample is generalized to the population.
Inferential statistics uses methods that takes
results obtained from a sample, extends them
to the population, and measures the reliability
of the result.
Take Note!
If the entire population is studied, then
inferential statistics is not necessary, because
descriptive statistics will provide all the
information that we need regarding the
population.
Example:
For the following statements, decide whether it
belongs to the field of descriptive statistics or
inferential statistics.
1. A badminton player wants to know his
average score for the past 10 games.
(Descriptive Statistics)
2. A car manufacturer wishes to estimate the
average lifetime of batteries by testing a
sample of 50 batteries. (Inferential
Statistics)
3. Janine wants to determine the variability of
her six exam scores in Algebra.
(Descriptive Statistics)
4. A shipping company wishes to estimate the
number of passengers traveling via their
ships next year using their data on the
number of passengers in the past three
years. (Inferential Statistics)
5. A politician wants to determine the total
number of votes his rival obtained in the
past election based on his copies of the
tally sheet of electoral returns.
(Descriptive Statistics)
DISTINCTION BETWEEN QUALITATIVE AND
QUANTITATIVE VARIABLES
Variables are the characteristics of the
individuals within the population. For example,
recently my mother and I planted a tomato
plant in our backyard. We collected information
about the tomatoes harvested from the plant.
The individuals we studied were the tomatoes.
The variable that interested us was the weight
of a tomato.My mom noted that the tomatoes
had different weights even though they came
from the same plant. She discovered that
variables such as weight may vary.
If variables did not vary, they would be
constants, and statistical inference would
not be necessary. Think about it this way: If
each tomato had the same weight, then
knowing the weight of one tomato would allow
us to determine the weights of all tomatoes.
However, the weights of the tomatoes vary.
One goal of research is to learn the causes of
the variability so that we can learn to grow
plants that yield the best tomatoes.
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It is helpful to divide variables into different
types, as different statistical methods are
applicable to each. The main division is into
qualitative (or categorical) or quantitative (or
numerical variables).
Variables can be classified into two groups:
1. Qualitative variables (Categorical) is
variable that yields categorical responses.
It is a word or a code that represents a
class or category.
2. Quantitative variables (Numeric) takes
on numerical values representing an
amount or quantity.
possible values. If you count to get the
value of a quantitative variable, it is
discrete.
2. A continuous variable is a quantitative
variable that has an infinite number of
possible values that are not countable. If
you measure to get the value of a
quantitative variable, it is continuous.
Example:
Determine whether the following quantitative
variables are discrete or continuous.
1. The number of heads obtained after
flipping a coin five times. (Discrete)
Example:
Determine whether the following variables are
qualitative or quantitative.
1. Haircolor (Qualitative)
2. Temperature (Quantitative)
3. Stages of breast cancer (Qualitative)
4. Number of hamburger sold (Quantitative)
5. Number of children (Quantitative)
6. Zip code (Qualitative)
2. The number of cars that arrive at a
McDonald’s drive-through between 12:00
P.M and 1:00 P.M. (Discrete)
3. The distance of a 2005 Toyota Prius can
travel in city conditions with a full tank of
gas. (Continuous)
4. Number of words correctly spelled.
(Discrete)
5. Time of a runner to finish one lap.
(Continuous)
LEVELS OF MEASUREMENT
7. Place of birth (Qualitative)
8. Degree of pain (Qualitative)
DISTINCTION BETWEEN DISCRETE AND
CONTINUOUS
Quantitative variables may be further classified
into:
1. A discrete variable is a quantitative
variable that either a finite number of
possible values or a countable number of
Levels of Measurement
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It is important to know which type of scale is
represented by your data since different
statistics are appropriate for different scales of
measurement. A characteristic may be
measured using nominal, ordinal, interval and
ration scales.
1. Nominal Level - They are sometimes
called categorical scales or categorical
data. Such a scale classifies persons or
objects into two or more categories.
Whatever the basis for classification, a
person can only be in one category, and
members of a given category have a
common set of characteristics.
3. Interval Level - This is a measurement level
not only classifies and orders the
measurements, but it also specifies that the
distances between each interval on the scale
are equivalent along the scale from low interval
to high interval. A value of zero does not mean
the absence of the quantity. Arithmetic
operations such as addition and subtraction
can be performed on values of the variable.
Example:
- Te m p e r a t u r e o n F a h r e n h e i t / C e l s i u s
Thermometer
- Trait anxiety (e.g., high anxious vs. low
anxious)
Example:
- IQ (e.g., high IQ vs. average IQ vs. low IQ)
- Method of payment (cash, check, debit card,
credit card)
- Type of school (public vs. private)
- Eye Color (Blue, Green, Brown)
2. Ordinal Level - This involves data that may
be arranged in some order, but differences
between data values either cannot be
determined or meaningless. An ordinal scale
not only classifies subjects but also ranks them
in terms of the degree to which they possess a
characteristics of interest. In other words, an
ordinal scale puts the subjects in order from
highest to lowest, from most to least. Although
ordinal scales indicate that some subjects are
higher, or lower than others, they do not
indicate how much higher or how much better.
4. Ratio Level - A ratio scale represents the
highest, most precise, level of measurement. It
has the properties of the interval level of
measurement and the ratios of the values of
the variable have meaning. A value of zero
means the absence of the quantity. Arithmetic
operations such as multiplication and division
can be performed on the values of the
variable.
Example:
- Height and weight
- Time
- Time until death
Operations that make sense for variables of
different scales.
Example:
- Food Preferences
- Stage of Disease
- Social Economic Class (First, Middle, Lower)
- Severity of Pain
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Both interval and ratio data involve
measurement. Most data analysis techniques
that apply to ratio data also apply to interval
data..Therefore, in most practical aspects,
these types of data (interval and ratio) are
grouped under metric data. In some other
instances, these type of data are also known
as numerical discrete and numerical
continuous.
2. Every year the PSA releases the Current
Population Report based on a survey of
50,000 households. The goal of this report
is to learn the demographic characteristics,
such as income, of all households within
the Philippines.
A. ______________________________
Example:
Categorize each of the following as nominal,
ordinal, interval or ratio measurement.
1. Ranking of college athletic teams.
(Ordinal)
2. Employee number. (Nominal)
3. Number of vehicles registered. (Ratio)
4. Brands of soft drinks. (Nominal)
5. Number of car passers along C5 on a
given day. (Ratio)
B. ______________________________
3. Researchers want to determine whether or
not higher folate intake is associated with a
lower risk of hypertension (high blood
pressure) in women (27 to 44 years of
age). To make this determination, they look
at 7373 cases of hypertension in these
women and find that those who consume
at least 1000 micrograms per day of total
folate had a decreased risk of hypertension
compared with those who consume less
than 200.
A. ______________________________
6. Zip code (Nominal)
7. Degree of pain (Ordinal)
ACTIVITIES/ASSESSMENTS:
Read each item carefully. Write the answer
on the yellow paper. Answers Only.
I.
B. ______________________________
A research objective is presented. For
each, identify the (A) population and (B)
sample in the study.
8. A polling organization contacts 2141 male
university graduates who have a whitecollar job and asks whether or not they had
received a raise at work during the past 4
months.
A. ______________________________
B. ______________________________
II. Indicate whether the following statements
require the use of descriptive or inferential
statistics.
______________1. A teacher wants to know
the attitudes of all students towards abortion.
______________2. A market analyst of a sales
firm draws a chart showing the sales figures of
a given product for the period 2006-2007.
______________3. A forecaster predicts the
results of an election using the number of
votes cast in 15 out of 25 barangays.
______________4. Men are better in math
than women.
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_____________5. Forty percent of the
employees of an organization were recorded
tardy for at least 15 working days.
______________10. Brands of soft drinks
______________6. There are very few
gender-related occupations.
______________12. Status Employment
____________ 7. An account predicts
accuracy rate of a client’s financial resources.
______________ 8. A quality control manager
wishes to check production output.
______________ 9. Records indicated that
75% of the faculty in the graduate school are
doctoral degree holders.
______________ 10. There is no relationship
between educational qualification of parents
and academic achievement of their children.
______________11. Socioeconomic status
______________13. Number of missing teeth
______________14. Number of vehicles
registered
______________15. Jersey Number
______________16. Number of employees
collecting retirement
benefits from GSIS
______________17. Duration of a seizure
______________18. Cause of death
______________19. Dividends
III. Identify the qualitative and quantitative
variables and indicate the highest level of
measurement required in each. If
quantitative, classify whether discrete or
continuous.
______________1. Occupation
______________20. Current assets list
______________21. Number of heart attacks
______________22. Account receivable
______________23. Clothing size
______________2. Number of government
officials
______________24. Blood type
______________3. Favorite color
______________25. Ethnic group
______________4. Temperature in Celsius
degrees
REFERENCES:
______________5. Type of school
Statistics. Informed Decision using Data by
Michael Sullivan, III,. Fifth Edition
______________6. Volume of mineral water
sold daily
Sampling: Design and Analysis by Sharon L.
Lhr. Second Edition
______________7. Employee number
______________8. Civil status
______________9. Equity accounts
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MODULE 2:
DATA COLLECTION
DATA COLLECTION
AND BASIC Concepts
in Sampling DESIGN
Everybody collects, interprets and uses information,
much of it in numerical or statistical forms in day-today life. It is a common practice that people receive
large quantities of information everyday through
conversations, televisions, computers, the radios,
newspapers, posters, notices and instructions. It is
just because there is so much information available
that people need to be able to absorb, select and
reject it. In everyday life, in business and industry,
certain statistical information is necessary and it is
independent to know where to find it how to collect it.
Objectives:
After successful completion of this
module, you should be able to:
• Determine the sources of data
(primary and secondary data).
• Distinguish the different methods
data collection under primary and
secondary data.
• Determine the appropriate
sample size.
• Differentiate various sampling
techniques.
• Know the sources of errors in
sampling.
Analysis of data can lead to powerful results. Data
can be used to offset anecdotal claims, such as the
suggestion that cellular telephones cause brain
cancer. Anecdotal means that the information being
conveyed is based on casual observation, not
scientific research. Because data are powerful, they
can be dangerous when misused. The misuse of
data usually occurs when data are incorrectly
obtained or analyzed. For example, radio or
television talk shows regularly ask poll questions for
which respondents must call in or use the Internet to
supply their vote. Most likely, the individuals who are
going to call in are those who have a strong opinion
about the topic. This group is not likely to be
representative of people in general, so the results of
the poll are not meaningful. Whenever we look at
data, we should be mindful of where the data come
from.
Even when data tell us that a relation exists, we
need to investigate. For example, a study showed
that breast-fed children have higher IQs than those
who were not breast-fed. Does this study mean that
a mother who breast-feeds her child will increase the
child’s IQ? Not necessarily. It may be that some
factor other than breast-feeding contributes to the IQ
of the children. In this case, it turns out that mothers
who breastfeed generally have higher IQs than
those who do not. Therefore, it may be genetics that
leads to the higher IQ, not breast-feeding.
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Data collection is the process of gathering
and measuring information on variables of
interest, in an established systematic fashion
that enables one to answer stated research
questions, test hypotheses, and evaluate
outcomes.
Without proper planning for data collection, a
number of problems can occur. If the data
collection steps and processes are not
properly planned, the research project can
ultimately end up with a data set that does not
serve the purpose for which it was intended.
For example, if more than one person is
involved in the data collection, but data
collectors do not follow consistent data
collection practices, they can end up with data
with different units, collection processes, and
variable names.
Consequences from Improperly Collected
Data
• Inability to answer research questions
accurately.
• Inability to repeat and validate the study.
• Distorted findings resulting in wasted
resources.
• Misleading other researchers to pursue
fruitless avenues of investigation.
• Compromising decisions for public policy.
• Causing harm to human participants and
animal subjects.
Steps in Data Gathering
1. Set the objectives for collecting data
2. Determine the data needed based on the
set objectives.
3. Determine the method to be used in data
gathering and define the comprehensive
data collection points.
4. Design data gathering forms to be used.
5. Collect data.
Choosing of Method of Data Collection
Decision-makers need information that is
relevant, timely, accurate and usable. The cost
of obtaining, processing and analyzing these
data is high. The challenge is to find ways,
which lead to information that is cost-effective,
relevant, timely and important for immediate
use. Some methods pay attention to timeliness
and reduction in cost. Others pay attention to
accuracy and the strength of the method in
using scientific.
The statistical data may be classified under
two categories, depending upon the sources.
approaches: Primary Data and Secondary
Data.
SOURCES OF DATA
Whether conducting research in the social
sciences, humanities arts, or natural sciences,
the ability to distinguish between primary and
secondary sources is essential.
Primary Sources - Provide a first-hand
account of an event or time period and are
considered to be authoritative. They
represent original thinking, reports on
discoveries or events, or they can share new
information. Often these sources are created
at the time the events occurred but they can
also include sources that are created later.
They are usually the first formal appearance
of original research.
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Primary Data - are data documented by the
primary source. The data collectors
documented the data themselves.
The first hand information obtained by the
investigator is more reliable and accurate since
the investigator can extract the correct
information by removing doubts, if any, in the
minds of the respondents regarding certain
questions. High response rates might be
obtained since the answers to various
questions are obtained on the spot. It permits
explanation of questions concerning difficult
subject matter.
Secondary Sources - offer an analysis,
interpretation or a restatement of primary
sources and are considered to be
persuasive. They often involve
generalisation, synthesis, interpretation,
commentary or evaluation in an attempt to
convince the reader of the creator's
argument. They often attempt to describe or
explain primary sources.
Secondary Data - are data documented by a
secondary source. The data collectors had the
data documented by other sources.
In secondary data, data are primary data for
the agency that collected them, and become
secondary for someone else who uses these
data for his own purposes.
Secondary data are less expensive to collect
both in money and time. These data can also
be better utilized and sometimes the quality of
such data may be better because these might
have been collected by persons who were
specially trained for that purpose.
On the other hand, such data must be used
with great care, because such data may also
be full of errors due to the fact that the purpose
of the collection of the data by the primary
agency may have been different from the
purpose of the user of these secondary data.
Secondly, there may have been bias
introduced, the size of the sample may have
been inadequate, or there may have been
arithmetic or definition errors, hence, it is
necessary to critically investigate the validity of
the secondary data.
The primary data can be collected by the
following five methods:
1. Direct personal interviews - The
researcher has direct contact with the
interviewee. The researcher gathers
information by asking questions to the
interviewee.
2. Indirect/Questionnaire Method - This
methods of data collection involve sourcing
and accessing existing data that were
originally collected for the purpose of the study.
Designing good “questioning tools” forms an
important and time consuming phase in the
development of most research proposals.
Once the decision has been made to use
these techniques, the following questions
should be considered before designing our
tools:
• What exactly do we want to know, according
to the objectives and variables we identified
earlier? Is questioning the right technique to
obtain all answers, or do we need additional
techniques, such as observations or
analysis of records?
• Of whom will we ask questions and what
techniques will we use? Do we understand
the topic sufficiently to design a
questionnaire, or do we need some loosely
structured interviews with key informants or
a focus group discussion first to orient
ourselves?
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• Are our informants mainly literate or
illiterate? If illiterate, the use of selfadministered questionnaires is not an
option.
• How large is the sample that will be
interviewed? Studies with many respondents
often use shorter, highly structured
questionnaires, whereas smaller studies
allow more flexibility and may use
questionnaires with a number of open-ended
questions.
Key Design Principles of a Good
Questionnaire
1. Keep the questionnaire as short as possible.
Example:
- Can you describe exactly what the
traditional birth attendant did when your
labor started?
- What do you think are the reasons for a high
drop-out rate of village health committee
members?
A closed-ended question is a type of
question that includes a list of response
categories from which the respondent will
select his answer. It is useful if the range of
possible responses is known. This type of
question is usually appropriate for collecting
objective data.
2. Decide on the type of questionnaire (Open
Ended or Closed Ended).
Example:
3. Write the questions properly.
Did you eat any of the following foods
yesterday?
4. Order the questions appropriately.
5. Avoid questions that prompt or motivate the
respondent to say what you would like to hear.
• Fish or meat
Yes
No
• Eggs.
Yes
No
• Milk or cheese
Yes
No
6. Write an introductory letter or an
introduction.
Take Note!
7. Write special instructions for interviewers or
respondents.
Question wording and question order have a
large effect on the responses obtained.
8. Translate the questions if necessary.
Example:
9. Always test your questions before taking the
survey. (Pre-test)
Two surveys were taken in late 1993/early
1994 about Elvis Presley.
An open-ended question is a type of question
that does not include response categories. The
respondent is not given any possible answers
to choose from. This type of question is usually
appropriate for collecting subjective data. It
permit free responses that should be recorded
in the respondent’s own words.
One survey asked: “In the past few years,
there have been a lot of rumors and stories
about whether Elvis Presley is really dead.
How do you feel about this? Do you think there
is any possibility that these rumors are true
and that Elvis Presley is still alive, or don’t you
think so?”
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Second survey asked: “A recent television
show examined various theories about Elvis
Presley’s death. Do you think it is possible that
Elvis is alive or not?”
8% of the respondents to the first question said
it is possible that Elvis is still alive and 16% of
respondents to the second question said it is
possible that Elvis is still alive.
3. A focus group is a group interview of
approximately six to twelve people who share
similar characteristics or common interests. A
facilitator guides the group based on a
predetermined set of topics.
- Unrealistic Controlled Environments
- Inability to Control for All Variables
5. Observation is a technique that involves
systematically selecting, watching and
recoding behaviors of people or other
phenomena and aspects of the setting in which
they occur, for the purpose of getting (gaining)
specified information. It includes all methods
from simple visual observations to the use of
high level machines and measurements,
sophisticated equipment or facilities such as:
- Radiographic
- biochemical
4. Experiment is a method of collecting data
where there is direct human intervention on the
conditions that may affect the values of the
variable of interest.
- X-ray machines
- Microscope
- Clinical examinations
Bear in mind that the experimental method has
several limitations that you should be aware of.
- Microbiological examinations
- Ethical, moral, and legal Concerns
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It gives relatively more accurate data on
behavior and activities but Investigators or
observer’s own biases, prejudice, desires, and
etc. and needs more resources and skilled
human power during the use of high level
machines.
size can produce accuracy of results.
Moreover, the results from the small sample
size will be questionable. A sample size that is
too large will result in wasting money and time
because enough sample will normally give an
accurate result.
The secondary data can be collected by the
following five methods:
The sample size is typically denoted by n and
it is always a positive integer. No exact sample
size can be mentioned here and it can vary in
different research settings. However, all else
being equal, large sized sample leads to
increased precision in estimates of various
properties of the population.
1. Published report on newspaper and
periodicals.
2. Financial Data reported in annual reports.
3. Records maintained by the institution.
Take Note!
4. Internal reports of the government
departments.
- Representativeness, not size, is the more
5. Information from official publications.
- Use no less than 30 subjects if possible.
Take Note!
- If you use complex statistics, you may need
• Always investigate the validity and reliability
of the data by examining the collection
method employed by your source.
important consideration.
a minimum of 100 or more in your sample
(varies with method).
• Do not use inappropriate data for your
research.
• The choice of methods of data collection is
largely based on the accuracy of the
information they yield.
SAMPLE SIZE
“How many participants should be chosen for a
survey”?
One of the most frequent problems in
statistical analysis is the determination of the
appropriate sample size. One may ask why
sample size is so important. The answer to this
is that an appropriate sample size is required
for validity. If the sample size it too small, it will
not yield valid results. An appropriate sample
Representative Sample
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Desired Confidence
Level
80%
85%
90%
95%
99%
Z - Score
1.28
1.44
1.65
1.96
2.58
3. Degree of Variability
Choosing of sample size depends on nonstatistical considerations and statistical
considerations.
• Non-statistical considerations – It may
include availability of resources, man power,
budget, ethics and sampling frame.
• Statistical considerations – It will include
the desired precision of the estimate.
Depending upon the target population and
attributes under consideration, the degree of
variability varies considerably. The more
heterogeneous a population is, the larger the
sample size is required to get an optimum level
of precision.
Methods in Determining the Sample Size
• Estimating the Mean or Average
The sample size required to estimate the
population mean µ to with a level of confidence
with specified margin of error e, given by
Three criteria need to be specified to
determine the appropriate sample size:
2
Zσ
n≥
( e )
1. Level of Precision
Also called sampling error, the level of
precision, is the range in which the true value
of the population is estimated to be.
where:
Z is the z-score corresponding to level of
confidence.
2. Confidence Interval
e is the level of precision.
It is statistical measure of the number of times
out of 100 that results can be expected to be
within a specified range. For example, a
confidence interval of 90% means that results
of an action will probably meet expectations
90% of the time.
Take Note:
To find the right z – score to use, refer to the
table:
If When σ is unknown, it is common practice to
conduct a preliminary survey to determine s
and use it as an estimate of σ or use results
from previous studies to obtain an estimate of
σ. When using this approach, the size of the
sample should be at least 30. The formula for
the sample standard deviation s is
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s=
which we know only after we have taken the
sample.
∑ (x − x̄)2
n−1
There are two ways to solve this dilemma:
Example:
A soft drink machine is regulated so that the
amount of drink dispensed is approximately
normally distributed with a standard deviation
equal to 0.5 ounce. Determine the sample size
needed if we wish to be 95% confident that our
sample mean will be within 0.03 ounce from
the true mean.
Solution: The z – score for confidence level
95% in the z – table is 1.96.
Example:
If last month 37% of all voters thought that
state taxes are too high, then it is likely that the
proportion with that opinion this month will not
be dramatically different, and we would use the
value 0.37 for p in the formula.
2. Simply to replace p in the formula by 0.5.
2
n≥
1. We could determine a preliminary value for
p based on a pilot study or an earlier study.
1.96(0.5)
= 1067.11
( 0.03 )
We need a 1068 sample for our study.
• Estimating Proportion (Infinite
Population)
When p = 0.5, the maximum value of
p(1- p)=0.25. This is called the most
conservative estimate, since it gives the
largest possible estimate of n.
The conservative formula using the strong law
of large number.
The sample size required to obtain a
confidence interval for p with specified margin
of error e is given by
2
Z
n≥
p(1 − p)
(e)
2
1 Z
≈ 385
n≥
4 (e)
Where:
Confidence level is 95%.
Where:
The level of precision is 0.05.
Z is the z-score corresponding to level of
confidence.
Example:
e is the level of precision.
P is population proportion.
There is a dilemma in this formula:
It dependents on
p=
x
N
Suppose we are doing a study on the
inhabitants of a large town, and want to find
out how many households serve breakfast in
the mornings. We don’t have much information
on the subject to begin with, so we’re going to
assume that half of the families serve
breakfast: this gives us maximum variability.
So p = 0.5. We want 99% confidence and at
least 1% precision.
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Solution: The z – score for confidence level
99% in the z – table is 2.58.
Where:
no is Cochran’s sample size recommendation.
2
2.58
n≥
0.5(1 − 0.5) = 16,641
( 0.01 )
We need a 16,641 sample for our study.
• Slovin’s Formula
This is the link for online calculator of sample
size:
https://select-statistics.co.uk/calculators/
sample-size-calculator-population-proportion/
Slovin’s formula is used to calculate the
sample size n given the population size and
error. It is computed as
n≥
N is the population size.
https://www.calculator.net/sample-sizecalculator.html
N
1 + Ne 2
Where:
N is the total population.
e is the level of precision.
Example:
A researcher plans to conduct a survey about
food preference of BS Stat students. If the
population of students is 1000, find the sample
size if the error is 5%.
Solution:
n≥
1000
= 285.71
1 + 1000(0.05)2
The researcher need to survey 286 BS stat
students.
• Finite Population Correction
If the population is small then the sample size
can be reduced slightly
n≥
n0
n −1
1+ o
N
BASIC SAMPLING DESIGN
The goal in sampling is to obtain individuals for
a study in such a way that accurate information
about the population can be obtained.
Reason for Sampling
- Important that the individuals included in a
sample represent a cross section of
individuals in the population.
- If sample is not representative it is biased.
You cannot generalize to the population from
your statistical data.
Some definitions are needed to make the
notion of a good sample more precise.
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- Deliberately or purposively selecting a
Definitions:
• Observation unit - An object on which a
measurement is taken. This is the basic unit
of observation, sometimes called an element.
In studying human populations, observation
units are often individuals.
• Target population - The complete collection
of observations we want to study.
• Sampled population - The collection of all
possible observation units that might have
been chosen in a sample; the population
from which the sample was taken.
• Sample - A subset of a population.
• Sampling unit - A unit that can be selected
for a sample.
We may want to study
individuals, but do not have a list of all
individuals in the target population. Instead,
households serve as the sampling units, and
the observation units are the individuals
living in the households.
“representative” sample.
Misspecifying the target population.
Failing to include all of the target population
in the sampling frame, called
undercoverage.
Including population units in the sampling
frame that are not in the target population,
called overcoverage.
- Having multiplicity of listings in the sampling
frame.
Substituting a convenient member of a
population for a designated member who is
not readily available.
- Failing to obtain responses from all of the
chosen sample. (Nonresponse)
- Allowing the sample to consist entirely of
volunteers.
Advantage of Sampling Over Complete
Enumeration
- Less Labor
• Sampling frame - A list, map, or other
specification of sampling units in the
population from which a sample may be
selected. For a survey using in-person
interviews, the sampling frame might be a list
of all street addresses.
• Sampling technique/Sampling Strategies It is a plan you set forth to be sure that the
sample you use in your research study
represents the population from which you
drew your sample.
• Sampling Bias - This involves problems in
your sampling, which reveals that your
sample is not representative of your
population.
- Reduced Cost
- Greater Speed
- Greater Scope
- Greater Efficiency and Accuracy
- Convenience
- Ethical Considerations
Two Type of Samples
1. Probability Sample
- Samples are obtained using some objective
chance mechanism, thus involving
randomization.
The following examples indicate some ways in
which selection bias can occur:
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- They require the use of a complete listing of
the elements of the universe called the
sampling frame.
- The probabilities of selection are known.
- They are generally referred to as random
- Most basic method of drawing a probability
sample.
- Assigns equal probabilities of selection to
each possible sample.
- Results to a simple random sample.
samples.
Advantage: It is very simple and easy to use.
- They allow drawing of valid generalizations
about the universe/population.
Disadvantage: The sample chosen may be
distributed over a wide geographic area.
2. Non - probability Sample
- Samples are obtained haphazardly, selected
purposively or are taken as volunteers.
- The probabilities of selection are unknown.
When to use: This is preferable to use if the
population is not widely spread geographically.
Also, this is more appropriate to use if the
population is more or less homogenous with
respect to the characteristics of the population.
- They should not be used for statistical
inference.
Sampling Procedure
- Identify the population.
- Determine if population is accessible.
- Select a sampling method.
- Choose a sample that is representative of
the population.
- Ask the question, can I generalize to the
Simple Random Sampling
general population from the accessible
population?
Sampling technique can be grouped into how
selections of items are made such as
probability sampling and non-probability
sampling.
Basic Sampling Technique of Probability
Sampling
• Systematic Random Sampling
- It is obtained by selecting every kth
individual from the population.
- The first individual selected corresponds to a
random number between 1 to k.
• Simple Random Sampling
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Obtaining a Systematic Random Sample
1. Decide on a method of assigning a unique
serial number, from 1 to N, to each one of
the elements in the population.
When to use: This is advisable to us if the
ordering of the population is essentially
random and when stratification with numerous
data is used.
2. Compute for the sampling interval
k=
N PopulationSize
=
n
SampleSize
3. Select a number, from 1 to k, using a
randomization mechanism. The element in
the population assigned to this number is
the first element of the sample. The other
elements of the sample are those assigned
to the numbers and so on until you get a
sample of size.
Example:
Systematic Random Sampling
• Stratified Random Sampling
We want to select a sample of 50 students
from 500 students under this method kth item
and picked up from the sampling frame.
Solution:
k=
500
= 10
50
- It is obtained by separating the population
into non-overlapping groups called strata
and then obtaining a simple random sample
from each stratum.
- The individuals within each stratum should
be homogeneous (or similar) in some way.
We start to get a sample starting form i and for
every kth unit subsequently. Suppose the
random number i is 6, then we select 15, 25,
35, 45, .. .
Advantage: Drawing of the sample is easy. It
is easy to administer in the field, and the
sample is spread evenly over the population.
Example:
A sample of 50 students is to be drawn from a
population consisting of 500 students
belonging to two institutions A and B. The
number of students in the institution A is 200
and the institution B is 300. How will you draw
the sample using proportional allocation?
Disadvantage: May give poor precision when
unsuspected periodicity is present in
the
population.
When to use: This is advisable to us if the
ordering of the population is essentially
random and when stratification with numerous
data is used.
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Solution:
There are two strata in this case.
Given:
N1 = 200
N2 = 300
N = 500
n = 50
n1 =
n
50
N1 =
200 = 20
(N)
( 500 )
n2 =
n
50
N2 =
300 = 30
(N)
( 500 )
The sample sizes are 20 from A and 30 from
B. Then the units from each institution are to
be selected by simple random sampling.
Advantage: Stratification of respondents is
advantageous in terms of precision of the
estimates of the characteristics of the
population. Sampling designs may vary by
stratum to adjust for the differences in the
conditions across strata. It is easy to use as a
random sampling design.
Disadvantage: Values of the stratification
variable may not be easily available for all
units in the population especially if the
characteristic of interest is homogeneous. It is
possible that there are not representative in
one or two strata. Also, transportation costs
can be high if the population covers a wide
geographic area.
When to use: If the population is such that the
distribution of the characteristics of the
respondents under consideration concentrated
in small and spread segment of the population.
Thus, this is preferred to use if precise
estimates are desired for stratified parts of the
population and if sampling problems differ in
the various strata of the population.
Stratified Random Sampling
• Cluster Sampling
- You take the sample from naturally occurring
groups in your population.
- The clusters are constructed such that the
sampling units are heterogeneous within the
cluster and homogeneous among the
clusters.
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Obtaining a Cluster Sample
1. Divide the population into non-overlapping
clusters.
2. Number the clusters in the population from 1
to N.
When to use: If the population can be
grouped into clusters where individual
population elements are known to be different
with respect to the characteristics under study,
this preferable to use.
3. Select n distinct numbers from 1 to N using
a randomization mechanism. The selected
clusters are the clusters associated with the
selected numbers.
4. The sample will consist of all the elements in
the selected clusters.
Example:
A researcher wants to survey academic
performance of high school students in
MIMAROPA.
1. He/She can divide the entire population into
different clusters.
2. Then the researcher selects a number of
clusters depending on his research through
simple or systematic random sampling.
3. Then, from the selected clusters the
researcher can either include all the high
school students as subject or he can select a
number of subjects from each cluster through
simple or systematic random sampling.
Cluster Sampling
• Multi - Stage Sampling
- Selection of the sample is done in two or
more steps or stages, with sampling units
varying in each stage.
- The population is first divided into a number
of first-stage sampling units from which a
sample is drawn. Smaller units, called the
secondary sampling units, comprising the
selected first-stage units then serve as the
sampling units for the next stage. If needed
additional stages may be added until the
units of observation for the survey are
clearly identified. The units comprising the
samples selected from the previous stage
constitute the frame for the stages.
Advantage: There is no need to come out with
a list of units in the population; all what is
needed is simply a list of the clusters. It is also
less costly since the elements are physically
closer together.
Obtaining a Multi-Stage Sampling
Disadvantage: In actual field applications,
adjacent households tend to have more similar
characteristics than households distantly apart.
1. Organize the sampling process into stages
where the unit of analysis is systematically
grouped.
2. Select a sampling technique for each
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3. Systematically apply the sampling
technique to each stage until the unit of
analysis has been selected.
Example:
Suppose we wish to study the expenditure
patterns of households in NCR. We can select
a sample of households for this study using
simple three-stage sampling.
- First, divide into smaller cities/municipalities
and a random sample of these cities/
municipalities is collected.
Multi-Stage Sampling
- Second, a random sample of smaller areas
such as barangays is taken from within each
of the cities/municipalities chosen in the first
stage.
Basic Sampling Technique of NonProbability Sampling
- Third, a random sample of even smaller
• Accidental Sampling - There is no system
areas such as households is taken from
within each of the areas chosen in the
second stage.
Advantage: It is easier to generate adequate
sampling frames. Transportation costs are
greatly reduced since there is some form of
clustering among the ultimate or final samples;
i.e., they are in the sample lower-stage units.
Disadvantage: Its complexity in theory may be
difficult to apply in the field. Estimation
procedures may be difficult for non-statisticians
to follow.
When to use: If no population list is available
and if the population covers a wide area.
Take Note!
Used probability sampling if the main objective
of the sample survey is making inferences
about the characteristics of the population
under study.
of selection but only those whom the
researcher or interviewer meets by chance.
• Quota Sampling - There is specified
number of persons of certain types is
included in the sample. The researcher is
aware of categories within the population
and draws samples from each category. The
size of each categorical sample is
proportional to the proportion of the
population that belongs in that category.
• Convenience Sampling - It is a process of
picking out people in the most convenient
and fastest way to get reactions
immediately. This method can be done by
telephone interview to get the immediate
reactions of a certain group of sample for a
certain issue.
• Purposive Sampling - It is based on certain
criteria laid down by the researcher. People
who satisfy the criteria are interviewed. It is
used to determine the target population of
those who will be taken for the study.
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• Judgement Sampling - selects sample in
ACTIVITIES/ASSESSMENTS:
accordance with an expert’s judgment.
I.
Cases wherein Non-Probability Sampling is
Useful
Determine if the source would be a primary
or a secondary source.
______________1. Government Records
- Only few are willing to be interviewed
______________2. Dictionary
- Extreme difficulties in locating or identifying
______________3. Artifact
subjects
- Probability sampling is more expensive to
implement
- Cannot enumerate the population elements.
______________4. A TV show explaining what
happened in Philippines.
______________5. Autobiography about
Rodrigo Duterte.
Sources of Errors in Sampling
1. Non-sampling Error
- Errors that result from the survey process.
- Any errors that cannot be attributed to the
______________6. Enrile diary describing
what he thought about the
world war II.
______________7. Audio and video
recordings
sample-to-sample variability.
______________8. Speeches
Sources of Non-Sampling Error
______________9. Newspaper
1. Non-responses
______________10. Review Articles
2. Interviewer Error
II. Determine the sample size of the following
problems. Show your solution.
3. Misrepresented Answers
4. Data entry errors
1. A dermatologist wishes to estimate the
proportion of young adults who apply
sunscreen regularly before going out in the
sun in the summer. Find the minimum
sample size required to estimate the
proportion with precision of 3%, and 90%
confidence.
5. Questionnaire Design
6. Wording of Questions
7. Selection Bias
2. Sampling Error
- Error that results from taking one sample
instead of examining the whole population.
- Error that results from using sampling to
estimate information regarding a population.
2. The administration at a college wishes to
estimate, the proportion of all its entering
freshmen who graduate within four years,
with 95% confidence. Estimate the
minimum size sample required. Assume
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1. that the population standard deviation is σ
= 1.3 and precision level is 0.05.
completed and returned at the end of the
program.
2. A government agency wishes to estimate
the proportion of drivers aged 16–24 who
have been involved in a traffic accident in
the last year. It wishes to make the
estimate to within 1% error and at 90%
confidence. Find the minimum sample size
required, using the information that several
years ago the proportion was 0.12.
______________4. 24 Hour Fitness wants to
administer a satisfaction survey to its current
members. Using its membership roster, the
club randomly selects 40 club members and
asks them about their level of satisfaction with
the club.
3. An internet service provider wishes to
estimate, to within one percentage error,
the current proportion of all email that is
spam, with 85% confidence. Last year the
proportion that was spam was 71%.
Estimate the minimum size sample
required if the total email that is spam is
10,000.
III. Determine the type of sampling. (ex.
Simple Random Sampling, Purposive
Sampling)
______________1. To determine customer
opinion of its boarding policy, Southwest
Airlines randomly selects 60 flights during a
certain week and surveys all passengers on
the flights.
______________2. A member of Congress
wishes to determine her constituency’s opinion
regarding estate taxes. She divides her
constituency into three income classes: lowincome households, middle-income
households, and upper-income households.
She then takes a simple random sample of
households from each income class.
______________3. The presider of a guestlecture series at a university stands outside the
auditorium before a lecture begins and hands
every fifth person who arrives, beginning with
the third, a speaker evaluation survey to be
______________5. A radio station asks its
listeners to call in their opinion regarding the
use of U.S. forces in peacekeeping missions.
______________6. A tax auditor selects every
1000th income tax return that is received.
______________7. For a survey, a sample of
municipalities was selected from every
province in the country and included all child
laborers in the selected municipalities.
______________8. To determine his DSL
Internet connection speed, Shawn divides up
the day into four parts: morning, midday,
evening, and late night. He then measures his
Internet connection speed at 5 randomly
selected times during each part of the day.
______________9. A college official divides
the student population into five classes:
freshman, sophomore, junior, senior, and
graduate student. The official takes a simple
random sample from each class and asks the
members opinions regarding student services.
______________10. In the game of lotto, 6
balls are selected from a container with 42
balls.
IV. Using proportional allocation, determine
the sample size needed for every school.
The total population of students is 10,679,
and the minimum sample is 2,450.
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School
Antipolo National
High School
Bagong Nayon
National
High School
Dela Paz National
High School
Sta. Cruz National
High School
Tubigan National
High School
Total
Population
per School
Sample
3,360
2,540
2,122
1,290
1,367
10,679
REFERENCES:
Statistics. Informed Decision using Data by
Michael Sullivan, III,. Fifth Edition
Sampling: Design and Analysis by Sharon L.
Lhr. Second Edition
http://www.economicsdiscussion.net/statistics/
sampling/advantages-of-sampling-overcompleteenumeration-in-statistics/11980
h t t p : / / w w w. n a t c o 1 . o r g / r e s e a r c
h / fi l e s /SamplingStrategies.pdf
https://data36.com/statistical-bias-typesexplained/
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MODULE 3: DESCRIPTIVE STATISTICS
OBJECTIVES:
After successful completion of this module, you should be
able to:
✦
✦
✦
✦
✦
✦
✦
✦
Distinguish the three main forms of data presentation.
Know the different parts of the table.
Choose appropriate diagrams/graphs to present a given set of
data.
Organize qualitative and quantitative data in tables.
Compute measures of central tendency, measures of variation and
measures of relative position of grouped and ungrouped data.
Describe the shape of a distribution.
Identify regions under the normal curve corresponding to
different standard normal values.
Compute probabilities using the standard normal table and Excel.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Data Presentation
Data are usually collected in a raw format and thus
the inherent information is difficult to understand.
Therefore, raw data need to be summarized,
processed, and analyzed to usefully derive
information from them. However, no matter how well
manipulated, the information derived from the raw
data should be presented in an effective format,
otherwise, it would be a great loss for both authors
and readers. Planning how the data will be presented
is essential before appropriately processing raw data.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Presentation of Data
Presentation of data refers to an exhibition
or putting up data in an attractive and useful
manner such that it can be easily interpreted.
The three main forms of presentation of data
are:
Textual Presentation
Tabular Presentation
Graphical Presentation
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Textual Presentation
•
All the data is presented in the form of text,
phrases, or paragraphs.
•
It involves enumerating important
characteristics, emphasizing significant figures
and identifying important features of data.
•
Text is the principal method for explaining
findings, outlining trends, and providing
contextual information.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example:
A researcher is asked to present the performance of a section in
the statistics test. The following are the test scores:
34
50
37
24
49
42
18
38
29
48
20
35
38
25
46
50
43
39
26
45
17
50
39
28
45
9
23
38
27
46
34
23
38
44
45
43
35
39
44
46
The data presented in textual form would be like this:
In the statistics class of 40 students, 3 obtained the perfect
score of 50. Sixteen students got a score 40 and above,
while only 3 got 19 and below. Generally, the students
performed well in the test with 23 or 70% getting a passing
score of 38 and above.
Polytechnic University of the Philippines
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Advantage of Textual Presentation
✦
✦
✦
The data would be more interpreted.
Can help in emphasizing some important points
in data.
Small sets of data can be easily presented.
Remember!
✦ Keep your paragraphs simple and short.
✦
Always make sure that the readers are provided
with additional explanations about the relevance
of the figures and its implications.
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Tabular Presentation:
•
It is a systematic and logical arrangement of
data in the form of Rows and Columns with
respect to the characteristics of data.
•
A table is best suited for representing individual
information and represents both quantitative
and qualitative information.
Polytechnic University of the Philippines
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Advantage of Tabular
Presentation
✦
✦
✦
✦
More information may be presented.
Exact values can be read from a table to
retain precision.
Flexibility is maintained without
distortion of data.
Less work and less cost are required in
the preparation.
Polytechnic University of the Philippines
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Preparing Tables
The making of a compact table itself is an art. This should
contain all the information needed within the smallest possible
space. What the purpose of tabulation is and how the tabulated
information is to be used are the main points to be kept in mind
while preparing for a statistical table. An ideal table should
consist of the following main parts:.
A. Title: The title must tell as simply as possible what is in the
table. It should answer the questions:
✦ Who? White females with breast cancer, black males with
lung cancer.
✦ What are the data? Counts, percentage distributions, rates.
✦ Where are the data from? Example: One hospital, or the
entire population covered by your registry.
✦ When? A particular year, time period.
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B. Boxhead: The boxhead contains the captions or
column headings. The heading of each column
should contain as few words as possible, yet
explain exactly what the data in the columns
represent.
C. Stubs: The row captions are known as the stub.
Items in the stub should be grouped to facilitate
interpretation of the data. For example, rows may
stand for score of classes and columns for data
related to sex of students. In the process, there will
be many rows for scores classes but only two
columns for male and female students.
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D. Footnotes: Footnotes are given at the foot of the
table for explanation of any fact or information
included in the table which needs some explanation.
Thus, they are meant for explaining or providing
further details about the data that have not been
covered in title, captions and stubs.
E. Sources of Data: We should also mention the source
of information from which data are taken. This may
preferably include the name of the author, volume,
page and the year of publication. This should also
state whether the data contained in the table is of
‘primary or secondary’ nature.
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Parts of the Table
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Construction of Data Tables
✦
✦
✦
✦
✦
✦
✦
✦
The title should be in accordance with the
objective of study
Comparison
Alternative location of stubs
Headings
Footnote
Size of columns
Use of abbreviations
Units
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Example:
Simple or One – Way Table
Optionally, the table may also include totals or
percentages.
Polytechnic University of the Philippines
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Example:
Compound Table
A compound table is just an extension of a simple in which
there are more than one variable distributed among its
attributes (subvariable). An attribute is just a quality, property
or component of a variable according to which it can be
differentiated with respect to other variables.
We may refer to a compound table as a cross tabulation or
even to a contingency table depending on the context in which
it is used.
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Organize Quantitative Variable in Table
Classes are categories into which data are grouped. When a
data set consists of a large number of different discrete data
values or when a data set consists of continuous data, we create
classes by using intervals of numbers.
Make sure that the classes do not overlap. This is necessary to
avoid confusion as to which class a data value belongs. Also,
make sure that the class widths are equal for all classes.
Upper Class
Lower Class Limit (LC)
Limit (UC)
Number
Age
The class width is the
(in thousands)
25 - 34
14,482
difference between
35
44
14,156
consecutive lower class
45 - 54
13,801
limits.
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55 - 64
65 - 74
One exception to the requirement of
equal class widths occurs in openended tables. A table is open ended if
the first class has no lower class limit
or the last class has no upper class
limit.
12,123
7,010
Scores
Frequency
10 - 19
25
20 - 29
36
30 - 39
40
40 and over
12
Guidelines for Determining the Lower Class Limit of the First
Class and Class Width
Choosing the Lower Class Limit of the First Class:
Choose the smallest observation in the data set or a
convenient number slightly lower than the smallest
observation in the data set.
For example, the smallest observation is 10.2. A convenient
lower class limit of the first class is 10.
Polytechnic University of the Philippines
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Guidelines for Determining the Lower Class Limit of the First
Class and Class Width
Determining the Class Width:
• Decide on the number of classes. Generally, there should be
between 5 and 20 classes. The smaller the data set, the fewer
classes you should have.
• Determine the class width by computing:
x − xmin
cw = max
cw is the class width
nc
nc is the number of classes
Round this value up to a convenient number.
Remember!
Creating the classes for summarizing continuous data is an art
form. There is no such thing as the correct frequency distribution.
However, there can be less desirable frequency distributions. The
larger the class width, the fewer classes a frequency distribution
will have.
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How to Construct Frequency
Distribution Table?
A frequency distribution list each
category of data and the number of
occurrences for each category of data.
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Example:
Use the “Sample Data file”.
Solution:
To answer this question we need to construct a frequency
distribution to determine how many female and male
respondents participated in the study.
Polytechnic University of the Philippines
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Procedure in Constructing
Frequency Table
✦
If the data is in the form of qualitative data
To construct the frequency distribution using
excel use the command:
=frequency(data_array,bins_array)
Then Ctrl
→ Shift → Enter
{=frequency(data_array,bins_array)}
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Final Output
Table 1 shows the frequency and percentage distribution of
the respondents in terms of sex. It can be gleaned from the
table that, out of 128 respondents considered in the study,
65 or 50.8% are male and 63 or 49.2% are female.
Polytechnic University of the Philippines
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Example:
Use the “Sample Data file”.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Procedure in Constructing
Frequency Table
If the data is in the form of quantitative data
Steps
1. Set an interval or range for your data. It is
needed for the “BIN RANGE”.
2. Click “DATA” on the menu bar and Click
“DATA ANALYSIS” on the tool bar
3. The dialog box “DATA ANALYSIS” will appear
and choose “HISTOGRAM” on the dialog box
then click OK.
✦
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Procedure in Constructing
Frequency Table
If the data is in the form of quantitative data
Steps
4. Highlight your data for the “INPUT RANGE”.
5. Highlight your data for the “BIN RANGE”.
6. Click the box of “LABELS IN FIRST ROW”
then click “OK”.
7. The result will appear on the new worksheet of
the excel file. Get the Percentage and total.
✦
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Department of Mathematics and Statistics
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Final Output
Polytechnic University of the Philippines
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Example:
Identify problems with the following
table.
Answer:
✦
✦
✦
Useless Information – Don’t show decimals if they are not
needed.
Poor Alignment – Make sure alignment makes sense.
• Don’t center numbers, always right justify – try to align
decimal points.
• Consider the appropriate placement of row titles.
Difficult to Read – Use commas used when the number exceeds
a thousand.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Graphical Presentation
✦
✦
✦
A graph is a very effective visual tool as it displays data at
a glance, facilitates comparison, and can reveal trends and
relationships within the data such as changes over time,
and correlation or relative share of a whole.
It is considered an important medium of communication
because we are able to create a pictorial representation of
the numerical figures.
Suited when we need to show the results of the study to
nonprofessionals and or people who dislike numbers and too
lengthy texts.
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Bar Graph
✦
✦
It is constructed by labeling each category
of data on either the horizontal or vertical
axis and the frequency or relative frequency
of the category on the other axis. Rectangles
of equal width are drawn for each category.
The height of each rectangle represents the
category’s frequency or relative frequency.
It is use to organize discrete data.
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Example: Simple Bar Graph
The simple bar chart is used for the case of one
variable only.
Polytechnic University of the Philippines
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Example:
Multiple Bar Graph\ Grouped
Column Chart
The multiple bar chart is an extension of a simple bar chart
when there are quantities of several variables to be
displayed. The bars representing the quantities for the
different variables are piled next to one another for each
attribute. The figure becomes very cumbersome when there
are too many variables and components.
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Example:
Component Bar Graph/ Subdivided
Column Chart
In this type of bar chart, the components (quantities) of each
variable are piled on top of one another. It saves space as
compared to a multiple bar chart. One of the disadvantage
of this graph is that it is not always easy to compare size of
the components, or parts. It is used to represent data in
which the total magnitude is divided into different or
components.
Polytechnic University of the Philippines
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Remember!
•
Bar graphs may also be drawn with horizontal
bars. Horizontal bars are preferable when
category names are lengthy.
•
In bar graphs, the order of the categories does
not usually matter. However, bar graphs that
have categories arranged in decreasing order
of frequency help prioritize categories for
decision-making purposes in areas such as
quality control, human resources, and
marketing.
Polytechnic University of the Philippines
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Department of Mathematics and Statistics
Histogram
✦
✦
✦
It is constructed by drawing rectangles for each class of
data. The height of each rectangle is the frequency or
relative frequency of the class. The width of each rectangle
is the same and the rectangles touch each other.
It is a graph used to present quantitative data, is similar to
the bar graph.
It is use to organize continuous data.
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Pie Chart
✦
It is a circle divided into sectors. Each sector represents a
category of data.The area of each sector is proportional to
the frequency of the category.
✦
Pie charts are typically used to present the relative
frequency of qualitative data. Inmost cases the data are
nominal, but ordinal data can also be displayed in a pie
chart.
Polytechnic University of the Philippines
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When should a bar graph or a
pie chart be used?
✦
Pie charts are useful for showing the
division of all possible values of a
qualitative variable into its parts.
✦
Bar graphs are useful when we want to
compare the different parts, not necessarily
the parts to the whole.
Polytechnic University of the Philippines
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Department of Mathematics and Statistics
Line Graph
✦
A graph that shows information that is
connected in some way (such as change over
time)
✦
Line segments are then drawn connecting the
points. It is use to organize continuous data.
✦
Very useful in identifying trends in the data
over time.
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Example:
Simple Line Graph
The simplest of line graphs is the single line graph, so
called because it displays information concerning one
variable only, in terms of its frequencies.
Polytechnic University of the Philippines
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Example:
Multiple Line Graph
Multiple line graphs illustrate information on
several variables so that comparison is possible
between them.
Polytechnic University of the Philippines
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Guidelines for Constructing
Good Graphics
✦
Title and label the graphic axes clearly,
providing explanations if needed. Include units
of measurement and a data source when
appropriate.
✦
Avoid distortion.
✦
Minimize the amount of white space in the
graph. Use the available space to let the data
stand out. If you truncate the scales, clearly
indicate this to the reader.
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Guidelines for Constructing
Good Graphics
✦
Avoid clutter, such as excessive gridlines and
unnecessary backgrounds or pictures.
✦
Don’t distract the reader.
✦
Avoid three dimensions.
✦
Do not use more than one design in the same
graphic. Let the data speak for themselves.
Polytechnic University of the Philippines
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Grouped and Ungrouped Data
Data is often described as ungrouped
or grouped.
Grouped data is the type of data
which is classified into groups after
collection.
Ungrouped data which is also known
as raw data is data that has not been
placed in any group or category after
collection.
Ungrouped data without a
frequency distribution
1, 5, 4, 7, 2, 4, 1, 3, 8, 2, 2, 9
Polytechnic University of the Philippines
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Scores
1 - 10
11 - 20
21 - 30
31 - 40
41 - 50
Total
Frequency
5
9
10
12
24
60
Ungrouped data with a
frequency distribution
No. of Television
Sets
0
1
2
3
4
5
Total
Frequency
7
15
12
4
5
2
45
Measures of Central Tendency:
MEAN
•
•
•
It is the sum of the data values divided by the number of
data values.
It is also called the average.
It is appropriate only for data under interval and ratio scale
measurement.
Advantage of Mean
✦ Simple to understand and easy to calculate.
✦ It is rigidly defined.
✦ It is least affected fluctuation of sampling.
✦ It takes into account all the values in the series.
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Formula for Mean:
Sample Mean
✦ For Grouped Data
For Ungrouped Data
where:
where:
r
n
xi = data values
xi = data values
∑i=1 fxi
∑i=1 xi
n = no. of
f = frequency
x̄ =
x̄ =
sample
n
n = no. of
n
observations
sample
observations
Population Mean
where:
where:
r
N
∑i=1 fxi
xi = data values
∑i=1 xi xi = data values
μ=
μ=
N = no. of
f = frequency
N
N
observations
N = no. of
observations
✦
Polytechnic University of the Philippines
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Measures of Central Tendency:
MEDIAN
It is the “middle observation” when the data set is sorted (in
either increasing or decreasing order).
• The median divides the distribution into two equal parts.
Advantage of Median
✦ The median is not affected by the size of extreme values but
by the number of observations.
✦ The median can be calculated even when the frequency
distribution contains “open-ended” intervals.
✦ It can also be used to define the middle of a number of
objects, properties, or quantities which are not really
quantitative in a nature.
✦ It can be easily interpreted.
•
Polytechnic University of the Philippines
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Formula for Median:
✦
For Ungrouped Data
1. Arrange the data from
lowest to highest (or highest
to lowest).
2. For an odd number of
data, the median of a data
set is the “middle
observation”. When the
number of data is even, the
median is the “average of
the two middle scores”.
Polytechnic University of the Philippines
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✦
For Grouped Data
n
− < cf i
(2
)
x̃ = LB +
f
where:
LB = lower boundary of the
median class
i = class width
n = no. of observations
< cf = less than the cumulative
frequency of the class
preceding the median class
f = frequency of the median
class
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Measures of Central Tendency:
MODE
•
•
•
•
It is the most frequently occurring value in a list of data.
It is sometimes called nominal average.
It is an appropriate measure of average for data using the
nominal scale of measurement.
It is the only measure of central tendency used in both
quantitative and qualitative data.
Advantage of Mode
The mode is easy to understand.
Like the median, it is not greatly affected by extreme
values.
Like the median, it can be computed even when the
frequency distribution contains “open-ended” intervals.
✦
✦
✦
Polytechnic University of the Philippines
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Formula for Mode:
✦
For Ungrouped Data
✦
For Grouped Data
d1
1.Obtain a frequency
i
x ̂ = LB +
( d1 + d2 )
distribution of the distinct
where:
values of the data.
LB = lower boundary of the
2.The mode is the most
frequently occurring data
(if there is one).
modal class
i = class width
d1 = difference between the
frequency of the modal class
and the class preceding it
d2 = difference between the
frequency of the modal class
and the class following it
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Remember!
•
Whenever you hear the word average, be aware that
the word may not always be referring to the mean.
One average could be used to support one position,
while another average could be used to support a
different position.
•
Mode is not always present in the data sets unlike
mean and median.
•
If you are interested in the “center of gravity” of your
data, then use the mean; if you are interested in the
“middle value” within your data, then use the median
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Choosing a Measure of Central Tendency:
We have discussed three types of central tendency-the
mode, the mean, and the median and examined how they
differ in terms of finding the center of a data distribution.
The next legitimate question to ask may be “When do we
use which measure?”
Consider the following data sets:
Data Set I
Data Set II
108 112 116 120 124
108 112 116 120 205
Determine the mean, median and mode.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
In both data sets, the median is 116, as it is the number that
divides the data set into two exact halves. However, you will
notice that the mean is not identical in both data sets. For the
first data set, the mean is equal to 116 where the mean of the
second data set is equal to 132.5
Notice how the mean of the second data set has been
influenced by the presence of an unusual case/outlier in the
data set. If we were to say the mean is equal to 132.5 for the
second data set and it represents a typical case, this will not
make much sense because the majority of data values are less
than 120. Therefore, the mean should not be used when
unusual, or outlying, data values are present in the data set, as
the mean tends to be extremely sensitive to the unusual
values. Rather, the median should be reported in this case.
Polytechnic University of the Philippines
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Department of Mathematics and Statistics
•
The mode is simply the most frequently occurring data
values in the data set. Therefore, it is mainly useful for the
nominal level of measurement. Both median and mean are
useful when the variable being measured can be quantified.
Also both data sets have no mode that’s why mode is not
appropriate measure to use in these data sets.
•
It is better to use the median than to use the mean when
the sample is small or asymmetrical (i.e., skewed) and
unusual cases/outliers is present in the data sets. This is
why the average housing price is always reported with the
median, since even one million-dollar house can distort the
average housing price when most of the houses are in
Php500,000–Php650,000 range.
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Example:
The data given below is the age of the residents in
Barangay 634, Sta. Mesa, Manila. Compute mean,
median and mode.
Class Interval
Frequency
55 - 59
55
50 - 54
23
45 - 49
37
40 - 44
37
35 - 39
48
30 - 34
42
25 - 29
27
Polytechnic University of the Philippines
College of Science
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Solution:
To compute mean of grouped data, first you need to
fill out this table.
Class
Interval
55 - 59
Frequency
(f)
3
50 - 54
6
45 - 49
7
40 - 44
35 - 39
9
6
30 - 34
4
25 - 29
5
Total
n=
x
fx
It is the midpoint of
every class interval.
To compute this:
LC + UP
x=
2
Ex:
7
∑
fxi =
i=1
Polytechnic University of the Philippines
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55 + 59
= 57
2
50 + 54
x=
= 52
2
x=
Solution:
x
fx
55 - 59
Frequency
(f)
3
57
171
50 - 54
6
52
312
45 - 49
40 - 44
7
9
47
42
329
378
35 - 39
6
37
222
30 - 34
4
32
128
25 - 29
5
27
Class Interval
135
7
Total
n = 40
∑
7
x̄ =
=
∑i=1 fxi
n
1,675
40
= 41.88
fxi = 1,675
i=1
The average age is 41.88
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Solution:
To compute median and mode of grouped data, first
you need to fill out this table.
Class
Interval
55 - 59
3
50 - 54
6
45 - 49
7
40 - 44
35 - 39
9
6
30 - 34
4
f
25 - 29
5
Total
n=
LB
< cf
To compute the lower
b o u n d a r y, a l w a y s
subtract 0.5 to lower
class limit (LC).
Ex:
55 − 0.5 = 54.5
50 − 0.5 = 49.5
45 − 0.5 = 44.5
Polytechnic University of the Philippines
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Solution:
Class
Interval
55 - 59
f
LB
3
54.5
50 - 54
6
49.5
45 - 49
7
44.5
40 - 44
35 - 39
9
6
39.5
34.5
30 - 34
4
29.5
25 - 29
5
24.5
Total
n = 40
< cf
5
If the arrangement of
the class interval is
descending order,
always start at the
bottom part.
Copy the frequency
of the lowest class
interval.
5 + 4 = 9 + 6 = 15 + 9 = 24 + 7 = 31 + 6 = 37 + 3 = 40
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
55 - 59
f
LB
< cf
3
54.5
40
50 - 54
6
49.5
37
45 - 49
40 - 44
7
9
44.5
39.5
31
24
35 - 39
6
34.5
15
30 - 34
4
29.5
9
25 - 29
5
24.5
5
Total
n = 40
x̃ = LB +
n
− < cf i
)
(2
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
f
First, compute
n
, it will help us to
2
determine the median class and the
< cf.
n 40
=
= 20
2
2
The median class is the class
containing the 20th item. Hence, the
median class is 40 - 44.
x̃ = 39.5 +
(20 − 15)5
= 42.28
9
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Solution:
Class
Interval
f
LB
< cf
55 - 59
3
54.5
40
50 - 54
6
49.5
37
45 - 49
7
44.5
31
40 - 44
9
39.5
24
35 - 39
6
34.5
15
30 - 34
4
29.5
9
25 - 29
5
24.5
5
x ̂ = LB +
d1
i
( d1 + d2 )
The modal class is the class interval
with the highest frequency. The
modal class is 40 - 44.
If there are two class interval that
contains the highest frequency,
always choose the highest class
interval.
d1 = 9 − 6 = 3
d2 = 9 − 7 = 2
x ̂ = 39.5 +
3
5 = 42.5
(3 + 2)
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Measures of Relative Position
Quantiles are statistics that describe
various subdivisions of a frequency
distribution into equal proportions.
Three special Quantiles:
1. Quartiles
2. Deciles
3. Percentiles
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Quartiles - split
the ordered data
into four quarters.
Deciles - split the
ordered data into
ten equal.
Percentiles - split
the ordered data
into 100 equal
parts.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Formula for Quartile:
✦
For Ungrouped Data
1. Arrange the data from
lowest to highest. Then use
this formula.
Qclass =
nk
+ 0.5
4
2. If the resulting positioning
point is an integer, the
particular numerical
observation corresponding
to that point is chosen for
the quartile. If not, use
interpolation.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
✦
For Grouped Data
nk
− < cf i
(4
)
Qk = LB +
f
where:
LB = lower boundary of the
quartile class
i = class width
n = no. of observations
k = quartile position
< cf = less than the cumulative
frequency of the class
preceding the quartile class
f = frequency of the quartile
class
Formula for Decile:
✦
For Ungrouped Data
1. Arrange the data from
lowest to highest. Then use
this formula.
Dclass =
nk
+ 0.5
10
2. If the resulting
positioning point is an
integer, the particular
numerical observation
corresponding to that point
is chosen for the decile.If
not, use interpolation.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
✦
For Grouped Data
nk
− < cf i
( 10
)
Dk = LB +
f
where:
LB = lower boundary of the
decile class
i = class width
n = no. of observations
k = decile position
< cf = less than the cumulative
frequency of the class
preceding the decile class
f = frequency of the decile class
Formula for Percentile:
✦
For Ungrouped Data
1. Arrange the data from
lowest to highest. Then use
this formula.
Pclass =
nk
+ 0.5
100
2. If the resulting
positioning point is an
integer, the particular
numerical observation
corresponding to that point
is chosen for the percentile.
If not, use interpolation.
For Grouped Data
nk
− < cf i
)
( 100
Pk = LB +
f
where:
LB = lower boundary of the
percentile class
i = class width
n = no. of observations
k = percentile position
✦
< cf = less than the cumulative
frequency of the class
preceding the percentile class
f = frequency of the percentile
class
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Example 1:
The data given below is the total number of hours
lost due to tardiness and absences of employees in a
company in a given year.
Find Q3, D4 and P55.
Month
Hour Lost (x)
January
February
March
April
May
June
July
August
September
October
November
December
55
23
37
37
48
42
27
20
30
32
24
40
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
To compute Q3 of ungrouped data:
1. Arrange the data from lowest to highest.
20
1
23
2
24
3
27
4
30
5
32
6
Qclass =
37
7
37
8
40
9
42
10
48
11
55
12
(12)(3)
= 9.5
4
2. Use interpolation since the computed Qclass is not an integer.
20
1
23
2
24
3
27
4
30
5
32
6
37
7
37
8
40
9
42
10
48
11
55
12
Q3 = 40 + 0.5(42 − 40)
= 41
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
To compute D4 of ungrouped data:
1. Arrange the data from lowest to highest.
20
23
24
27
30
32
37
37
40
42
48
55
1
2
3
4
5
6
7
8
9
10
11
12
Dclass =
(12)(4)
+ 0.5 = 5.3
10
2. Use interpolation since the computed Dclass is not an integer.
20
23
24
27
30
32
37
37
40
42
48
55
1
2
3
4
5
6
7
8
9
10
11
12
D4 = 30 + 0.3(32 − 30)
= 30.6
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Solution:
To compute P55 of ungrouped data:
1. Arrange the data from lowest to highest.
20
23
24
27
30
32
37
37
40
42
48
55
1
2
3
4
5
6
7
8
9
10
11
12
Pclass =
(12)(55)
+ 0.5 = 7.1
100
2. Use interpolation since the computed Pclass is not an integer.
20
23
24
27
30
32
37
37
40
42
48
55
1
2
3
4
5
6
7
8
9
10
11
12
P55 = 37 + 0.1(37 − 37)
= 37
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example 2:
The data given below is the age of the residents in
Barangay 634, Sta. Mesa, Manila. Compute Q1, D7, and
P10.
Class Interval
Frequency
55 - 59
55
50 - 54
23
45 - 49
37
40 - 44
37
35 - 39
48
30 - 34
42
25 - 29
27
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
To compute Q1, D7, and P10 of grouped data, first you
need to fill out this table.
Class
Interval
55 - 59
3
50 - 54
6
45 - 49
7
40 - 44
35 - 39
9
6
30 - 34
4
f
25 - 29
5
Total
n=
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
LB
< cf
To compute the lower
b o u n d a r y, a l w a y s
subtract 0.5 to lower
class limit (LC).
Ex:
55 − 0.5 = 54.5
50 − 0.5 = 49.5
45 − 0.5 = 44.5
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Solution:
Class
Interval
55 - 59
f
LB
3
54.5
50 - 54
6
49.5
45 - 49
7
44.5
40 - 44
35 - 39
9
6
39.5
34.5
30 - 34
4
29.5
25 - 29
5
24.5
Total
n = 40
< cf
5
If the arrangement of
the class interval is
descending order,
always start at the
bottom part.
Copy the frequency
of the lowest class
interval.
5 + 4 = 9 + 6 = 15 + 9 = 24 + 7 = 31 + 6 = 37 + 3 = 40
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
55 - 59
f
LB
< cf
3
54.5
40
50 - 54
6
49.5
37
45 - 49
40 - 44
7
9
44.5
39.5
31
24
35 - 39
6
34.5
15
30 - 34
4
29.5
9
25 - 29
5
24.5
5
Total
n = 40
nk
− < cf i
(4
)
Qk = LB +
f
First, compute
nk
, it will help us to
4
determine the quartile class and the
< cf.
nk (40)(1)
=
= 10
4
4
The quartile class is the class
containing the 10th item. Hence, the
quartile class is 35 - 39.
Q1 = 34.5 +
(10 − 9)5
= 35.33
6
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
55 - 59
f
LB
< cf
3
54.5
40
50 - 54
6
49.5
37
45 - 49
40 - 44
7
9
44.5
39.5
31
24
35 - 39
6
34.5
15
30 - 34
4
29.5
9
25 - 29
5
24.5
5
Total
n = 40
Dk = LB +
nk
− < cf i
)
( 10
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
f
First, compute
nk
, it will help us to
10
determine the decile class and the
< cf.
nk
(40)(7)
=
= 28
10
10
The decile class is the class
containing the 28 item. Hence, the
decile class is 45 - 49.
D7 = 44.5 +
(28 − 24)5
= 47.36
7
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Solution:
Class
Interval
55 - 59
f
LB
< cf
3
54.5
40
50 - 54
6
49.5
37
45 - 49
40 - 44
7
9
44.5
39.5
31
24
35 - 39
6
34.5
15
30 - 34
4
29.5
9
25 - 29
5
24.5
5
Total
n = 40
Pk = LB +
nk
− < cf i
( 100
)
First, compute
nk
, it will help us to
100
determine the percentile class and
the
nk
(40)(10)
< cf.
=
=4
100
100
The percentile class is the class
containing the 4th item. Hence, the
percentile class is 25 - 29.
P10 = 24.5 +
f
(5 − 0)5
= 29.5
5
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example 2:
The ages of the town’s people in a certain community
is as follows:
Class Interval
Frequency
18 - 24
28
25 - 31
54
32 - 38
38
39 - 45
20
46 - 52
17
53 - 59
3
Find Q2, D5, and P50.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
To compute Q2, D5, and P50 of grouped data, first you
need to fill out this table.
Class
Interval
f
18 - 24
28
25 - 31
54
32 - 38
38
39 - 45
20
46 - 52
17
53 - 59
3
Total
n=
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
LB
< cf
To compute the lower
b o u n d a r y, a l w a y s
subtract 0.5 to lower
class limit (LC).
Ex:
18 − 0.5 = 17.5
25 − 0.5 = 24.5
32 − 0.5 = 31.5
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Solution:
Class
Interval
f
LB
< cf
18 - 24
28
17.5
28
25 - 31
54
24.5
32 - 38
38
31.5
39 - 45
20
38.5
46 - 52
17
45.5
53 - 59
3
52.5
Total
n = 160
If the arrangement of
the class interval is
a s c e n d i n g o r d e r,
always start at the
upper part.
Copy the frequency
of the lowest class
interval.
28 + 54 = 82 + 38 = 120 + 20 = 140 + 17 = 157 + 3 = 160
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
f
LB
< cf
18 - 24
28
17.5
28
25 - 31
54
24.5
82
32 - 38
38
31.5
120
39 - 45
20
38.5
140
46 - 52
17
45.5
157
53 - 59
3
52.5
160
Total
n = 160
nk
− < cf i
(4
)
Qk = LB +
f
First, compute
nk
, it will help us to
4
determine the quartile class and the
< cf.
nk (160)(2)
=
= 80
4
4
The quartile class is the class
containing the 80th item. Hence, the
quartile class is 25 - 31.
Q2 = 24.5 +
(80 − 28)7
= 31.24
54
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
f
LB
< cf
18 - 24
28
17.5
28
25 - 31
54
24.5
82
32 - 38
38
31.5
120
39 - 45
20
38.5
140
46 - 52
17
45.5
157
53 - 59
3
52.5
160
Total
n = 160
Dk = LB +
nk
− < cf i
)
( 10
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
f
First, compute
nk
, it will help us to
10
determine the decile class and the
< cf.
(160)(5)
nk
=
= 80
10
10
The decile class is the class
containing the 80th item. Hence, the
decile class is 25 - 31.
D5 = 24.5 +
(80 − 28)7
= 31.24
54
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Solution:
Class
Interval
f
LB
< cf
18 - 24
28
17.5
28
25 - 31
54
24.5
82
32 - 38
38
31.5
120
39 - 45
20
38.5
140
46 - 52
17
45.5
157
53 - 59
3
52.5
160
Total
n = 160
Pk = LB +
nk
− < cf i
( 100
)
f
First, compute
nk
, it will help us to
100
determine the percentile class and
the
nk
(160)(50)
< cf.
=
= 80
100
100
The percentile class is the class
containing the 80th item. Hence, the
percentile class is 25 - 31.
P50 = 24.5 +
(80 − 28)7
= 31.24
54
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Sample Interpretation:
1. Jennifer just received the results of her SAT exam. Her
SAT Mathematics score of 600 is in the 74th percentile. What
does this mean?
A percentile rank of 74% means that 74% of SAT
Mathematics scores are less than or equal to 600 and 26%
of the scores are greater. So 26% of the students who took
the exam scored better than Jennifer.
2. Time taken to finish a test is 35 minutes. This time was the
first quartile. What does this mean?
25% of the learners finished the exam in 35 minutes or
less, and 75% of the learners finished the exam in more
than 35 minutes.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Measures of Dispersion/Variability
Based on the figure below, determine which between the
two scatter diagram illustrate larger variability?
Figure 1
Figure 2
Since the data points in figure 2 is more scattered than the
data points in figure 1, then the data set depicted in figure 2
is more varied.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Measures of Dispersion/Variability:
RANGE
It is the difference between the largest and the smallest
observations or items in a set of data.
R = Xmax. − Xmin.
Range is simple to calculate. However, we should be
cautious about using range as a measure of variability.
Range is a very crude measure of variability as it only
uses the highest and lowest values in computation.
Therefore, it does not accurately capture information
about how data values in the set differ if the data set
contains an unusual cases/outliers.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Measures of Dispersion/Variability:
STANDARD DEVIATION
•
It is a measure of how far away items in a data set are from
the mean.
•
The larger the standard deviation, the more variation there
is in the data set.
•
The standard deviation can never be a negative number,
due to the way it’s calculated and the fact that it measures a
distance (distances are never negative numbers).
•
The smallest possible value for the standard deviation is 0,
and that happens only in contrived situations where every
single number in the data set is exactly the same (no
deviation).
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Formula for Standard Deviation:
Sample Standard Deviation
✦ For Grouped Data
For Ungrouped Data
where:
where:
r
n
∑i=1 f(xi − x̄)2
xi = data
∑i=1 (xi − x̄)2 xi = data
values s =
values s =
n−1
n−1
x̄ = mean
x̄ = mean
f = frequency
n = no. of sample observations
n = no. of sample observations
Population Standard Deviation
where:
where:
r
N
xi = data
2 xi = data
∑i=1 f(xi − μ)2
∑i=1 (xi − μ)
values σ =
values σ =
N
N
μ = mean
μ = mean
f = frequency
N = no. of observations
N = no. of observations
✦
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Measures of Dispersion/Variability:
VARIANCE
It represents all data points in a set and is calculated
by averaging the squared deviation of each mean.
Variance is not easy to read as it is the squared format
and hence not easily interpretable. However,
Standard deviation being in the same units as the
mean we can easily understand the spread of data.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Formula for Variance:
Sample Variance
✦ For Grouped Data
For Ungrouped Data
where:
where:
r
n
∑i=1 f(xi − x̄)2
∑i=1 (xi − x̄)2 xi = data
xi = data
2
2
values s =
values s =
n−1
n−1
x̄ = mean
x̄ = mean
f = frequency
n = no. of sample observations
n = no. of sample observations
Population Variance
where:
where:
r
N
xi = data
∑i=1 f(xi − μ)2
∑i=1 (xi − μ)2 xi = data
2
values σ 2 =
values σ =
N
N
μ = mean
μ = mean
f = frequency
N = no. of observations
N = no. of observations
✦
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example 1:
The data given below is the age of the residents in
Barangay 634, Sta. Mesa, Manila. Compute sample
standard deviation and sample variance.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Class Interval
Frequency
55 - 59
55
50 - 54
23
45 - 49
37
40 - 44
37
35 - 39
48
30 - 34
42
25 - 29
27
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Solution:
To compute SD and Var of grouped data, first you
need to fill out this table.
Class
Interval
55 - 59
50 - 54
45 - 49
40 - 44
35 - 39
30 - 34
25 - 29
f
x
fx
(xi − x̄)2
f(xi − x̄)2
3
6
7
9
6
4
5
7
Total
∑
n=
7
∑
fxi =
i=1
f(xi − x̄)2 =
i=1
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
55 - 59
f
x
fx
(xi − x̄)2
3
57
171
228.61
50 - 54
45 - 49
40 - 44
35 - 39
30 - 34
25 - 29
6
7
9
6
4
5
52
47
42
37
32
27
312
329
378
222
128
135
102.41
26.21
0.01
23.81
97.61
221.41
7
Total
1,675
40
= 41.88
7
fx =
∑ i
i=1
1,675
n = 40
f(xi − x̄)2
∑
f(xi − x̄)2 =
i=1
(x1 − x̄)2 = (57 − 41.88)2 = 228.61
(x2 − x̄)2 = (52 − 41.88)2 = 102.41
(x3 − x̄)2 = (47 − 41.88)2 = 26.21
x̄ =
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Solution:
Class
Interval
55 - 59
f
x
fx
(xi − x̄)2
f(xi − x̄)2
3
57
171
228.61
685.83
50 - 54
45 - 49
40 - 44
35 - 39
30 - 34
25 - 29
6
7
9
6
4
5
52
47
42
37
32
27
312
329
378
222
128
135
102.41
26.21
0.01
23.81
97.61
221.41
614.46
183.47
0.09
142.86
390.44
1107.05
7
Total
n = 40
fx =
∑ i
i=1
1,675
7
∑
i=1
f(xi − x̄)2 =
3,124.20
f(x1 − x̄)2 = 3(228.61) = 685.83
f(x2 − x̄)2 = 6(102.41) = 614.46
f(x3 − x̄)2 = 7(26.21) = 183.47
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Solution:
7
∑i=1 f(xi − x̄)2
s=
2
n−1
2
Class
Interval
55 - 59
50 - 54
45 - 49
40 - 44
35 - 39
(xi − x̄)
f(xi − x̄)
228.61
102.41
26.21
0.01
23.81
685.83
614.46
183.47
0.09
142.86
s=
30 - 34
25 - 29
97.61
221.41
390.44
1107.05
s2 =
3,124.20
40 − 1
= 8.95
7
∑i=1 f(xi − x̄)2
n−1
7
∑
Total
i=1
f(xi − x̄)2 =
3,124.20
s2 =
3,124.20
40 − 1
= 80.11
Polytechnic University of the Philippines
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How to interpret variance and standard
deviation?
Consider the following data set of toddler
weights in an outpatient clinic, assuming that the
data values were taken:
Data Set
15 13 20 19 14
Computed variance for this data set is 9.7.
Computed standard deviation for this data set is
3.11.
What does this mean?
Polytechnic University of the Philippines
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We cannot use variance as a measure of variability. Let us
assume that the values represent weight losses measured in
pounds taken from five subjects. Because the deviation of each
observation from the mean has been squared, the unit for the
variance is now in (pound)2 . What does (pound)2 mean? If we
were to say that data values differ from the mean on average
about 9.7 (pound)2, would this claim make sense? Probably not,
since there is no such a unit as a (pound)2.
Why do we then take the square of the deviation if the (unit)2
will not make sense to interpret at the end? The answer is
simple: If you do not square the deviation and sum each
deviation, it will always add up to zero no matter what data
set you work with.
n
n
∑
i=1
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(xi − x̄) = 0 →
∑
(xi − x̄)2 ≠ 0
i=1
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How can we then talk about variability if the measure of
variability comes out to be equal to zero? This is why we take
square of the deviation to compute the variance first and
then take square root of it to compute the standard
deviation, bringing us back to the original unit of
measurement.
We get the standard deviation of 3.11 by taking square root of
9.7; we can then say that the data values differ from the mean
(16.2 lbs.) on an average of about 3.11 pounds. We can
interpret this finding to mean that, on average, the weights fall
between 13.09 and 19.31 pounds. This makes more sense
when you look at the data set, compared to the variance. Note
that the mean and standard deviation should always be
reported together!
16.2 − 3.11 = 13.09
16.2 + 3.11 = 19.31
Polytechnic University of the Philippines
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Choosing a Measure of Dispersion/Variability:
We have discussed four types of dispersion/variability - the
range, the interquartile range, the variance,
and the
standard deviation and examined how they differ. The next
legitimate question to ask may be “When do we use which
measure?”
You should use the range only as a crude measure, since it
is extremely sensitive to unusual values in the data set.
Interquartile range is not as sensitive to unusual data values,
where standard deviation is very sensitive to unusual values.
Therefore, the interquartile range should be used with the
median when the data contain unusual data values.
However, the standard deviation should be used with the
mean when the data are free of unusual data values.
Polytechnic University of the Philippines
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Shape of Distribution
These two statistics give you insights into the shape of
the distribution.
✦
Skewness is the degree of distortion from the
symmetrical bell curve or the normal distribution. It
measures the lack of symmetry in data distribution.
✦
Kurtosis is a measure of the combined sizes of the
two tails. It tells you how tall and sharp the central
peak is, relative to a standard bell curve.
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Skewness
A symmetrical distribution will have a skewness of 0.
So, a normal distribution will have a skewness of 0.
In a symmetrical distribution, the Mean, Median and
Mode are equal to each other and the ordinate at
mean divides the distribution into two equal parts.
Polytechnic University of the Philippines
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There are two types of Skewness:
• Negatively Skewed/Skewed Left is when the tail of the left
side of the distribution is longer or fatter than the tail on the
right side. The mean and median will be less than the mode.
• Positively Skewed/Skewed Right means when the tail on the
right side of the distribution is longer or fatter. The mean and
median will be greater than the mode.
Skewness < 0
Skewness > 0
Skewness = 0
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Karl Pearson’s Measure of
Skewness
Noticed that the mean, median and mode are not
equal in a skewed distribution.
The Karl Pearson's measure of skewness is based
upon the divergence of mean from mode in a skewed
distribution. Karl Pearson’s Coefficient of Skewness
(Sk), given by
where:
x̄ is the mean
x ̂ is the median
Sk =
x̄ − x ̂
s
s is the sample standard deviation
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So far we have seen that Sk is strategically dependent
upon mode. If mode is not defined for a distribution
we cannot find Sk .But empirical relation between
mean, median and mode states that, for a moderately
symmetrical distribution, we have
Mean − Mode ≈ 3(Mean − Median)
Hence Karl Pearson's coefficient of skewness is
defined in terms of median as
where:
x̄ is the mean
x̃ is the median
Sk =
3(x̄ − x̃)
s
s is the sample standard deviation
Polytechnic University of the Philippines
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Kurtosis
It is actually the measure of outliers present in the
distribution. The outliers in a sample, therefore, have
even more effect on the kurtosis than they do on the
skewness.
Higher kurtosis means more of the variance is the
result of infrequent extreme deviations, as opposed to
frequent modestly sized deviations. In other words, it’s
the tails that mostly account for kurtosis, not the
central peak.
The kurtosis decreases as the tails become lighter. It
increases as the tails become heavier.
Polytechnic University of the Philippines
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•
Mesokurtic (Kurtosis=3): This distribution has
kurtosis statistic similar to that of the normal
distribution.
•
Leptokurtic (Kurtosis>3): Peak is higher and
sharper than normal distribution, which means that
data are heavy-tailed or profusion of outliers.
•
Platykurtic (Kurtosis<3):
Compared to a normal
distribution, its tails are shorter
and thinner, and often its central
peak is lower and broader.
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Percentile Coefficient of Kurtosis
A measure of kurtosis based on quartiles and
percentiles is
k=
QD
P90 − P10
where:
QD is semi-interquartile range QD =
Q3 − Q1
2
Polytechnic University of the Philippines
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How to Calculate Measures of Central Tendency,
Measures of Variation, Skewness and Kurtosis for
Ungrouped and Sample Data Using Excel?
Example:
The data given below are the scores of randomly
selected applied statistics undergraduate students in
Section A and Section B. Compare the scores of Section
A and Section B based on measures of central tendency,
and measures of variation and determine which section
performed better in their final examination. Also,
describe the shape of the distribution of these two data
sets using skewness and kurtosis
Data Set A
Data Set B
40 38 42 40 39 39 43 40 39 40
46 37 40 33 42 36 40 47 34 45
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1. Click “DATA” on the menu bar and Click “DATA
ANALYSIS” on the tool bar. The Dialog box will appear.
2. Select “Descriptive Statistics” then click “OK”.
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3. Highlight your data for the “INPUT RANGE” and click
the box of “LABELS IN FIRST ROW” then click “OK”.
4. Click “Summary statistics” and then click “OK”. Repeat the
process for Data Set B.
Polytechnic University of the Philippines
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When comparing distributions, it is better to use a measure of
variation/dispersion in addition to a measure of central tendency
but because in this example Data set A and Data set B have the
same value for measures of central tendency, we will just used
measure of variation/dispersion to compare these two data set.
Polytechnic University of the Philippines
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Based on the result, Data set B has a larger variability since it
has larger value computed based on different measures of
variation. This means that Data Set B is much more spread
out than the Data Set A.
In this example, we want a data set with a large mean value
and a small standard deviation so we can say that this is the
section that performed better. Section A and Section B have
the same mean value but in terms of standard deviation
Section A have smaller value compared to Section B,
therefore, Section A performed better in their final
examination.
In terms of the shape of the distribution, these two data sets
have the shape in terms of Skewness and kurtosis. It shows
that Data Set A and Data Set B have platykurtic shaped and it
is skewed to the right.
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Normal Distribution
✦
The normal distribution is sometimes called the bell curve
because the graph of its probability density looks like a
bell.
✦
It is also known as the Gaussian distribution, after the
German mathematician Carl Friedrich Gauss who first
described it.
✦
It is a probability function that describes how the values
of a variable are distributed.
Polytechnic University of the Philippines
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Normal Curve
50
100
150
The red curve is a model called the normal curve ,
which is used to describe continuous random variables
that are said to be normally distributed.
A continuous random variable is normally distributed,
or has a normal probability distribution, if its relative
frequency histogram has the shape of a normal curve.
Polytechnic University of the Philippines
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No data will ever be exactly/perfectly normally
distributed in reality. If so, how do we know
whether or not a collected data set is normally
distributed?
We can begin with a visual display of the data in a
histogram to see if the data set is normally
distributed. However, a visual check, alone, may not
be sufficient to know whether the data are normally
distributed. There are statistical measures,
skewness and kurtosis, which, along with a
histogram, allow us to determine whether the set is
normally distributed.
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Why is it important to know if the data follows
a normal distribution?
The most important reason is that many human
characteristics fall into an approximately normal
distribution and that the measurement scores are
assumed to be normally distributed when
running most statistical analyses. Therefore, the
statistical results you get at the end may not be
trustworthy if the variable is not normally
distributed.
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Properties of Normal Curve
1. The normal curve is bell-shaped and symmetric
about the mean, μ.
2. Because mean, median and mode are equal, the
normal curve has a single peak and the highest
point occurs at x = μ.
3. The normal curve has
inflection points at μ − σ
and μ + σ.
Inflection point
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Inflection point
μ−σ μ μ+σ
Properties of Normal Curve
4. The area under the normal curve is 1.
5. The area under the normal curve to the right
of μ equals the area under the curve to the
left of μ, which equals 0.50
6. The normal curve approaches,
but never touches the x-axis
as it extends farther and
farther away from the mean.
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area = 1
0.50 0.50
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μ1 < μ2, σ1 < σ2
μ1 = μ2, σ1 < σ2
Mean:
Changing the mean shifts the entire
curve left or right on the X-axis.
Standard Deviation:
✦ Changing
the standard deviation
either tightens or spreads out the
width of the distribution along the Xμ1 < μ2, σ1 = σ2
axis.
Larger standard deviations produce distributions that are more
spread out.
✦
Polytechnic University of the Philippines
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Determine whether the graph represent a normal
curve.
A.
C.
B.
D.
All of them did not represent the normal curve.
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Role of Area under a Normal
Curve
Suppose that a random variable X is normally
distributed with mean μ and standard deviation σ . The
area under the normal curve for any interval of values of
the random variable X represents either
✦
the proportion of the population with the characteristic
described by the interval of values or
✦
the probability that a randomly selected individual
from the population will have the characteristic
described by the interval of values.
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Standard Normal Distribution
A normal random variable having mean
value μ = 0 and standard deviation σ = 1 is
called a standard normal random variable,
and its density curve is called the standard
normal curve.
It will always be denoted by the letter Z.
Polytechnic University of the Philippines
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Standardizing a Normal Random Variable
The normal random variable of a standard
normal distribution is called a standard
score or a z-score. Every normal random
variable X can be transformed into a z score
via the following equation:
z=
x−μ
σ
where X is a normal random variable, μ is the mean of X, and
σ is the standard deviation of X.
Probabilities for a standard normal
random variable are computed
using Standard Normal
Distribution Table which shows
a cumulative probability associated
with a particular z-score.
Polytechnic University of the Philippines
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Remember!
Positive values of z-score indicate how far above
the mean a score falls and negative values
indicate how far below the mean a score falls.
Whether positive or negative, larger z-scores
mean that scores are far away from the mean and
smaller z-scores means that scores are close to
the mean.
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Standard Normal Distribution Table 1 (Positive Side P(Z < z))
Polytechnic University of the Philippines
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Standard Normal Distribution Table 2 (Negative Side P(Z < − z))
Polytechnic University of the Philippines
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Patterns for Finding Areas under a Standard Normal Curve
Using Table 1
A. Area to the right of a negative z value or to the left of a
positive z value.
Use Table 1 directly
0
z1
z1 0
B. Area between z values on either side of 0.
=
0 z2
z1 0 z2
z1 0
1 − Area
C. Area between z values on same side of 0.
=
z1 z2
0 z1
1 − Area
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0 z2
1 − Area
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Patterns for Finding Areas under a Standard Normal Curve
Using Table 1
D. Area to the right of a positive z value or to the left of a
negative z value.
=
0 z1
0 z1
0
Area = 1
E. Area between a given z value and 0.
=
0
z1
0
z1
0
Area = 0.50
Polytechnic University of the Philippines
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Patterns for Finding Areas under a Standard Normal Curve
Using Table 2
A. Area to the right of a positive z value or to the left of a
negative z value.
Use Table 2 directly
z1 0
0 z1
B. Area between z values on same side of 0.
=
0 z1
z1 z2
0
z2
C. Area between z values on either side of 0.
=
z1 0
z2
+
0 z2
z1 0
0.50 − Area
0.50 − Area
Polytechnic University of the Philippines
College of Science
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Patterns for Finding Areas under a Standard Normal Curve
Using Table 2
D. Area to the right of a negative z value or to the left of a
positive z value.
=
z1 0
+
z1 0
0
0.50 − Area
Area = 0.50
E. Area between a given z value and 0.
=
0
z1
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0
Area = 0.50
0
z1
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Example 1:
Scores on a standardized college entrance examination (CEE)
are normally distributed with mean 510 and standard
deviation 60. A selective university considers for admission
only applicants with CEE scores over 560. Find proportion of
all individuals who took the CEE who meet the university's
CEE requirement for consideration for admission.
Solution:
Given: μ = 510,σ = 60 and x = 560
Step 1: Draw a normal curve and
shade the desired area.
Area = P(X > 560)
450
510
X
570
560
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Using Table 1 By-hand Approach!
Step 2: Convert the value of x to a z-score.
P(X > 560) = P (Z > z)
560 − 510
=P Z>
)
(
60
= P(Z > 0.83)
= 1 − P(Z ≤ 0.83)
= 1 − 0.7967
= 0.2033
Area = P(Z > 0.83)
= 0.2033
−2 −1
0
1
2
Z
0.83
Use the Complement Rule
and determine one minus
the area.
The proportion of all CEE scores that exceed 560 is
0.2033 or 20.33%.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Using Table 2 By-hand Approach!
Step 2: Convert the value of x to a z-score.
Area = P(Z > 0.83)
P(X > 560) = P (Z > z)
560 − 510
=P Z>
(
)
60
= P(Z > 0.83)
= 0.2033
= 0.2033
−2 −1
The proportion of all CEE
scores that exceed 560 is
0.2033 or 20.33%.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
0
1
2
0.83
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Z
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Step 2: Used Excel to determine the area under
Technology Approach!
any normal curve.
Use “TRUE” for
cumulative since we
want the area under the
normal curve.
The proportion of all CEE
scores that exceed 560 is
0.2033 or 20.33%.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example 2:
A pediatrician obtains the heights of her three-year-old female
patients. The heights are approximately normally distributed,
with mean 38.72 inches and standard deviation 3.17 inches.
Determine the proportion of the three-year-old females that
have a height less than 35 inches.
Solution:
Given: μ = 38.72,σ = 3.17 and x = 35
Step 1: Draw a normal curve and shade
the desired area.
Area = P(X < 35)
Polytechnic University of the Philippines
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35.55 38.72 41.89
35
X
Using Table 1 By-hand Approach!
Step 2: Convert the value of x to a z-score.
Area = P(Z < − 1.17) = 0.1210
P(X < 35) = P (Z < z)
35 − 38.72
=P Z<
(
3.17 )
= P(Z < − 1.17)
= 1 − P(Z ≥ − 1.17)
= 1 − 0.8790
Z
2
−2 −1 0 1
= 0.1210
−1.17
Use the Complement Rule
and determine one minus
the area.
The proportion of the pediatrician’s three-year-old
females who are less than 35 inches tall is 0.1210 or
12.10%.
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Using Table 2 By-hand Approach!
Step 2: Convert the value of x to a z-score.
Area = P(Z < − 1.17) = 0.1210
P(X < 35) = P (Z < z)
35 − 38.72
=P Z<
(
3.17 )
= P(Z < − 1.17)
= 0.1210
−2 −1
0
1
Z
2
−1.17
The proportion of the pediatrician’s three-year-old
females who are less than 35 inches tall is 0.1210 or
12.10%.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Step 2: Used Excel to determine the area under
any normal curve.
Technology Approach!
Use “TRUE”
for cumulative
since we want
the area under
the normal
curve.
The proportion of the
pediatrician’s threeyear-old females who
are less than 35 inches
tall is 0.1210 or 12.10%.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example 3:
A pediatrician obtains the heights of her three-year-old female
patients. The heights are approximately normally distributed,
with mean 38.72 inches and standard deviation 3.17 inches.
Determine the probability that a randomly selected three-yearold girl is between 35 and 40 inches tall, inclusive.
Solution:
Given: μ = 38.72,σ = 3.17, and 35 ≤ X ≤ 40
Area = P(35 ≤ X ≤ 40)
Step 1: Draw a normal curve and
shade the desired area.
Polytechnic University of the Philippines
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35.55 38.72 41.89
40
35
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X
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Using Table 1 By-hand Approach!
Step 2: Convert the value of x to a z-score.
P(35 ≤ X ≤ 40) = P(z ≤ Z ≤ z)
40 − 38.72
35 − 38.72
≤Z≤
=P
( 3.17
3.17 )
= P(−1.17 ≤ Z ≤ 0.40)
= P(Z ≤ 0.40) − [1 − P(Z ≥ − 1.17)]
= 0.6554 − [1 − 0.8790] Area = P(−1.17 ≤ Z ≤ 0.40)
= 0.6554 − 0.1210
= 0.5344
The probability a randomly
selected three-year-old female
is between 35 and 40 inches tall
is 0.5344.
X
−2 −1 0 1 2
−1.17
0.40
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Using Table 2 By-hand Approach!
Step 2: Convert the value of x to a z-score.
P(35 ≤ X ≤ 40) = P(z ≤ Z ≤ z)
35 − 38.72
40 − 38.72
=P
≤Z≤
( 3.17
3.17 )
= P(−1.17 ≤ Z ≤ 0.40)
= [0.50 − P(Z ≥ 0.40) + [0.50 − P(Z ≤ − 1.17)]
= [0.50 − 0.3446] + [0.50 − 0.1210]
= 0.1554 + 0.3790
Area = P(−1.17 ≤ Z ≤ 0.40)
= 0.5344
The probability a randomly selected
three-year-old female is between 35
and 40 inches tall is 0.5344.
−2 −1
Polytechnic University of the Philippines
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−1.17
0
1
2
X
0.40
Step 2: Used Excel to determine the area under
Technology Approach!
any normal curve.
Use “TRUE” for
cumulative since
we want the area
under the normal
curve.
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ACTIVITIES/ASSESSMENTS:
1. Which one do you think is more informative?
Why?
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ACTIVITIES/ASSESSMENTS:
2. What features
of the ‘Good
Presentation’
make it better
than the ‘Bad
Presentation’?
A.
Polytechnic University of the Philippines
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B.
ACTIVITIES/ASSESSMENTS:
3. Review the table and consider questions such as the
following.
Needs
Satisfactory
Improvement
Origin / Rating
Poor
V Good
Excellent
Total
External
0%
2%
12%
19%
9%
41%
Internal
4%
8%
15%
23%
9%
59%
Grand Total
4%
10%
27%
41%
17%
100%
1. What percentage of the employees originated from within the
organization?
2. What percentage of the employees are both internal and rated
‘Very Good’?
3. What percentage of the employees received ‘Needs Improvement’
or ‘Poor’?
4. What category contains the greatest number of employees?
5. Do you see any notable differences in the percentage by category?
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ACTIVITIES/ASSESSMENTS:
4. Consider the above Frequency Distribution of
Salaries.
Salary
41,000 - 50,000
51,000 - 60,000
61,000 - 70,000
71,000 - 80,000
81,000 - 90,000
91,000 - 100,000
101,000 - 110,000
Total
Frequency
1
20
53
43
26
6
1
150
Percentage
1%
13%
35%
29%
17%
4%
1%
100%
1.What percentage of the employees earns less than or
equal 80,000?
2.What is the salary range of values?
3.What salary categories have percentage less than 5?
4.What salary category includes the most employees?
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ACTIVITIES/ASSESSMENTS:
5. The length of life of an instrument produced by a machine has a normal
distribution with a mean of 12 months and standard deviation of 2 months.
Find the probability that an instrument produced by this machine will last
A. less than 7 months.
B. between 7 and 12 months.
Be sure to draw a normal curve with the area corresponding to the
probability shaded.
6. The lengths of human pregnancies are approximately normally distributed,
with mean μ = 266 days and standard deviation σ = 16 days.
What proportion of pregnancies lasts more than 270 days?
B. What proportion of pregnancies lasts less than 250 days?
C. What proportion of pregnancies lasts between 240 and 280 days?
D. What is the probability that a randomly selected pregnancy?
lasts more than 280 days?
Be sure to draw a normal curve with the area corresponding to the
probability shaded.
Polytechnic University of the Philippines
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ACTIVITIES/ASSESSMENTS:
7. Construct frequency distribution table
scores of 75 randomly selected students.
37 46 37 26 30 41 28 49 29 34 46
35 46 45 27 41 26 45 39 43 46 36
49 47 30 43 31 34 38 41 39 45 28
38 30 29 38 26 31 42 44 48 43 37
42 33 42 42 43 39 39 31 46 46 48
Scores
26 to 30
31 to 35
36 to 40
41 to 45
46 to 50
Total
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based on the
50
32
43
46
48
38
46
37
38
50
Frequency Percentage (%)
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35
36
39
27
45
42
48
26
50
31
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ACTIVITIES/ASSESSMENTS:
A. Based on the frequency distribution, compute measures of
central tendency, measures of variation, Q1, D9, P10 , Skewness
and kurtosis.
B. Based on the raw data, compute measures of central
tendency, measures of variation, Skewness and kurtosis using
Excel.
C. Compute Skewness and kurtosis of grouped and ungrouped
data. Make sure to describe the shape of the distribution
D. Do you think that computed value for grouped and
ungrouped data are the same?
8. Begin with the following set of data, call it Data Set I.
5, −2, 6, 14, −3, 0, 1, 4, 3, 2, 5
A. Compute the sample standard deviation and sample mean of
Data Set I.
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ACTIVITIES/ASSESSMENTS:
B. Form a new data set, Data Set II, by adding 3 to each
number in Data Set I. Calculate the sample standard deviation
and sample mean of Data Set II.
C. Form a new data set, Data Set III, by subtracting 6 from
each number in Data Set I. Calculate the sample standard
deviation and sample mean of Data Set III.
D. Comparing the answers to parts (a), (b), and (c), can you
guess the pattern? State the general principle that you expect
to be true.
9.Using “Encoded Data file”, construct frequency distribution
table for age, sex, marital status and educational attainment
and interpret the table.
Polytechnic University of the Philippines
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References
https://prezi.com/rirrca9ckuiz/textualpresentation-of-data/
https://www.toppr.com/guides/economics/
presentation-of-data/textual-and-tabularpresentation-of-data/
Statistics. Informed Decision using Data by
Michael Sullivan, III,. Fifth Edition
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MODULE 4: INFERENTIAL STATISTICS
OBJECTIVES:
After successful completion of this module, you should be
able to:
✦ Differentiate the null and alternative hypotheses.
✦ Formulates
the appropriate null and alternative
hypotheses.
✦ Explain the logic of hypothesis testing.
✦ Assess and test if the data follows a normal distribution.
✦ Distinguish
between independent and dependent
sampling.
✦ Identify
the appropriate test statistics for normally
distributed data.
✦ Conduct test for two categorical variables.
Polytechnic University of the Philippines
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What is HYPOTHESIS TESTING?
Hypothesis testing is a procedure on sample
evidence and probability, used to test claims
regarding a characteristic of one or more populations.
What is HYPOTHESIS?
•A
statement or claim regarding a characteristic of
one or more populations.
•A
preconceived idea, assumed to be true but has to
be tested for its truth or falsity.
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Procedures for Testing
Hypothesis
1. State the null and alternative hypothesis.
2. Set the level of significance or alpha level (α).
3. Determine the test distribution to use.
4. Calculate test statistic or p - value.
5. Make statistical Decision
6. Draw Conclusion
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1. State the Null and Alternative Hypothesis
Two Types of Hypothesis
1. Null Hypothesis
Denoted by
The statement being tested.
Assumed true until evidence indicates otherwise.
Must contain the condition of equality and must be written
with the symbol = , ≤ , or ≥.
•
•
•
•
2. Alternative Hypothesis
•
•
•
•
Denoted by
Statement that must be true if the null hypothesis is false
Sometimes referred to as the research hypothesis
Must contain the condition of equality and must be written
with the symbol ≠, < or >.
Polytechnic University of the Philippines
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Example Hypothesis:
✦
✦
✦
Null Hypothesis:
Students who eat and not eat breakfast will perform the same on
a math exam.
Students who experience and not experience test anxiety prior to
an English exam will get the same scores.
Motorists who talk and not talk on the phone while driving will
get the same errors on a driving course.
Alternative Hypothesis:
✦
✦
✦
Students who eat breakfast will perform better on a math exam
than students who do not eat breakfast.
Students who experience test anxiety prior to an English exam
will get higher scores than students who do not experience test
anxiety.
Motorists who talk on the phone while driving will be more likely
to make errors on a driving course than those who do not talk on
the phone.
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Reminders:
If you are conducting a research study and you want
to use a hypothesis test to support your claim, the
claim must be stated in such a way that it becomes
the alternative hypothesis, so it cannot contain the
condition of equality.
Two Types of Alternative Test
1. One - tailed test
✦ Left tailed
✦ Right tailed
2. Two - tailed test
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2. Set the Level of Significance or Alpha Level (α)
You should establish a predetermined level of
significance, below which you will reject the null
hypothesis.
• The generally accepted levels are 0.10, 0.05, and 0.01.
• Be as rigorous as possible.
Two Types of Error
•
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Example:
H0: The defendant is innocent.
Ha: The defendant is not innocent.
What happen to the defendant if the jury made type I
and type II error?
Answer:
A type I error is like putting an innocent person in
jail.
A type II error is like letting a guilty person go free.
Polytechnic University of the Philippines
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Reminders:
It is important to note that we want to set
( α ) before we start our study because the
Type I error is the more ‘grevious’ error to
make.
The smaller (α ) is, the smaller the region
of rejection.
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3. Determine the Test Distribution to Use.
Determine the appropriate statistical test to
be used.
✦
Dependent Sample t - Test
✦
Independent Sample t - Test
✦
One Way Analysis of Variance
(ANOVA) Test
Pearson r
✦
✦
Chi - Square Test
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4. Calculate Test Statistic or p - value.
Performing statistical analysis using statistical
software such as Excel, SPSS, R, Minitab, SAS,
etc.
5. Make Statistical Decision
✦
Using confidence interval
✦
Using p-value approach
✦
Using traditional method
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Decision Rule:
✦ Using Confidence Interval
Reject the null hypothesis if the test statistic is not within
the range specified by the confidence interval.
✦
Using Traditional Approach
Reject Ho if the computed value of the test statistic falls in
the region of rejection.
✦
Using P-value Approach
Reject the null hypothesis if the computed p-value is less
than or equal to the set significance level , otherwise do not
reject the null hypothesis.
Example: If the level of significance (α = 0.05),
P-value
Decision
Reject H0
0.01
0.05
Reject H0
Failed to Reject H0
0.10
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Traditional Approach
Rejection of region
or critical region is
the set of all values of
the test statistic
which will lead to the
rejection of H0.
Acceptance Region is
the set of all values of
the test statistic that
leads the researcher to
retain H0.
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One-tailed and Left tailed One-tailed and Right tailed
Ha : μ1 < μ2
Ha : μ1 > μ2
Rejection Region
Rejection Region
-2
0
2
-2
0
2
Two-tailed
Ha : μ1 ≠ μ2
Rejection Region
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-2
Rejection Region
0
2
In stating your decision you can use:
✦ Fail to reject the null hypothesis/ Do not reject
the null hypothesis/ Retain the null hypothesis
✦ Reject the null hypothesis.
It is important to recognize that we never accept
the null hypothesis. We are merely saying that the
sample evidence is not strong enough to warrant
rejection of the null hypothesis.
6. Draw Conclusion
Record conclusions and recommendations in a report,
and associate interpretations to justify your
conclusion or recommendations.
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Assessing and Testing Normality
of the Data
To determine if the data is follows a normality
distribution, we can use the graphical or
numerical method.
Graphical:
Normal Q-Q Plot
Histogram
Numerical:
Shapiro Wilk Test
Kolmogorov Smirnov Test
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How to Check Normality?
Histogram plots the observed values against their
frequency, states a visual estimation whether the
distribution is bell shaped or not.
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How to Check Normality?
Q-Q probability plots display the observed values
against normally distributed data (represented by the
line).
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Reminders:
Graphical methods are typically not
very useful when the sample size is
small.
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Hypotheses of Normality Test
The hypotheses used are:
Ho: The sample data follows a normal distribution.
Ha: The sample data does not follow a normal
distribution.
When we are testing normality:
• If P value > alpha, it means that the data are
normal.
• If P value ≤ alpha, it means that the data are NOT
normal.
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How to Calculate Shapiro - Wilk Test in Excel?
Sample Data
STEP 1:
Rearrange
the data in
ascending
order.
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STEP 2: Calculate SS as follows:
n
SS =
x − x̄)
∑( i
2
i=1
Use "=DEVSQ( )”
function in excel
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and Statistics
SS means Sum of Square
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and Statistics
m
STEP 3: Calculate b as follows: b =
a x
− xi)
∑ i ( n+1−i
i=1
n is the number of
observation
If n is even:
n
m=
2
If n is odd:
n−1
m=
2
Since n is even in this
example, m=8. That’s
why we used a1 to a8
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Taking the ai weights from
the table of Shapiro -Wilk
(based on the value of n)
Shapiro - Wilk Table
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Note that if n is odd, the median
data value is not used in the
calculation of b.
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STEP 4: Calculate the test statistic:
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b2
W=
SS
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STEP 5:
Find the value in the table of Shapiro - Will (for a
given value of n) that is closest to W, interpolating if
necessary. This is the p-value for the test.
Polytechnic University of the Philippines
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We choose this
interval in the table of
Shapiro - Wilk,
because our n=16 and
our test statistic
(W=0.955) is within
this interval.
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
We used interpolation to get the
p-value of Shapiro-Wilk Test
Result
Since the computed p-value is greater than the set
level of significance, we failed to reject the null
hypothesis. Therefore, the sample data follows a
normal distribution.
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Inferential Statistics
1. Parametric Tests
Assume underlying statistical distributions in the data.
Therefore, several conditions of validity must be met
so that the result of a parametric test is reliable.
✦ Apply to data in ratio scale, and some apply to data in
interval scale.
2. Non Parametric Test
✦ Refer to a statistical method in which the data is not
required to fit a normal distribution.
✦ Most non-parametric tests apply to data in an ordinal
scale, and some apply to data in nominal scale.
✦
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Inference About Two Means
To perform inference on the difference of two
population means, we must first determine whether the
data come from an independent or dependent sample.
Distinguish between Independent and Dependent Sample
✦
✦
A sampling method is independent when the
individuals selected for one sample do not dictate
which individuals are to be in a second sample.
A sampling method is dependent when the individual
selected to be in one sample are used to determine the
individuals to be in the second sample.
Polytechnic University of the Philippines
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Example:
Determine whether the sample is independent or dependent.
1. An urban economist believes that commute times to
work in the South are less than commute times to work
in the Midwest. He randomly selects 40 employed
individuals in the south and 45 employed individuals in
the Midwest and determines their commute times.
Answer: Independent
2. In an experiment conducted in biology class, Prof.
Rhea measured the time required for 12 students to
catch a failing meter stick using their dominant hand
and nondominant hand. The goal of the study was to
determine whether the reaction time in an individual’s
dominant hand is different from the reaction time in
the non dominant hand. Answer: Dependent
Polytechnic University of the Philippines
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Example:
Determine whether the sample is independent or
dependent.
3. A researcher wants to know if the mean
length of stay in for-profit hospitals is different
from the mean length of stay in not-for-profit
hospitals. He randomly selected 20 individuals in
the for-profit hospital and matched them with 20
individuals in the not-for-profit by diagnosis.
Answer:
Dependent
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Dependent Sample t - Test
The dependent sample t-test (also called
the paired t-test or paired-samples t-test)
compares the means of two related groups
to determine whether there is a statistically
significant difference between these
means.
H0 : μ1 ≥ μ2 and Ha : μ1 < μ2
H0 : μ1 ≤ μ2 and Ha : μ1 > μ2
H0 : μ1 = μ2 and Ha : μ1 ≠ μ2
Polytechnic University of the Philippines
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Assumptions
1. Your dependent variable should be measured at
the interval or ratio level (i.e., they are
continuous).
2. Your independent variable should consist of two
categorical, "related groups" or "matched pairs”.
3. There should be no significant outliers in the
differences between the two related groups.
4. The distribution of the differences in the
dependent variable between the two related
groups should be approximately normally
distributed.
Polytechnic University of the Philippines
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Example:
A teacher is interested to know if the new learning program
will help to increase the number of correct remembered
words. 10 Subjects learn a list of 50 words. Learning
performance is measured using a recall test.
After the first test all subjects
are instructed how to use the
learning program and then
learn a second list of 50 words.
Learning performance is again
measured with the recall test. In
the following table the number
of correct remembered words
are listed for both tests.
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1. State the Null and Alternative
Hypothesis
Null hypothesis: Ho : μ1 ≥ μ2
The new learning program will not help to increase
the number of correct remembered words.
Alternative hypothesis: Ha : μ1 < μ2
The new learning program will help to increase the
number of correct remembered words.
2. Set the Level of Significance or Alpha
Level (α)
α = 0.05
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3. Determine the Test
Distribution to Use.
Dependent Variable:
Number of correct remembered words
Independent Variable:
Treatment (Before and After)
Since we are comparing the means of two
related groups, we will use the dependent
sample t-test.
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4. Calculate Test Statistic or
p - value.
Click “Data”, then click “Data Analysis”
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5. Make Statistical Decision
Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Reject Ho
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6. Draw Conclusion
There is sufficient evidence to support that the new
learning program help to increase the number of
correct remembered words.
Proper Presentation of Results
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Exercises:
Apply the procedure in testing the hypothesis.
Professor Rhea measured the time (in second) required to
catch a falling meter sticks for 10 randomly selected
students' dominant hand and non-dominant hand. Professor
Rhea claims that the reaction time in an individual's
dominant hand is less than the reaction time in
their non-dominant hand.
Test the claim at the level
of significance. The data
obtained are presented:
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Result
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Independent Sample t - Test
The independent sample t - test allows
researchers to evaluate or to compare the mean
difference between two populations using the data
from two separate samples. It is used to test
whether population means are significantly
different from each other, using the means from
randomly drawn samples.
H0 : μ1 ≥ μ2 and Ha : μ1 < μ2
H0 : μ1 ≤ μ2 and Ha : μ1 > μ2
H0 : μ1 = μ2 and Ha : μ1 ≠ μ2
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Assumptions
1.
2.
3.
4.
5.
6.
Your dependent variable should be measured on a
continuous scale (i.e., it is measured at the interval or
ratio level).
Your independent variable should consist of two
categorical, independent groups.
You should have independence of observations, which
means that there is no relationship between the
observations in each group or between the groups
themselves.
There should be no significant outliers.
Your dependent variable should be approximately
normally distributed for each group of the independent
variable.
There needs to be homogeneity of variances.
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Example:
Researchers wanted to know whether there was a difference in
comprehension among students learning a computer program
based on the style of the text. They randomly divided 18
students into two groups of 9 each. The researchers verified
that the 18 students were similar in terms of educational level,
age, and so on. Group 1 individuals learned the software using
visual manual (multimodal
instruction), while Group 2
individual learned the software
using textual manual (Unimodal
instruction). The following data
represent scores the students
received on an exam given to them
they studied from the manuals.
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1. State the Null and Alternative
Hypothesis
Null hypothesis: Ho : μ1 = μ2
There is no significant difference between the scores of the
students learning computer program using textual and
visual style.
Alternative hypothesis: Ha : μ1 ≠ μ2
There is significant difference between the scores of the
students learning computer program using textual and
visual style.
2. Set the Level of Significance or Alpha
Level (α)
α = 0.05
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3. Determine the Test
Distribution to Use.
Dependent Variable:
Scores
Independent Variable:
Style of the Text (Visual and Textual)
Since we are comparing the means of two
independent groups, we will use the
independent sample t-test.
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Click “Data”, then click “Data Analysis”
Determine if the
variances are equal
or not equal.
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Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Ho: Equal Variances Assumed
Ha: Equal Variances Not Assumed
Failed to
Reject Ho
Since we failed to reject Ho, we will proceed to t-test: Two
Sample Assuming Equal Variances.
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4. Calculate Test Statistic or
p - value.
Click “Data”, then click “Data Analysis”
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Result
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5. Make Statistical Decision
Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Polytechnic University of the Philippines
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Failed to
Reject Ho
6. Draw Conclusion
There is no enough evidence to support that
there is a difference in comprehension among
students learning a computer program based on
the style of the text.
Proper Presentation of Results
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Exercises:
Apply the procedure in testing the hypothesis.
Twenty participants were given a list of 20 words to
process. The 20 participants were randomly assigned to
one of two treatment conditions. Half were instructed to
count the number of vowels in each word (shallow
processing). Half were instructed to judge whether the
object described by each word would be useful if one
were stranded on a desert island (deep processing).
After a brief distractor task, all subjects were given a
surprise free recall task. Did the instruction affect the
level of recall?The number of words correctly recalled
was recorded for each subject. Here are the data:
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Polytechnic University of the Philippines
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Since the result of F-test conclude that the
variances of the two groups are equal, we will
apply “Assuming Equal Variances”.
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Result
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One - Way Analysis of Variance
(ANOVA)
One-way analysis of variance (ANOVA)
is a method of test ing the equality of
three or more population means by
analyzing sample variances.
Ho : μ1 = μ2 = . . . = μk
Ha : At least one of the population means
is different from the others.
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Assumptions
1. Your dependent variable should be measured at the
interval or ratio level (i.e., they are continuous).
2. Your independent variable should consist of two or more
categorical, independent groups.
3. You should have independence of observations, which
means that there is no relationship between the
observations in each group or between the groups
themselves.
4. There should be no significant outliers.
5. Your dependent variable should be approximately
normally distributed for each category of the independent
variable.
6. There needs to be homogeneity of variances.
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Example:
A Researchers wanted to compare math test scores of
students at the end of secondary school from various cities.
Eight randomly selected students from Makati, Manila,
and Quezon City each were administered the same exam;
the results are presented in the following table. Can the
researchers conclude
that the distribution of
exam scores is different
for each city at the
level of significance?
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
1. State the Null and Alternative
Hypothesis
Null hypothesis:
There is no significant difference between the
mathematics scores of students at various city.
Alternative hypothesis:
There is significant difference between the
mathematics scores of students at various city.
2. Set the Level of Significance or Alpha
Level (α)
α = 0.10
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
3. Determine the Test
Distribution to Use.
Dependent Variable:
Mathematics Scores
Independent Variable:
Cities (Makati, Manila, Quezon City)
Since we are comparing the means of one
independent variable that consist of two
or more categorical groups, we will use
the one-way ANOVA.
Polytechnic University of the Philippines
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Click “Data”, then click “Data Analysis”
Determine if the
variances are equal
or not equal.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Ho: Equal Variances Assumed
Ha: Equal Variances Not Assumed
Failed to
Reject Ho
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
E q u a l
Variances
Assumed
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Ho: Equal Variances Assumed
Ha: Equal Variances Not Assumed
Failed to
Reject Ho
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
E q u a l
Variances
Assumed
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Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Ho: Equal Variances Assumed
Ha: Equal Variances Not Assumed
Failed to
Reject Ho
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
E q u a l
Variances
Assumed
4. Calculate Test Statistic or
p - value.
Click “Data”, then click “Data Analysis”
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Result
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
5. Make Statistical Decision
Using p-value approach: If pvalue ≤ α , reject Ho,
otherwise failed to reject Ho
Reject Ho
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
6. Draw Conclusion
There is enough evidence to support that the
distribution of exam scores of students in
mathematics is different for each city.
Proper Presentation of Results
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Exercises:
Apply the procedure in testing the hypothesis.
A teacher is concerned about the level of
knowledge possessed by PUP students regarding
Philippine history. Students completed a high
school senior level standardized history exam.
Academic major of the students was also recorded.
Data in terms of percent correct is recorded below
for 24 students. Is there a significant difference
between the levels of knowledge possessed by PUP
students regarding Philippine history when
grouped according to their academic major?
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Result
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Pearson Product Moment
Correlation
The Pearson product moment correlation
coefficient (Pearson r) is a measure of the
strength of a linear association between
two variables and is denoted by r.
Ho: There is no significant relationship
between two continuous variables.
Ha: There is significant relationship between
two continuous variables.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Features of r
•
Unit free
•
Range between -1 and 1
•
The closer to -1, the stronger the negative
linear relationship.
•
The closer to 1, the stronger the positive
linear relationship.
•
The closer to 0, the weaker the linear
relationship.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Pearson Product Moment
Correlation
If r is positive, the correlation is direct.
If r is negative, the correlation is inverse.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Sample of Observations from
Various r Values
Y
Y
Y
X
r = -1
X
r = -.6
Y
X
r =0
Y
r = .6
r=1
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Reminders:
•
Correlation does not imply causation.
•
Watch out for hidden (lurking) variables.
Lurking Variable
•
A variable that is not included as an explanatory
or response variable in the analysis but can affect
the interpretation of relationships between
variables.
•
Can falsely identify a strong relationship between
variables or it can hide the true relationship.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Assumptions
1. Your two variables should be measured at the
interval or ratio level (i.e., they are
continuous).
2. There is a linear relationship between your
two variables.
3. There should be no significant outliers.
4. Your variables should be approximately
normally distributed.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Significance Testing of Pearson r
Test Statistic:
t=r
df
1 − r2
where:
df = degrees of freedom
r = correlation coefficient of Pearson r
Note:
df = n − 2
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example:
A dietetics student wanted to look at the
relationship between calcium intake and
knowledge about calcium in sports
science students. Table shows the data
she collected. Is there a relationship
between calcium intake and knowledge
about calcium in sports science
students?
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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1. State the Null and Alternative
Hypothesis
Null hypothesis:
There is no significant relationship between the
calcium intake and knowledge about calcium in sports
science students.
Alternative hypothesis:
There is significant relationship between the calcium
intake and knowledge about calcium in sports science
students.
2. Set the Level of Significance or Alpha
Level (α)
α = 0.0.5
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
3. Determine the Test
Distribution to Use.
Dependent Variable:
Calcium Intake
Independent Variable:
Knowledge about Calcium
Since we are testing the significant
relationship of two variables, we will use
Pearson r.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
4. Calculate Test Statistic or p - value.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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t=r
df
1 − r2
df = n − 2
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Result
Polytechnic
University
of the Philippines
Polytechnic
University
of the Philippines
College
of Science
College
of Science
Department
of Mathematics
and Statistics
Department
of Mathematics
and Statistics
5. Make Statistical Decision
Using p-value approach: If pvalue ≤ α ,
reject Ho, otherwise failed to reject
Strong and
Ho
D i r e c t
Correlation
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Reject Ho
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6. Draw Conclusion
There is sufficient evidence to conclude that there
is significant relationship between the calcium
intake and knowledge about calcium in sports
science students.
Proper Presentation of Results
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Exercises:
Apply the procedure in testing the hypothesis.
A group of twelve children participated in a
psychological study designed to assess the
relationship, if any, between age (years)
and average total sleep time (minutes). To
obtain a measure for average total sleep
time, recordings were taken on each child
on five consecutive nights and then
averaged. The results obtained are shown in
the table.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Result
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Chi-Square Distribution
Definition:
The chi-square distribution is
written as χ 2 distribution.
The symbol χ is the Greek letter
“chi”, pronounced as “ki”.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Chi - Square: Test for
Independence
✦
Used to discover if there is association
between two categorical variables.
✦
Used when you want to decide whether
two variables are independent or
dependent.
✦
A contingency table will be constructed.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Chi - Square: Test for
Independence
H0: The two categorical variables are
independent.
Ha: The two categorical variables are
dependent.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Chi - Square: Test for
Independence
The test statistic for a test of independence is given
by
(O − E)2
2
χ =
∑
E
where:
O is the observed frequency for a category
E is the expected frequency for a category
E=
(row total)(column total)
grand total
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Observed and Expected Frequencies
The frequencies obtained from the performance of an
experiment are called the observed frequencies and are
denoted by O.
The expected frequencies, denoted by E, are the
frequencies that we expect to obtain if the null hypothesis is
true.
Example of Contingency Table:
Observed Values
Some College
Bachelor's Degree
Masters Degree
Column Total
Low
20
17
11
48
Medium High
35
20
33
25
18
21
86
66
Row Total
80
70
50
200
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Assumptions
1. There are 2 variables, and both are measured as
categories, usually at the nominal level.
2. The two variables should consist of two or more
categorical, independent groups.
3. The data in the cells should be frequencies, or counts
of cases rather than percentages or some other
transformation of the data.
4. For a 2 by 2 table, all expected frequencies > 5.
5. For a larger table, all expected frequencies > 1 and
no more than 20% of all cells may have expected
frequencies < 5.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Example:
1. A doctor who knows that hypertension depends
on smoking habits can tell his smoking patients what
they should do.
2. If the traffic condition (light, moderate, heavy,
standstill) is found to be dependent on vehicle plate
numbers (odd, even) a traffic officer may decide to
revise traffic law enforcement.
3. If poverty status of households is found to be
correlated with family size, government ought to
adopt a viable poverty management program
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Reminders:
The word contingency refers to
dependence, but this is only a
statistical dependence and cannot be
used to establish a direct cause-andeffect link between the two variables in
question.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Example:
Educators are always looking for novel ways in
which to teach statistics to undergraduates as part
of a non-statistics degree course (e.g., psychology).
With current technology, it is possible to present
how-to guides for statistical programs online
instead of in a book. However, different people
learn in different ways. An educator would like to
know whether gender (male/female) is associated
with the preferred type of learning medium (online
vs. books). Use “Data_Example and Exercises file”.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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1. State the Null and Alternative
Hypothesis
Null hypothesis:
Gender is independent with the preferred type of
learning medium.
Alternative hypothesis:
Gender is dependent with the preferred type of
learning medium.
2. Set the Level of Significance or Alpha
Level (α)
α = 0.0.5
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
3. Determine the Test
Distribution to Use.
Two Categorical Variables
Gender (Male and Female)
Preferred type of learning medium
(online vs. books)
Since we are testing the significant
relationship of two categorical variables,
we will use Chi-square test.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
4. Calculate Test Statistic or
p - value.
Click “Insert”, then click “Pivot Table”
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Row Total
Grand Total
Column Total
E=
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
(row total)(column total)
grand total
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Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
5. Make Statistical Decision
Using p-value approach: If pvalue ≤ α, reject Ho,
otherwise failed to reject Ho
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Reject Ho
6. Draw Conclusion
There is sufficient evidence to conclude that there
gender is associated with the preferred type of
learning medium.
Proper Presentation of Results
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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Exercises:
Apply the procedure in testing the hypothesis.
A survey was conducted at a community college of 102
randomly selected students who dropped a course in the
current semester to learn why students drop courses.
Personal drop reasons include financial, transportation,
family issues, health issues, and lack of child care. Course
drop reasons include reducing ones load, being unprepared
for the course, the course was not what was expected,
dissatisfaction with teaching, and not getting the desired
grade. Work drop reasons include an increase in hours, a
change in shift, and obtaining full-time employment. Test
whether gender is independent of drop reason at the 1%
level of significance. Use “Data_Example and Exercises
file”.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Result
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
ACTIVITIES/ASSESSMENTS:
Determine whether the sampling is dependent or independent.
________1. A researcher wishes to compare academic
aptitudes of married mathematicians and their spouses. She
obtains a random sample of 287 such couples who take an
academic aptitude test and determines each spouses academic
aptitude.
________2. A political scientist wants to know how a random
sample of 18- to 25-year-olds feel about Democrats and
Republicans in Congress. She obtains a random sample of
1030 registered voters 18 to 25 years of age and asks, Do you
have favorable/unfavorable opinion of the Democratic/
Republican party? Each individual was asked to disclose his
or her opinion about each party.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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ACTIVITIES/ASSESSMENTS:
________3.
An educator wants to determine whether a new
curriculum significantly improves standardized test scores for third
grade students. She randomly divides 80 third-graders into two
groups. Group 1 is taught using the new curriculum, while group 2 is
taught using the traditional curriculum. At the end of the school year,
both groups are given the standardized test and the mean scores are
compared.
________4. A stock analyst wants to know if there is difference
between the mean rate of return from energy stocks and that from
financial stocks. He randomly select 13 energy stocks and computes
the rate of return for the past year. He randomly selects 13 financial
stocks and compute the rate of return for the past year.
________5. An urban economist believes that commute times to work
in the South are less than commute times to work in the Midwest. He
randomly selects 40 employed individuals in the south and 45
employed individuals in the Midwest and determines their commute
times.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
ACTIVITIES/ASSESSMENTS:
Solve the following problems. Make sure to follow the 6 steps
procedure.
1. A study is designed to test whether there is a difference in mean daily
calcium intake in adults with normal bone density, adults with
osteopenia (a low bone density which may lead to osteoporosis) and
adults with osteoporosis. Adults 60 years of age with normal bone
density, osteopenia and osteoporosis are selected at random from
hospital records and invited to participate in the study. Each
participant's daily calcium intake is measured based on reported food
intake and supplements. The data are shown below.
I s t h e r e a s t a t i s t i c a l l y Normal Bone Osteopenia Osteoporosis
Density
1200
1000
890
significant difference in mean
1000
1100
650
calcium intake in patients
with normal bone density as
980
700
1100
compared to patients with
900
800
900
osteopenia and osteoporosis?
750
500
400
800
700
350
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
ACTIVITIES/ASSESSMENTS:
2. Some studies have shown that in the United
States, men spend more than women buying gifts
and cards on Valentine’s Day. Suppose a researcher
wants to test this hypothesis by randomly sampling
nine men and 10 women with comparable
demographic characteristics from various large cities
across the United States to be in a study. Each study
participant is asked to keep a log beginning one
month before Valentine’s Day and record all
purchases made for Valentine’s Day during that onemonth period. The resulting data are shown below.
Use these data and a 1% level of significance to test
to determine if, on average, men actually do spend
significantly more than women on Valentine’s Day.
Assume that such spending is normally distributed
in the population and that the population variances
are equal.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
Men
(in $)
Women
(in $)
107.48 125.98
143.61
45.53
90.19
56.35
125.53
80.62
70.7
46.37
83
44.34
129.63
75.21
154.22
68.48
93.8
85.82
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126.11
lOMoARcPSD|29729911
ACTIVITIES/ASSESSMENTS:
3. A researcher is interested whether a training course increases
the teaching performance of the teachers who attended the
training courses. Test at 10% level of significance. The data are
shown below:
Case Before After
1
85
95
2
84
98
3
86
97
4
87
92
5
89
96
6
82
93
7
80
94
8
84
95
9
86
90
10
82
82
Case Before After
11
89
97
12
87
98
13
82
95
14
81
95
15
86
92
16
89
91
17
89
94
18
84
95
19
85
96
20
88
97
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
ACTIVITIES/ASSESSMENTS:
4. A pediatrician wants to
determine the relation that may
exist between a child’s height
and head circumference. She
randomly selects eleven 3yearold children from her
practice, measures their heights
and head circumference, and
obtains the data shown in the
table below.
Height
(inches)
Head
Circumference
(inches)
27.75
24.5
25.5
26
25
27.75
26.5
27
26.75
26.75
27.5
17.5
17.1
17.1
17.3
16.9
17.6
17.3
17.5
17.3
17.5
17.5
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
ACTIVITIES/ASSESSMENTS:
5. The following data represent the smoking status from a
random sample of 1054 U.S. residents 18 years or older by
level of education.
Smoking Status
No. Of Years
of Education
Current
Former
Never
Less than 12
178
88
208
12
137
69
143
13 - 15
44
25
44
16 or more
34
33
51
Test whether smoking status and level of education are
independent at the α = 0.05 level of significance.
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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ACTIVITIES/ASSESSMENTS:
6. A pediatrician wants to
determine the relation that may
exist between a child’s height
and head circumference. She
randomly selects eleven 3yearold children from her
practice, measures their heights
and head circumference, and
obtains the data shown in the
table below.
Height
(inches)
Head
Circumference
(inches)
27.75
24.5
25.5
26
25
27.75
26.5
27
26.75
26.75
27.5
17.5
17.1
17.1
17.3
16.9
17.6
17.3
17.5
17.3
17.5
17.5
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
References
h t t p s : / / w o l f w e b . u n r. e d u / h o m e p a g e / a n i a /
stat352f12lectures/352lecture21f12.pdf
Statistics. Informed Decision using Data by
Michael Sullivan, III,. Fifth Edition
http://www.real-statistics.com/tests-normalityand-symmetry/statistical-tests-normalitysymmetry/shapiro-wilk-test/
Polytechnic University of the Philippines
College of Science
Department of Mathematics and Statistics
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