Lesson 2 Statistics Benefits, Risks, and MeasurementsAssignmen.docx

Lesson 2: Statistics: Benefits, Risks, and Measurements
Assignments
· See your Course Syllabus for the reading assignments.
· Work through the Lesson 2 online notes that follow.
· Complete the Practice Questions and Lesson 2 Assignment.
Learning Objectives
Chapters 1 and 3
After successfully completing this lesson, you should be able
to:
· Identify the three conditions needed to conduct a proper study.
· Apply the seven pitfalls that can be encountered when asking
questions in a survey.
· Distinguish between measurement variables and categorical
variables.
· Distinguish between continuous variables and discrete
variables for those that are measurement variables.
· Distinguish between validity, reliability, and bias.
Terms to Know
From Chapter 1
· statistics
· population
· sample
· observational study
· experiment
· selection bias
· nonresponse bias
From Chapter 3
· data (variable)
· categorical variables
· measurement variables
· measurement (discrete) variables
· measurement (continuous) variables
· validity
· reliability

· bias
2.1 What is Statistics?
Section 2.1. Chapter 1
Overview
What is statistics? If you think statistics is just another math
course with many formulas and lifeless numbers, you are not
alone. However, this is a myth that hopefully will be debunked
as you work through this course. Statistics is about data. More
precisely, statistics is a collection of procedures and principles
for gaining and processing information from collected data.
Knowing these principles and procedures will help you make
intelligent decisions in everyday life when faced with
uncertainty. The following examples are meant to illuminate the
definition of statistics.
Example 2.1. Angry Women
Who are those angry women? (Streitfield, D., 1988 and Wallis,
1987.) In 1987, Shere Hite published a best-selling book
called Women and Love: A Cultural Revolution in Progress.
This 7-year research project produced a controversial 922-page
publication that summarized the results from a survey that was
designed to examine how American women feel about their
relationships with men. Hite mailed out 100,000 fifteen-page
questionnaires to women who were members of a wide variety
of organizations across the U.S. These organizations included
church, political, volunteer, senior citizen, and counseling
groups, among many others. Questionnaires were actually sent
to the leader of each organization. The leader was asked to
distribute questionnaires to all members. Each questionnaire
contained 127 open-ended questions with many parts and
follow-ups. Part of Hite’s directions read as follows: “Feel free
to skip around and answer only those questions you choose.”
Approximately 4500 questionnaires were returned. Below are a
few statements from this 1987 publication.
· 84% of women are not emotionally satisfied with their
relationships
· 95% of women reported emotional and psychological

harassment from their partners
· 70% of women married 5 years or more are having
extramarital affairs
You should notice that this study is an example of a sample
survey. The sample is comprised of individuals who actually
provided the data while the population is the larger group from
which the sample is chosen and whom the sample is to
represent. In this example, the population is all American
women (although some people may say all American women
who have relationships with men), while the sample is the 4500
respondents who returned the questionnaire.
As you might expect, a sample should appropriately represent
the population. However, in this instance, the sample does not
represent the population because of two problems. The first
problem found in this study is that only “joiners” were allowed
to be a part of the sample. Even though Shere Hite tries to
defend her methods by saying she sampled from a variety of
organizations, the fact remains that only people who were
involved in some organization had a chance to be in the sample.
This problem is an example of selection bias.
The other problem found with this sample is nonresponse bias.
Nonresponse bias can occur when a large number of people who
are selected for the study elect to not respond to the survey or
key questions on the survey. This is clearly evident because the
response rate was only 4500/100,000 = 4.5%. (Note: most
researchers like response rates to be at least 60% or 70%).
Moreover, the directions encouraged the participants to“skip
around” and answer only questions that they liked. As you
might expect, only people with strong opinions would take the
time to answer a questionnaire that contained 127 open-ended
questions. In fact, Shere Hite estimated that, on the average,
participants took about 4.4 hours to answer the questionnaire.
Also the use of the group leader to distribute the questionnaires
meant that there was a gatekeeper who had power to affect both
who responded and the response rate for each organization.
So with Example 2.1, the overall conclusion is that even though

the sample size is quite large, the sample does not adequately
represent the population. Unfortunately, the results from this 7-
year study are of little value.
Example 2.2 Pets and Marriage
Does owning a pet lead to less marital problems? (Rubin, 1998)
Karen Allen, a researcher at the University of Buffalo,
conducted a study to determine whether or not couples who own
cats or dogs have more satisfying marriages and experience less
stress than couples who don’t own pets. Allen compared 50 pet-
owning couples with 50 pet-free couples. The volunteers
completed a standard questionnaire that assessed both their
relationships and attachments to pets. Each couple also kept
track of their social contacts for two weeks. Allen examined
stress levels by monitoring the heart rates and blood pressure
readings while couples discussed sensitive topics. Pet-owning
couples not only started out with lower heart rates and lower
blood pressure readings, but also had smaller increases in heart
rates and blood pressure readings when they quarreled.
The study described above is an example of a comparative
study. In this instance, the couples who owned pets are
compared with couples who do not own pets as shown in Figure
2.1.
Figure 2.1. Illustration of a Comparative Study
A comparative study can either be an observational study or
an experiment. Observational studies collect data on
participants in their naturally occuring settings/groupings, while
withexperiments, the participants are randomly assigned to one
of two groups before the data is collected.
The study found in this example is an observational study
because participants are observed in their naturally occuring
groupings as either a pet-owning or pet-free couple. It would
be difficult to conduct an experiment because the researcher
would have to randomly assign couples to either have or not
have pets. It is not ethical to impose pet-ownership on the
couples, nor would it necessarily be good for the pet. So what

are the statistical differences between observational studies and
experiments? With an experiment, appropriate evidence can
support cause and effect conclusions. This is not possible with
observational studies.
In this study one cannot say that owning pets causes married
couples to have less stress and more satisfying marriages
because randomization was not used to cancel out other factors
that may affect stress level and marital satisfaction. Factors
such as income, number of hours spent working, where you live
(i.e., suburbs versus inner city), whether or not there are
children, etc., may also be responsible for changes in stress
level and marital satisfaction. We will never know because the
study is not a randomized experiment.
The researcher correctly stated the conclusion by indicating that
there was a difference in the two groups when considering
stress level and marital satisfaction. Appropriately, no “cause
and effect” language was used. However, it is not uncommon
for people who have no statistical background to incorrectly
infer a “cause and effect” conclusion from observational
studies. So as you examine other studies that are found in the
daily news, first determine if the study is an experiment or an
observational study. Next decide if the conclusions are
appropriate for the type of study that was conducted.
Example 2.3. Heights of Males and Females
One of the major points brought out in Chapter 1 is that the
number of people in a study is an important factor to consider
when designing a comparative study. To help you understand
this concept, consider the following two samples of five heights
in inches.
Sample of Female Heights in Inches: 61 64 68 66 63
Sample of Male Heights in Inches: 76 64 70 68 71
Do we have enough evidence to say that there is a difference in
heights when comparing a sample of five female heights with a
sample of five male heights? In order to answer this question,
look at Figure 2.2.

Figure 2.2. Heights for Sample Size 5
As you examine the graph you will probably decide that the
evidence may not be strong enough to clearly say that there is
difference in the two genders with regard to height. So suppose
we instead obtain a sample of 15 female heights and 15 male
heights. The results are found in Figure 2.3.
Figure 2.3. Heights for Sample Size of 15
What you should notice is that it is easier to distinguish
between the two groups with the larger sample size. If the
sample size were increased to a value even greater than 15, the
differences in the two groups would be easier to detect.
Therefore, sample size is an important factor to consider when
trying to detect differences between groups.
The overall conclusion from Chapter 1 is that in order to
conduct a proper study, one must:
· Get a representative sample
· Get a large enough sample
· Decide whether the study should be an observational study or
an experiment
2.2 Asking Research Questions
Overview
Suppose you desire to do a study or administer a survey. As an
investigator, the most challenging task that you will confront is
to decide what questions to ask and/or what measurements to
obtain. In this chapter you will be introduced to some key
definitions associated with obtaining measurements. You will
also learn about possible pitfalls found with survey questions.
It’s All in the Wording
Chapter 3 lists seven possible pitfalls that can occur when
asking questions in a survey or study. Of all the possible
pitfalls, the one that is most commonly found is deliberate bias.
People who use a form of deliberate bias often desire to gather
support for a specific cause or opinion. It is also possible that
more than one type of pitfall can happen at the same time.

Examine the following examples.
Example 2.4. Deliberate Bias (One-Sided Statements)
Consider two different wordings for a particular question:
Wording 1: It is hard for today’s college graduates to have a
bright future with the way things are today in the world.
a. agree
b. disagree
Wording 2: Today’s college graduates will have a bright future.
a. agree
b. disagree
Although Wording 1 and Wording 2 are contradictory
statements, when both questions are used in the same survey, it
is not uncommon to find that people answer “agree” to both
questions. This is because respondents tend to agree to one-
sided statements. Listed below are revised wordings for these
two questions. These choices are preferred because the
statements are now at least two-sided.
Revised Wording 1: Do you agree or disagree that it is hard for
today’s college graduates to have a bright future with the way
things are today in the world?
Revised Wording 2: Do you agree or disagree that today’s
college graduates will have a bright future?
Example 2.5. Deliberate Bias (Filtering)
Consider two different choices of answers for a particular
question:
Choice 1: What is your opinion of our current president?
a. favorable
b. unfavorable
Choice 2: What is your opinion of our current president?
a. favorable
b. unfavorable
c. undecided
This example illustrates the problem of “filtering.” Filtering
exists when certain choices such as “undecided” or “don’t
know” are not included in the list of possible answers. People
tend to provide an answer of “undecided” or “don’t know” only

when these choices are included in the list of possible answers.
Example 2.6. Deliberate Bias (Importance of Order)
Wording 1: Pick a color: red or blue?
Wording 2: Pick a color: blue or red?
The results in Table 2.1 are from a study conducted in a
Statistics class. As you can see the results vary somewhat
based on the order in which the colors are presented. Even
though many people probably have a preference for one color
over the other, if order does not matter, the percents should be
same with each wording.
Table 2.1. Deliberate Bias (Order of Comparisons)
Color Choice
Wording 1
Wording 2
Red
59%
45%
Blue
41%
55%
Example 2.7. Deliberate Bias (Anchoring)
Wording 1: Knowing that the population of the U.S. is 270
million, what is the population of Canada?
Wording 2: Knowing that the population of Australia is 15
This survey was conducted in Stat 100 classes where both
wordings of the question were randomly distributed. The
students did not know that there were two versions of this
question so each only answered the question that they received.
The results for this survey are found in Figure 2.4.
Figure 2.4. STAT 100 Survey Results
As you can see, the students were influenced by the wording of
the question that they were asked to answer. People’s

perceptions can be severely distorted when they are provided
with a reference point or an anchor. People tend to say close to
the anchor because of either having limited knowledge about the
topic or being distracted by the anchor. You should also
consider the following three points:
· The sample sizes were large enough to detect a difference in
the two groups (recall the point made in Chapter 1)
· Canada’s population is about 30 million
· The anchor might be less distracting if the following wording
were used: “What is the population of Canada, when knowing
that the population of the U.S. is 270 million?”
Example 2.8. Unintentional Bias
Wording 1: Do you favor or oppose an ordinance
that forbids surveillance cameras to be placed on Beaver
Avenue?
Wording 2: Do you favor or oppose an ordinance that does not
allow surveillance cameras to be placed on Beaver Avenue?
People will tend to answer “oppose” or “no” to a question that
contains words such as forbid, control, ban, outlaw,
and restrain regardless of what question is actually being asked.
People do not like to be told that they can’t do something. So
the responses to the two questions would not provide similar
results. Wording 2 would be preferred over Wording 1.
Example 2.9. Unnecessary Complexity (“Double-Barreled”
Problem)
Consider the following question.
Question: Do you think that health care workers and military
personnel should first receive the smallpox vaccination?
The problem with this question is that the respondent must
consider both health care workers and military personnel at the
same time. The following rewording is much better.
Revised Question: Who should first receive the smallpox
vaccination?
a. health care workers
b. military personnel

c. both Health care workers and Military Personnel
d. other
Example 2.10. Asking the Uninformed and Unnecessary
Complexity (Double Negative Problem and List Problem)
Consider the following question.
Question: Do you agree or disagree that children who have a
Body Mass Index (BMI) at or above the 95th percentile should
not be allowed to spend a lot of time watching television,
playing computer games, and listening to music?
The first concern with this question is that many people may not
clearly understand what the Body Mass Index (BMI) represents.
BMI is a measure that is used to identify obesity and is
calculated by dividing a person's weight (in kilograms) by the
square of their height (in meters). (Note: many Web sites have
BMI calculators.) In children and adolescents, obesity is
defined as a BMI for age and gender at or above the 95th
percentile. This definition should be included prior to the listing
of the question on a survey.
This question can also cause problems because of a possible
“double negative”. Specifically, the problem is with the
“disagree” choice. This choice produces a double negative
because “disagree” and “should not” are both in the statement.
Many respondents will not understand what they are really
saying. (It is easy to make the mistake of the double negative).
Revised Question-First Revision: Do you agree or disagree that
children who have a Body Mass Index (BMI) at or above the
95th percentile should spend less time watching television,
playing computer games, and listening to music?
As you examine this revised question you should also note that
there still is a list of three choices embedded in the questions.
Since respondents sometimes can get hung upon the list of
choices; the second revision would be preferred.
Revised Question-Second Revision: Do you agree or disagree
that children who have a Body Mass Index (BMI) at or above
the 95th percentile should spend less time in sedentary
activities?

A follow-up question could be asked to clarify which sedentary
activities should be reduced.
2.3 Defining a Common Language
In the previous examples we mostly considered problems
associated with questions that measure opinion. In order to
discern what we want to measure, we also need to understand
some basic definitions. Data is a collection of a number of
pieces of information. Each specific piece of information is
called an observation. The observations are measurements of
certain characteristics which we call "variables". The word
“variable” is used because the pieces of information, the
observations, vary from one person to the next.
Figure 2.5. Types of Data
Example 2.11. Variables
Consider the following variables:
Table 2.2. Classification of Variables
Number
Variable
Type of Variable
1
Which are you? Near-sighted, far-sighted, neither
Categorical
2
What is your height?
Measurement and Continuous
3
How many phone calls do you typically make in a day on a cell
phone?
Measurement and Discrete
4
What is your cholesterol level?
?
Hopefully, you find the classification of the first three variables
easy to understand.
Variable #1 is a categorical variable because the possible

choices are “words” or“categories.”
Variable #2 is a measurement variable because the possible
choices are “numbers.” This variable is also called a continuous
variable because it can assume a range of values. You need an
instrument, such as a tape measure or a ruler, to determine
height. With measurement variables that are continuous, it is
often necessary to use an instrument to determine the value of
the variable. Measurement variables that are continuous can be
subdivided into fractional parts (subdivided into smaller and
smaller units of measurement). Typically, a measurement-
continuous variable is expressed as "an amount of " something.
Variable #3 is a measurement variable because the possible
choices are numbers. It is also a discrete variable because one
can simply count the number of phone calls made on a cell
phone in any given day. The possible numbers are only integers
such as 0, 1, 2,….50, etc. (Some of you probably make a lot of
cell phone calls.) Measurement-Discrete variables cannot be
subdivided into fractional parts (smaller and smaller units of
measurement). Typically, a measurement-discrete variable is
expressed as "a number of " something.
Variable #4 is somewhat ambiguous. Obviously the variable is a
measurement variable. But the question that remains is whether
this variable is discrete or continuous. One person may state
that their cholesterol level is 169 points, while a physician may
report the cholesterol level as 169 milligrams per deciliter
(mg/dL). Which is correct? Cholesterol levels must be
determined by a blood test where an instrument is used to
determine the final value. The reported value represents the
concentration of cholesterol in the blood. The appropriate units
are milligrams per deciliter (mg/dL). What typically happens is
that the value of the cholesterol level is rounded to the nearest
whole number. Consequently, the cholesterol level starts to be
viewed as a discrete variable. However, this perception is
incorrect because cholesterol levels cannot be counted.
Example 2.12. Best Way to Determine Heart Rate
Consider an experiment where heart rate (heart beats/minute) is

measured by three different methods. Let’s consider three
different methods to determine heart rate.
Method 1: Count heart beats for 6 seconds & multiple by 10 to
get heart beats/minute
Method 2: Count heart beats for 30 seconds & multiply by 2 to
get heart beats/minute
Method 3: Count heart beats for 60 seconds
We collected six measurements on an individual for each of the
three methods. These results are found in Table 2.3.
Table 2.3. Results from the Heart Rate Experiment
Method
Six Results
Heart Rate (HeartBeats/Minute)
Minimum and Maximum Heart Rate
Average Heart Rate
1
7, 7, 7, 7, 7, 7
70, 70, 70, 70, 70, 70
70, 70
70
2
36, 35, 37, 38, 37, 37
72, 70, 74, 76, 74, 74
70, 76
73
3
73, 76, 74, 75, 74, 75
73, 76, 74, 75, 74, 75
73, 76
74.5
In this example, we will not explore whether or not heart rate is
a valid measure of overall health and fitness. Obviously, it does
provide some information about whether or not a person may
have some health problems. But by itself, it usually does not
provide a complete picture. The questions that we should pose
are the following:

Question 1: Which method is the most reliable?
Question 2: Which method is the most biased?
What may surprise you is that the answer to both questions
is method 1. Method 1 is the most reliable because every time
we took the measurement we observed 7 beats in 6 seconds. The
results are consistent. Results from method 1 are also the most
biased because it consistently underestimates the individual's
true heart rate. If you look at the results from method 3, which
is really the best method to determine heart rate, you find that
the individual's average heart rate is 74.5 beats/minute. The
results from method 1 always fell below this value. What this
means is that even though method 1 is reliable, it still can have
other problems, which in this case, is biasedness.
Example 2.13. Validity or Reliability
Suppose you are interested in knowing whether the average
price of homes in a certain county had gone up or down this
year in comparison with last year. Would you be more
interested in having a valid measure or a reliable measure of
sales?
Ideally you would like the measure to be both valid and
reliable. However, a reliable measure that is not valid, can still
often provide some meaningful information. Since the goal is to
make a comparison of the average price of homes over two
years, the measure must be reliable. So, even if the measure is
not the most valid, the amount of change from one year to the
next may be sufficient information to make a
comparison.Lesson 2 Practice Questions
Answer the following Practice Questions.Think About It!
Come up with an answer to these questions by yourself and then
click the icon on the left to reveal the answer.
Use the following scenario with Questions 1 and 2. Suppose you
have 30 blueberry bushes and want to know if fertilizing them
will help them produce more fruit. You randomly assign fifteen
of them to receive fertilizer and the remaining fifteen to receive
none.
1. What type of study is being implemented?

a. observational study
b. experiment
c. neither a or b
d. both a and b
2. If the fertilized bushes produce significantly more fruit than
the unfertilized bushes, can you conclude that the fertilizer
caused the bushes to produce more fruit?
a. yes
b. no
3. Consider the following research question. "Do you agree that
constituents should be allowed to recall elected officials?"
Which of the following pitfalls associated with research
questions applies in this instance?
a. deliberate bias
b. unintentional bias
c. ordering of questions
d. unnecessary complexity
4. Consider the following variable: the number of words in a
textbook. What type of variable is this?
a. categorical
b. measurement & discrete
c. measurement & continuous
5. Which of the following characteristics could be measured as
either discrete or continuous depending on the units used?
a. brand of car you own
b. caffeine consumption
c. your favorite color
Lesson 3: Measurement Data: Summaries, Displays, and Bell-
Shaped CurvesAssignments
· Complete the Practice Questions and Lesson 3
Assignment.Learning Objectives
Chapters 7 and 8

to:
· Interpret any of the four graphs used with measurement data.
· Distinguish between a measure of center and a measure of
spread.
· Determine when sensitive statistics or resistant statistics
should be used to describe a data set.
· Interpret a five-number summary.
· Apply the empirical rule to variables that are normally
distributed.Terms to Know
From Chapter 7
· dotplot
· stemplot
· histogram
· symmetric
· skewed
· boxplot
· mean
· median
· percentiles
· five-number summary
· outlier
· interquartile range (IQR)
· standard deviation (SD)
· sensitive measure
· resistant measure
From Chapter 8
· normal (bell-shaped) distribution
· empirical rule
· Z-scoreCommentary
Overview
The goal of this lesson is to learn about different ways to
display and summarize measurement data. These methods will
be appropriate for all measurement variables regardless of
whether the variable is discrete or continuous.

Figure 3.1. Flow Chart for Display and Summarization of
Measurement Data3.1 Graphs: Displaying Measurement Data
Here are four different graphs that can be used to describe
measurement data. These graphs include:
1. Dotplots
2. Stemplots (Stem and Leaf Plot)
3. Histograms
4. Boxplots
Example 3.1. Graphs
Consider the following sample.
Sample:The ages of forty selected PSU Tenured Faculty (n = 40
ages)Ages
45 59 51 62 58 54 56 42 59 49 47 52 63 40 53 61 47 54 58 53
32 61 39 51 37 43 53 46 59 56 58 48 55 50 57 60 54 63 60
55Dotplots
The dotplotis the first graph that will be used to display this
sample of 40 ages. The purpose of the dotplot is to represent
each observation as a dot. On this dotplot you will find that the
ages range from 32 to 63 years. You should also notice that
there are more tenured faculty at older ages.
Figure 3.2. Dotplot (Ages of 40 PSU Tenured Faculty)Stemplots
(Stem and Leaf Plot)
The second graph that is possible with measurement data is
a stemplot. Stemplots concisely display the data in order from
smallest to largest. Below is the list of the 40 ages in order from
youngest to oldest.
Ages (Sorted)
32 37 39 40 42 43 45 46 47 47 48 49 50 51 51 52 53 53 53 54
54 54 55 55 56 56 57 58 58 58 59 59 59 60 60 61 61 62 63 63
These forty observations are displayed in the stemplot found
in Figure 3.3. In this stemplot we again find that the range of
ages span from 32 year to 63 years. We also find that there are
more tenured PSU faculty at older ages. Stemplots can provide
useful information about small data sets.

Figure 3.3. Stemplot (Ages of PSU Tenured Faculty)Histograms
The third graph is called a histogram. Of all the graphs
presented so far, the histogram may be the most valuable.
A histogramis essentially a bar graph for measurement data. The
difference, however, between a histogram and a bar graph, is
that with a histogram the categories are a range of numbers
rather than words. The requirement is that each numerical
category must have the same width.
Recall that the ages span from 32 to 63 years. The rangeof these
ages is (63-32) = 31 years. Looking at the stemplot, one finds
that there are more tenured faculty at the older ages. We want to
be able to show this trend. There needs to be enough categories
to properly display any trend. If we only choose 4 categories
(since we have tenured faculty in their 30s, 40s, 50s, and 60s)
we would not be able to detect this trend as well. If we choose 9
categories the trend will become more obvious. Statistical
software will usually make this determination for you.
Width of each Category= Range/ number of categories = 31/9 =
3.4 (rounded to 4 years)
The starting point is always below our lowest observed value
and the ending point is always above our highest observed
value. In this instance, our starting point of 30 years is below
the lowest observed age which is 32 years. The ending point of
66 years is above our highest observed age which is 63 years. In
order to make sure that an observation only falls into one
category, we construct our 4 year categories as shown in Table
3.1 below. Our first category includes ages starting at 30 years
and ending just below 34 years. An observed age of 34 is not
included in the first category, but rather is included in the
second category. Using this method, no observed age can fall
into more than one category. The observed ages are then placed
into the appropriate category and the histogram is constructed.
Recall Ages (Sorted)
32 37 39 40 42 43 45 46 47 47 48 49 50 51 51 52 53 53 53 54
54 54 55 55 56 56 57 58 58 58 59 59 59 60 60 61 61 62 63 63
Table 3.1. Summary of Ages for 40 PSU Tenured Faculty

Numerical
Category
Ages that Fall into the Category
Number of observed Ages in that Category
Percents
1. 30≤ Age < 34
Ages 30 to 33
1
1/40 = .025 (2.5%)
2. 34≤ Age < 38
Ages 34 to 37
1
1/40 = .025 (2.5%)
3. 38≤ Age < 42
Ages 38 to 41
2
2/40 = .05 (5.0%)
4. 42≤ Age < 46
Ages 42 to 45
3
3/40 = .075 (7.5%)
5. 46≤ Age < 50
Ages 46 to 49
5
5/40 = .125 (12.5%)
6. 50≤ Age < 54
Ages 50 to 53
7
7/40 = .175 (17.5%)
7. 54≤ Age < 58
Ages 54 to 57
8
8/40 = .20 (20.0%)
8. 58≤ Age < 62
Ages 58 to 61
10

10/40 = .25 (25.0%)
9. 62≤ Age < 66
Ages 62 to 65
3
3/40 = .075 (7.5%)
n = 40
40/40 = 1.0 (100%)
The information from Table 3.1 is used make the histogram
found in Figure 3.4. The horizontal axis displays the categories
while the vertical axis displays the percent of the observations
(ages) found in each category. (Note: You will not be asked to
make histograms. You will only be asked to interpret them.
However, it is important to see how one is made so that you
understand the interpretation.)
Figure 3.4. Histogram (Ages of PSU Tenured Faculty)
As you look at the histogram, you should notice that there are
more ages on the upper half of the graph. In statistics, when
data on a histogram is off-center, the data is labeled as skewed.
In this case, the data is skewed to the left because a larger
percent of the ages are found in the upper tail.
Table 3.2. Possible Histogram
Interpretations
Figure 3.5 Data Shapes
Histogram Interpretations
Explanation
Graph
Skewed to the right
Larger percent of data is found on lower tail of the histogram
Skewed to the left
Larger percent of data is found on upper tail of the histogram
Symmetric

Equal percent of data on each tail of the histogram
3.2 Numbers: Summarizing Measurement Data
There are two general ways to summarize measurement data.
These include:
1. measures of center
2. measure of spread (variation)
There are two ways to represent the center of measurement data.
These include:
1. mean (won’t be asked to calculate)
2. median
Example 3.2. Measures of Center
Consider the following sample:
Sample:The Number of Movie Rentals/Month for 5 Selected
PSU Students (n = 5)
1 5 1 4 2
Suppose you want to find a number to represent the center of
the data. The first choice would be the mean. The mean is also
known as the average. The mean is found by obtaining a sum of
all the observations and dividing by the sample size (n). In this
instance:
mean= (1 + 5 + 1 + 4 + 2) / 5 = 13 / 5 = 2.6 movie
rentals/month
Another possibility is the median. The median is the middle
value of a sample when the observations are sorted from
smallest to largest.
Sorted Sample:
1 1 2 4 5
In this example, the middle observation is 2 so themedian = 2.0
movie rentals/month.
As you examine how the mean and median were calculated,
hopefully you notice that the two methods are very different.
The mean is an example of a sensitive measure because all
observations were used in the calculation. In contrast, the
median is an example of a resistant measure because only the
middle observation was used to determine its value.
Example 3.3. Which Measure of Center to Use

Consider the following sample:
Sample: The Annual Salaries ($) for 20 Selected Employees at a
Local CompanySalaries (Sorted)
30000 32000 32000 33000 33000 34000 34000 38000 38000
38000 42000
43000 45000 45000 48000 50000 55000 55000 65000 110000
The mean for this sample is $45,000 while the media is $40,000.
(Note: because the sample size is an even number, the median is
the average of the middle two numbers, which in this case
are $38,000 and $42,000). Even though we can always
determine both the mean and median, one must determine which
measure is more appropriate to use when there is a large
difference between the two measures of center. In this instance,
there is a difference of $5,000 between the two measures, so
one should decide which measure of center is more appropriate
to use. To help you understand what is happening, look at the
histogram found in Figure 3.6.
Figure 3.6. Histogram (Salaries)
As you can see, the histogram is right-skewed because a larger
percent of the salaries are located on the lower tail. The very
large salary of $110,000 is largely responsible for the histogram
being right skewed. With right-skewed histograms, the mean
will be greater than the median, because the mean is sensitive to
the large salary of $110,000 and is pulled in the direction of the
unusually large observation. In contrast, the median, which is
the middle value of the data set, is resistant to any extreme
observations because these observations are not used to
determine its value. Table 3.3 summarizes the link between the
two measures of center and histogram shape.
Table 3.3. Link between Measures of Center and Histogram
Shape
Compare Two Measures Of Centers
Histogram Shape
If mean and median are approximately equal
symmetric

If mean is greater than median
right skewed
If mean is less than median
left skewed
Figure 3.7. Different Distributions
So, getting back to the question of which measure of center is
more appropriate to use. When you have skewed data, the mean
is somewhat misleading. The mean can be pulled in one
direction or the other by outliers. Generally, when the data is
skewed, the median is more appropriate to use as the more
typical measure of center. We generally use the mean for the
typical measure of center when the data is symmetric.
However, it is important to be given both measures of center.
The difference between the mean and median is important since
the direction and magnitude of that difference will determine
the shape of the data as indicated in Table 3.3. and in the plots
shown in Figure 3.7.
The question being asked can also affect which measure of
center can be considered more typical and therefore, more
appropriate. Although we would normally use the median with
skewed data, there may be cases where we might use the mean
as a more typical measure of center. It all depends on the
question being asked and on the shape of the data. For example,
given the right-skewed data for the company in Example 3.3:
· If you are applying for a higher level position at the company
in Example 3.3, the mean might represent the typical salary
figure better than the median since it accounts for some of the
higher salaries in the company. In this case, the median salary
figure may not be as appropriate as the mean salary figure.
· However, if you are applying for an entry level position within
the company in Example 3.3, the median salary figure would
represent the typical salary figure better than the mean and be
more appropriate to use.3.3 Five Useful Numbers (Percentiles)
A percentileis the position of an observation in the data set
relative to the other observations in the data set. Specifically

the percentile represents the percentage of the sample that falls
below this observation. For example, the median is also known
as the
50th percentile because half of the data or 50% of the
observations lie below the median. Table 3.4displays three
percentiles that will be of interest to us. Figure 3.8 shows these
percentiles (quartiles) graphically.
Table 3.4. Percentiles of Interest
Percentile
Alternate Names
Interpretation
25th percentile
· Lower Quartile (QL)
· First Quartile (Q1)
25% of the data falls below this percentile
50th percentile
· Median
· Second Quartile ( Q2)
75th percentile
· Upper Quartile (QU)
· Third Quartile (Q3)
Figure 3.8. Quartiles for a Distribution
A five-number summary is a useful summary of a data set that is
partially based on selected percentiles. Below are the five
numbers that are found in a five-number summary.
Figure 3.9. Five-Number Summary Flow Chart
Example 3.4. Five-Number Summary
Recall the sample that was used in the previous example.
Sample:The Annual Salaries ($) for 20 Selected Employees at a
Local CompanySalaries (Sorted)
30000 32000 32000 33000 33000 34000

34000 38000 38000 38000 42000
43000 45000 45000 48000 50000 55000
55000 65000 110000
Table 3.5. Five-Number Summary of Salaries
Lowest
Lower Quartile (QL)
Median
Upper Quartile (QU)
Highest
$30,000
$33,500
$40,000
$49,000
$110,000
Below are possible questions that can be answered with this five
number summary.
1. What percent of the salaries lie below $49,000?
Answer: 75%
Reason: $49,000 represents the 75th percentile or upper quartile
2. What percent of the salaries lie above $40,000?
Answer: 50%
Reason: $40,000 represents the 50th percentile so 50% of the
observations lie below this percentile and 50% lie above this
percentile
3. What percent of the salaries lie between $33,500
and $49,000?
Answer: 50%
Reason: asking for percent of observations that lies between the
25th percentile and the 75th percentile (75% - 25% =
50%)Boxplots
The five-number summary is also of value because it is the
basis of the boxplot. Figure 3.10 is a vertical boxplot of the
variable salaries. The most important part of this graph is the
box. The ends of the box locate the lower quartile and upper
quartile, which in this case are $33,500 and $49,000
respectively. The line in the middle of the box is the median. As

you examine the box portion of the box, you should notice that
the median is closer to the lower quartile than to the upper
quartile. This suggests that data set is skewed and specifically
skewed to the right. In this instance the largest observation is
represented with an asterisk. Since this observation is an
unusually large salary of $110,000, the graph identifies this
observation as an outlier or unusual observation. Appropriate
statistical criterion is used to determine whether or not an
observation is an outlier. Lines called 'whiskers' extend from
the box out to the lowest and highest observations that are not
outliers.
Figure 3.10. Horizontal Boxplot of Salaries
One of the most important uses of the boxplot is to compare two
or more samples of one measurement variable.
Example 3.5. Using Boxplots for Comparisons
Recall Example 2.7 from Lesson 2. Consider two different
wordings for a particular question:
Wording 1: Knowing that the population of the U.S is 270
Wording 2: Knowing that the population of Australia is 15
million, what is thepopulation of Canada?
The results from these questions are displayed on side-by-side
boxplots found in Figure 3.11.
Figure 3.11. Boxplots of Canada’s Population by Wording
Four comparisons can be made with side-by-side boxplots. One
can compare the
1. centers: medians
2. amount of spread (variation): lengths of the box
3. shape: position of the median in the box relative to the
quartiles
4. number of outliers
With this example, the median for those who had Wording 1 is
larger than the median found with Wording 2. One also finds
that the length of the box for Wording 1 is also larger than that

found with Wording 2. This suggests that there is more spread
or variation in the responses for Wording 1. The median is also
not positioned in the same place in each box that indicates that
the two samples do not have the same shape. Finally, there are
two outliers with Wording 2 while there are none with Wording
1. Overall, these findings suggest that the wording of the
question does affect the responses that are obtained.3.4
Measures of Spread or Variation
Two ways to represent the spread or variation are:
1. Interquartile Range (IQR)
2. Standard Deviation (SD)
Example 3.6. Measures of Spread or Variation
Recall the five-number summary from Example 3.4.
Table 3.6. Five-Number Summary of Salaries
Lowest
Lower Quartile (QL)
Median
Upper Quartile(QU)
Highest
$30,000
$33,250
$40,000
$49,500
$110,000
With the five-number summaryone can easily determine
the Interquartile Range (IQR). The IQR = QU- QL. In our
example,
IQR = QU- QL = $49,500 - $33,250 = $16,250
What does this IQR represent? With this example, one can say
that the middle 50% of the salaries spans $16,250 (or spans
from $33,250 to $49,500). The IQR is the length of the box on
a boxplot. Notice that only a few numbers are needed to
determine the IQR and those numbers are not the extreme
observations that may be outliers. The IQR is a type
of resistant measure.
The second measure of spread or variation is called the standard

deviation (SD). The standard deviation is roughly the average
distance that the observations in the sample fall from the mean.
The standard deviation is calculated using every observation in
the data set. Consequently it is called a sensitive
measure because it will be influenced by outliers. The standard
deviation for the variable “salaries” is $17,936 (Note: you will
not be asked to calculate a SD). What does the standard
deviation represent? With this example, one can say that the
average distance of any individual salary from the mean salary
of $45,000 is about $17,936. Figure 3.12 shows how far each
individual salary is from the mean.
Figure 3.12. Dotplot of Salaries
What you notice in Figure 3.12 is that many of the observations
are reasonably close to the sample mean. But since there is an
outlier of $110,000 in this sample, the standard deviation is
inflated such that average distance is about $17,936. In this
instance, the IQR is the preferred measure of spread because the
sample has an outlier.
Table 3.7shows the numbers that can be used to summarize
measurement data.
Table 3.7. Numbers used to Summarize Measurement Data
Numerical Measure
Sensitive Measure
Resistant Measure
Measure of Center
Mean
Median
Measure of Spread (Variation)
Standard Deviation (SD)
Interquartile Range (IQR)
· If a sample has outliers and/or skewness, resistant measures
are preferred over sensitive measures. This is because sensitive
measures tend to overreact to the presence of outliers.
· If a sample is reasonably symmetric, sensitive measures
should be used. It is always better to use all of the observations

in the sample when there are no problems with skewness and/or
outliers.3.5 Predictable Patterns
Many measurement variables found in nature follow a
predictable pattern. The predictable pattern of interest is a type
of symmetry where much of the data is clumped around the
center and few observations are found on the extremes. Data
that has this pattern are said to be bell-shaped or have a normal
distribution.
Example 3.7. Normal Curves
Consider the following three variables from data that was
collected from a sample of Stat 100 students:
· Variable #1: Heights (inches)
· Variable #2: Grade Point Average
· Variable #3: Number of Tattoos
Figure 3.13. Histogram of Height (Mean = 66.3 inches &
Median = 66 inches)
Variable #1 is a great example of a normal distribution as
shown in Figure 3.13. Since a normal distribution is a type of
symmetric distribution, you would expect the mean and median
to be very close in value. With this example, the mean is 66.3
inches and median is 66 inches.
Figure 3.14. Histogram of GPA (Mean = 3.25 & Median = 3.3)
Variable #2 reasonably follows a normal distribution as shown
in Figure 3.14. The only problem is that found with the upper
tail where the data is clumped which is partially explained by
the fact that GPAs at Penn State cannot exceed 4.0. However,
since the sample size is large (n = 198 students) and the mean
and median are very close, one can assume that this sample is
reasonably normal. It also helps that this variable is continuous.
Figure 3.15. Number of Tattoos (Mean = .23 & Median = 0)
Variable #3 is not normally distributed as shown in Figure
3.15. The major problem with this variable is that it is discrete

rather than continuous. Ideally, normal distributions should be
based on measurements of variables that are continuous. As you
can see, the graph has gaps because this variable is discrete.
Even when ignoring this fact, the distribution is skewed because
most people do not have any tattoos. The only reason that the
mean and median are so close is because of the large sample
size.
Empirical Rule
The empirical rule is a guideline that can be applied when you
know that the sample is normally distributed. The empirical
rule helps one to understand what the standard deviation
represents.
The empirical rule says that for any normal (bell-shaped) curve,
approximately:
· 68% of the values (data) fall within 1 standard deviation of the
mean in either direction
· 95% of the values (data) fall within 2 standard deviations of
the mean in either direction
· 99.7% of the values (data) fall within 3 standard deviations of
the mean in either direction
Figure 3.16 The Empirical Rule
Example 3.8. Empirical Rule
Recall the variable heights used in Example 3.7. Since the
histogram shows that this data is normally distributed, the
empirical rule can be applied. The mean and standard deviation
(SD) for this sample are 66.3 inches and 4 inches, respectively.
Below are the calculations for the sample of heights.
Mean ± 1(SD) = 66.3 ± 4 inches = (62.3 to 70.3 inches)
Mean ± 2(SD) = 66.3 ± 2(4) inches = 66.3 ± 8 inches = (58.3 to
74.3 inches)
Mean ± 3(SD) = 66.3 ± 3(4) inches = 66.3 ± 12 inches = (54.3
to 78.3 inches)
Because the sample of heights is normally distributed, one can
say that approximately
· 68% of the heights lie between 62.3 and 70.3inches

· 95% of the heights lie between 58.3 and 74.3 inches
· 99.7% of the heights lie between 54.3 and 78.3 inches
One would not expect someone in this sample to be smaller than
54.3 inches or taller than 78.3 inches.
Standardized Scores (Z-Scores) used with Normal Distributions
A standardized score is simply a way to “standardize” data that
is normally distributed. By “standardize”, we mean that we
convert the normal data into normal data that has a mean of 0
and a standard deviation of 1.0. This normal distribution is
then called the “Standard Normal” distribution. Standardizing
the data enables you to use the Z-Table (Table 8.1 on page 157
of the text) to determine percentiles for data values found in the
data set.
Example 3.9 Standardized Scores (Z-Scores)
If we know that the gas mileage for compact SUVs follows a
normal distribution with a population mean of 28 mpg and a
population standard deviation of 2 mpg. What is the percentile
for a compact SUV that gets 30 mpg? The percentile will give
us an idea of how the gas mileage of this compact SUV
compares to the gas mileage of all other compact SUVs.
We first have to compute the Z-Score (standardized
score) which is found by using the following formula:
Z-Score = (observed value - population mean)/population
standard deviation
In our problem, the observed value we want to find the
percentile for is 30, the population mean is 28, and the
population standard deviation is 2.
Our Z-Score is: Z = (30-28)/2 = 2/2 = +1.0
Our Z-Score is +1.0 which indicates that our observed value is 1
standard deviation above the population mean. Negative Z-
Scores indicate that the observed value is below the population
mean.
We go to Table 8.1 in the text and look for our Z-Score. Since
the Z-Score is positive, we look at the fourth column from the
left. We go down the rows and see that a Z-Score of +1.0 is at
the 84th percentile. This indicates that 84% of compact SUVs

have lower gas mileage than this particular brand of compact
SUV. Conversely, only 16% of compact SUVs get better gas
mileage, so our brand of interest is in the top 16% for gas
mileage.Lesson 3 Practice Questions
Answer the following Practice Questions to check your
understanding of the material in this lesson.Think About It!
1. The following question was asked of a sample of STAT 100
students: How many times a month do you usually drink at least
two beers? Which of the following graphs cannot be used to
describe the resulting data?
a. boxplot
b. dotplot
c. stem and leaf
d. pie chart
e. histogram
2. The following histogram displays the number of CDs owned
from a sample of STAT 100 students. What shape is displayed
on this histogram?
a. symmetric
b. right skewed
c. left skewed
3. Find the median for the following sample of five numbers: 2
7 6 4 3
a. 2
b. 7
c. 6
d. 4
d. 3
4. Which of the following cannot be determined from a five-
number summary of a data set?
a. lowest (minimum value)
b. lower quartile QL
c. mean (or average)

d. interquartile range
5. Which of the following is not a measure of spread or
variation?
a. interquartile range
b. median
c. standard deviation
6. Given a population mean for September temperatures in State
College, PA of 75 degrees and a population standard deviation
of 3 degrees. What percentile is a value of 72 degrees?
a. 84th
b. 16th
c. 99.87th
d. .13thLesson 4: How to Get a Good SampleAssignments
Assignment.Learning ObjectivesChapter 4
to:
· Distinguish between a population, sample, and sampling
frame.
· Interpret and identify the factors that affect the margin of
error.
· Identify types of probability samples and judgment samples.
· Apply the “Difficulties and Disasters” in sampling to real
world problems.
· Identify all steps used and issues addressed by the Gallup
Poll.Terms to KnowChapter 4
· sample surveys
· experiments
· observational studies
· case studies
· unit (sampling unit)
· population
· sample
· sampling frame

· census
· margin of error (ME)
· sample size (n)
· probability sampling
· judgment sampling
· simple random sample
· stratified sampling
· cluster sampling
· systematic sampling
· voluntary sample
· haphazard (convenience) sample
· gallup poll
· nonresponse (no response or voluntary response)
· random-digit dialing
· selection bias
· sample percent
· population percentCommentary
Section 4.1. Chapter 4 in Textbook
Overview
In this lesson, we will add to our knowledge base by explaining
ways to obtain appropriate samples for statistical studies.4.1
Common Research StrategiesChapter 4 Section 4.1
The following research strategies are described in this section
of the textbook.
1. Sample Surveys
2. Experiments
3. Observational Studies
4. Meta-Analyses(also covered in Chapter 25--not required for
the course)
5. Case Studies
Terms Used with Sample Surveys (Chapter 4 Section 4.2 in
Textbook)
It is first necessary to distinguish between a census and
a sample survey. A census is a collection of data from every
member of the population, while a sample survey is a collection
of data from a subset of the population. A sample survey is a

type of observational study. Obviously, it is much easier to
conduct a sample survey than a census. The remaining sections
of this lesson (Chapter 4) will discuss issues about sample
surveys.
Of the many terms that are used with sample surveys, the
following four need the most clarification because of how they
are connected to each other.
· Sampling Unit: The individual person or object that has the
measurement (observation) taken on them / it
· Population: The entire group of individuals or objects that we
wish to estimate some characteristic's (variable's) value
· Sampling Frame: The list of the sampling units from which
those to be contacted for inclusion in the sample is obtained.
The sampling frame lies between the population and sample.
Ideally the sampling frame should match the population, but
rarely does because the population is not usually small enough
to list all members of the population.
· Sample: Those individuals or objects who provided the data
collected
Figure 4.1 Relationship between Population, Sampling Frame
and Sample
Example 4.1. Who are those angry women?
(Streitfield, D., 1988 and Wallis, 1987)
Recalling some of the information from Example 2.1 in Lesson
2, in 1987, Shere Hite published a best-selling book
called Women and Love: A Cultural Revolution in Progress.
This 7-year research project produced a controversial 922-page
publication that summarized the results from a survey that was
designed to examine how American women felt about their
relationships with men. Hite mailed out 100,000fifteen-page
questionnaires to women who were members of a wide variety
of organizations across the U.S. Questionnaires were actually
sent to the leader of each organization. The leader was asked to
distribute questionnaires to all members. Each questionnaire
contained 127 open-ended questions with many parts and

follow-ups. Part of Hite’s directions read as follows: “Feel free
to skip around and answer only those questions you choose.”
Approximately 4500 questionnaires were returned.
In Lesson 2, we determined that the
· population was all American women.
· sample was the 4,500 women who responded.
It is also easy to identify that the sampling unit was an
American woman. So, the key question is “What is
the sampling frame?” Most people think the sampling frame was
the 100,000 women who received the questionnaires. However,
this answer is not correct because the sampling frame was the
list from which the 100,000 who were sent the survey was
obtained. In this instance, the sampling frame included all
American women who had some affiliation with an
organization. There is no statistical term to attach to the
100,000 women who received the questionnaire. However, if
the response rate had been 100%, the sample would have been
the 100,000 women who responded to the survey.
You should also remember that ideally the sampling frame
should include the entire population. If this is not possible, the
sampling frame should appropriately represent the desired
population. In this case, the sampling frame of all American
women who were “affiliated with some organization” did not
appropriately represent the population of all American women.
InLesson 2, we called this problem selection bias.
Chapter 4 of your text also lists three difficulties that are
possible when samples are obtained for surveys. These three
difficulties, which happen to be possible with this example,
include:
1. Using the wrong sampling frame. We just discussed this
problem in the
preceding paragraph. This problem is also called selection bias.
2. Not reaching the individuals selected. Because the
questionnaire was sent to leaders of organizations, there is no
guarantee that these questionnaires actually reached the women
who were supposed to be in the sample.

3. Getting “no response” or a “volunteer response.” In Lesson
2, we learned that this survey has a problem with nonresponse
bias because of the low response rate. This problem can also be
called “no response” or “volunteer response.”4.2 The Beauty of
Sampling
Sample surveys are generally used to estimate the percentage of
people in the population that have a certain characteristic or
opinion. If you follow the news, you will probably recall that
most of these polls are based on samples of size 1000 to 1500
people. So, why is a sample size of around 1000 people
commonly used in surveying? The answer is based on
understanding what is called the margin of error.
The margin of error:
· measures the accuracy of the percent estimated in the survey
· is calculated using a formula that includes the sample size (n)
For a sample size of n = 1000, the margin of error
is1n√=11000√=0.03 , or about 3%.
Even though you will not be asked to calculate a margin of
error in this course, you should remember the margin of error
formula and that the margin of error formula depends only on
the size of the sample. The size of the population is not used in
the calculation of the margin of error. So, a percentage
estimated by a selected sample size will have the same margin
of error (accuracy), regardless of whether the population size is
5,000 or 5 billion. It also helps that pollsters believe that an
accuracy of ± 3% is reasonable with surveys.
So what does the margin of error represent? The following
statement represents the generic interpretation of a margin of
error.
Generic Interpretation: If one obtains many samples of the
same size from a defined population, the difference between the
sample percent and the true population percent will be within
the margin of error, at least 95% of the time.
Key Features of the Interpretation of the Margin of Error
· Statistical theory is often based on what would happen if the
survey were repeated many times. So, even though a pollster

usually obtains only one sample, the pollster must remember
that the margin of error interpretation is based on doing the
survey repeatedly under identical conditions.
· The margin of error represents the largest distance that would
occur between the sample percent, which is the percent obtained
by the poll, and the true population percent, which is unknown
because we have not sampled the entire population.
· In statistics, when talking about the margin of error, it is just
not possible to say that we are 100% certain that with all
samples the difference between the sample percent and the
population percent will be within the margin of error. So,
statisticians work with reasonable conditions so that one can say
that at least 95% of the time, the difference between the sample
percent and the population percent will be within the margin of
error.
Example 4.2. Margin of Error
Suppose a recent poll based on 1000 Americans finds that 55%
approve of the president’s current educational plan. Since the
sample size is 1000, the margin of error is about 3%. These
poll results suggest that 55% ± 3% of all Americans approve of
the president’s current economic plan. What is the correct
interpretation of this margin of error?
Margin of Error Interpretation
The difference between our sample percent and the true
population percent will be within 3%, at least 95% of the time.
This means that we are almost certain that 55% ± 3% or (52% to
58%) of all Americans approve of the president’s current
educational plan. Because the range of possible values from
this poll all fall above 50%, we can also say that we are pretty
sure that a majority of Americans support the president’s
current educational plan. If any of the range of possible values
would have been 50% or less, then we would not have been able
to say that the majority supported the plan. The range of values
(52% to 58%) is called a95% confidence interval. We will go
into further detail about confidence intervals in Lesson 7.4.3
Relationship between Sample Size and Margin of Error

There is a predictable relationship between sample size and
margin of error. The numbers found in Table 4.1 help to explain
this relationship.
Table 4.1. Calculated Margins of Error for Selected Sample
Sizes
Sample Size (n)
Margin of Error (M.E.)
200
7.1%
400
5.0%
700
3.8%
1000
3.2%
1200
2.9%
1500
2.6%
2000
2.2%
3000
1.8%
4000
1.6%
5000
1.4%
From this table, one can clearly see that as sample size
increases, the margin of error decreases. In order to add
additional clarity to this finding, the information from Table
4.1 is also displayed inFigure 4.2.
Figure 4.2 Relationship Between Sample Size and Margin of
Error
In Figure 4.2, you again find that as the sample size increases,
the margin of error decreases. However, you should also notice

that the amount by which the margin of error decreases is
substantial between samples sizes of 200 and 1500. This
implies that the accuracy of the estimate is strongly affected by
the size of the sample. In contrast, the margin of error does not
substantially decrease at sample sizes above 1500. Therefore,
pollsters have concluded that it is not worth it to spend
additional time and money for samples that contain more than
1500 people.4.4 Simple Random Sampling and Other Sampling
Methods
Sampling Methods can be classified into one of two categories:
· Probability Sampling: Sample has a known probability of
being selected
· Judgment Sampling: Sample does not have known probability
of being selected
Probability Sampling
In probability sampling it is possible to both determine which
sampling units belong to which sample and the probability that
each sample will be selected. The following sampling methods,
which are listed in Chapter 4, are types of probability sampling:
1. Simple Random Sampling (SRS)
2. Stratified Sampling
3. Cluster Sampling
4. Multistage Sampling
5. Random-Digit Dialing
6. Systematic Sampling
Of the five methods listed above, students have the most trouble
distinguishing between stratified sampling and cluster sampling.
Stratified Sampling is possible when it makes sense to partition
the population into groups based on a factor that may influence
the variable that is being measured. These groups are then
called strata. An individual group is called a stratum.
With stratified samplingone should:
· partition the population into groups (strata)
· obtain a simple random sample from each group (stratum)
· collect data on each sampling unit that was randomly sampled
from each group (stratum)

Stratified sampling works best when a heterogeneous population
is split into fairly homogeneous groups. Under these
conditions, stratification generally produces more precise
estimates of the population percents than estimates that would
be found from a simple random sample. Table 4.2 shows some
examples of ways to obtain a stratified sample.
Table 4.2. Examples of Stratified Samples
Example 1
Example 2
Example 3
Population
All people in U.S.
All PSU intercollegiate athletes
All elementary students in the local school district
Groups (Strata)
4 Time Zones in the U.S. (Eastern,Central, Mountain,Pacific)
26 PSU intercollegiate teams
11 different elementary schools in the local school district
Obtain a Simple Random Sample
500 people from each of the 4 time zones
5 athletes from each of the 26 PSU teams
20 students from each of the 11 elementary schools
Sample
4 × 500 = 2000 selected people
26 × 5 = 130 selected athletes
11 × 20 = 220 selected students
Cluster Sampling is very different from Stratified Sampling.
With cluster sampling one should
· divide the population into groups (clusters).
· obtain a simple random sample of so many clusters from all
possible clusters.
· obtain data on every sampling unit in each of the randomly
selected clusters.
It is important to note that, unlike with the strata in stratified
sampling, the clusters should be microcosms, rather than

subsections, of the population. Each cluster should be
heterogeneous. Additionally, the statistical analysis used with
cluster sampling is not only different, but also more
complicated than that used with stratified sampling.
Table 4.3. Examples of Cluster Samples
Example 1
Example 2
Example 3
Population
All people in U.S.
All PSU intercollegiate athletes
All elementary students in a local school district
Groups (Clusters)
4 Time Zones in the U.S. (Eastern,Central, Mountain,Pacific.)
26 PSU intercollegiate teams
11 different elementary schools in the local school district
Obtain a Simple Random Sample
2 time zones from the 4 possible time zones
8 teams from the 26 possible teams
4 elementary schools from the l1 possible elementary schools
Sample
every person in the 2 selected time zones
every athlete on the 8 selected teams
every student in the 4 selected elementary schools
Each of the three examples that are found in Tables
4.2 and 4.3were used to illustrate how both stratified and cluster
sampling could be accomplished. However, there are obviously
times when one sampling method is preferred over the other.
The following explanations add some clarification about when
to use which method.
· With Example 1: Stratified sampling would be preferred over
cluster sampling, particularly if the questions of interest are
affected by time zone. Cluster sampling really works best when
there are a reasonable number of clusters relative to the entire
population. In this case, selecting 2 clusters from 4 possible

clusters really does not provide much advantage over simple
random sampling.
· With Example 2: Either stratified sampling or cluster sampling
could be used. It would depend on what questions are being
asked. For instance, consider the question “Do you agree or
disagree that you receive adequate attention from the team of
doctors at Sports Medicine when injured?” The answer to this
question would probably not be team dependent, so cluster
sampling would be fine. In contrast, if the question of interest
is “Do you agree or disagree that weather affects your
performance during an athletic event?” The answer to this
question would probably be influenced by whether or not the
sport is played outside or inside. Consequently, stratified
sampling would be preferred.
· With Example 3: Cluster sampling would probably be better
than stratified sampling if each individual elementary school
appropriately represents the entire population. Stratified
sampling could be used if the elementary schools had very
different locations (i.e., one elementary school is located in a
rural setting while another elementary school is located in an
urban setting.) Again, the questions of interest would affect
which sampling method should be used.
Judgment Sampling
The following sampling methods that are listed in your text are
types of judgment sampling:
1. volunteer samples
2. haphazard (convenience) samples
Since judgment sampling is based on human choice rather than
random selection, statistical theory cannot explain what is
happening. In your textbook, the two types of judgment
samples listed above are called “sampling disasters.”
Section 4.2. Article: “How Polls are Conducted”
The article is exceptional and provides great insight into how
major polls are conducted. When you are finished reading this
article you may want to go to the Gallup Poll Web site and see
the results from recent Gallup polls. Check your Course

Schedule for the address.
It is important to be mindful of the final point that is made in
this article. We all need to remember that public opinion on a
given topic cannot be appropriately measured with one question
that is only asked on one poll. Such results only provide a
snapshot at that moment under certain conditions. The concept
of repeating procedures over different conditions and times
leads to more valuable and durable results.Lesson 4 Practice
Questions
1. Which of the following is not an example of probability
sampling?
a. simple random sampling
b. cluster sampling
c. convenience sampling
d. stratified sampling
2. Which of the following surveys would have the smallest
margin of error?
a. a sample size of n = 1,600 from a population of 50 million
b. a sample size of n = 500 from a population of 5 billion
c. a sample size of n = 100 from a population of 10 million
3. Suppose a recent survey finds that 80% of Penn State
students prefer that fall semester begins after Labor Day. The
results of this survey were based on opinions expressed by 200
Penn State students. Which of the following represents the
calculation of the margin of error for this survey?
a. 200
b. 1/200
c. 1/ √200
d. √200
4. Suppose a margin of error for a poll is 4%. What is the
correct interpretation of the margin of error for this poll? In
about 95% of all samples of this size, the ________________.

a. difference between the sample percent and the population
percent will be within 4%.
b. probability that the sample percent does not equal the
population percent is 4%.
c. probability that the sample percent does equal the population
percent is 4%.
d. difference between the sample percent and the population
percent will exceed 4%.
5. In order to survey the opinions of its customers, a restaurant
chain obtained a random sample of 30 customers from each
restaurant in the chain. Each selected customer was asked to fill
out a survey. Which one of the following sampling plans was
used in this survey?
a. cluster sampling
b. stratified sampling
Lesson 7: Categorical Variables: Graphs and
RelationshipsAssignments
· See your Course Syllabus for the reading assignment.
Assignment.Learning ObjectivesChapters 9, 12, and 13
to:
· Interpret graphs used with categorical data.
· Distinguish between a descriptive result and an inferential
result.
· Apply what it means to be statistically significant.
· Distinguish between an actual (observed) count and an
expected count.
· Distinguish between and interpret: chi-squared statistic, risk,
relative risk, and increased risk.Terms to KnowFrom Chapter 9
· categorical data (variables)
· pie chart
· bar graph
· cluster bar graphFrom Chapters 12 and 13

· contingency table
· 2 × 2 (contingency) table
· cell
· sample percent
· population percent
· conditional percent
· margin of error
· 95% confidence interval (C.I.)
· descriptive method
· inferential method
· statistically significant
· association (relationship)
· chi-squared statistic or chi-squared test
· actual (observed) counts
· expected counts
· p-value
· risk
· relative risk
· increased risk 7.1 Overview Part I:Section 7.1. Chapter 9
Section 9.2 in Textbook
We have learned that variables (observed characteristics) can be
classified as either categorical variables or measurement
variables. We have also learned different ways to describe
measurement variables. We will now learn how to describe
categorical variables.
Figure 7.1 Categorical Data ChartGraphs: Displaying
Categorical Variables
There are two graphs that can be used to describe categorical
data. These graphs include:
1. Pie Chart
2. Bar Graph
These graphs are commonly found in the newspaper, so I
suspect that you have seen them before. Categorical data must
be numerically summarized in a table before it can be displayed
on a graph.

Example 7.1. Graphs (One Sample of One Categorical Variable)
Consider the following question that was asked on a STAT 100
Survey.
Survey Question: How would you describe your hometown?
Rural
Suburban
Small Town
Big City
The results from this question are summarized in Table 7.1.
Table 7.1. Numerical Summary of Hometown Description
Hometown
Count
Proportion
Percent
Rural
75
75/555 = .14
.14 × 100% = 14%
Suburb
296
296/555 = .53
.53 × 100% = 53%
Small Town
139
139/555 = .25
.25 × 100% = 25%
Big City
45
45/555 = .08
.08 × 100% = 8%
Total
n = 555
555/555 = 1.0
1.0 x 100% = 100%
The percents from Table 7.1 are used to make the pie chart
found in Figure 7.2.

Figure 7.2. Pie Chart of Hometown Description
As you can see in Figure 7.2, the majority of Penn State
students who were enrolled in STAT 100 during this semester
were from the suburbs. This data can also be displayed on a bar
graph, as shown in Figure 7.3.
Figure 7.3. Bar Graph of Hometown Description
In this case, both graphs do an equally good job of displaying
the data. In general, bar graphs are preferred over pie charts
when the question (variable) has more than five categories
(choices). Otherwise, it really does not matter which graph is
used.7.2 Example 7.2. Graphs (More than One Sample of One
Categorical Variable)
Consider the following survey question that was asked of four
different samples of Penn State students: 100 freshman (Fr),
100 sophomores (So), 100 juniors (Jr), and 100 seniors (Sr).
Question: Do you currently own at least one credit card?
Yes
No
The results to this question are found in Table 7.2.
Table 7.2. Responses to Credit Card Ownership by Year in
School
Credit Card Response
Fr
So
Jr
Se
Yes
42
55
76
81
No
58
45

24
19
Total
100
100
100
100
Since there is more than one sample of categorical data, the bar
graph is the only possibility. In this instance, the bar graph will
be called a cluster bar graph, because there will be a cluster of
bars for each sample. A cluster bar graph works best when the
counts are converted to percents. Percents will allow us to
compare the results from the four samples. Table 7.3 shows the
conversion of counts to percents for this sample. Each of these
percents are called conditional percents because each
calculation is restricted to or contingent on the year in school.
In this case, it was not really necessary to convert the counts
into percents because the sample size is the same for each
sample. However, since this doesn’t always happen, the
conversion to percents must be done so that a meaningful
comparison can be made. Figure 7.4 is an example of a cluster
bar graph that displays the conditional percents for the data
found in Table 7.3.
Table 7.3. Conditional Percents for Data in Table 7.2
Credit Card Response
Fr
So
Jr
Se
Yes
42/100 = .42(42%)
55 (55%)
76 (76%)
81 (81%)
No
58/100 = .58(58%)

45 (45%)
24 (24%)
19 (19%)
Total
100/100 = 1.00(100%)
100 (100%)
100 (100%)
100 (100%)
Figure 7.4. Credit Card Ownership by Year in School
The graph in Figure 7.4 does suggest that there is a difference
in the percent of Penn State students who own at least one
credit card when considering year in school. Specifically, as a
Penn State student progresses from freshman to senior year, it is
more likely that he or she will own at least one credit card.
You should also notice that there is redundant information on
the graph because the question allows for only a "yes" or "no"
response. As the percent who say "yes" increases from freshman
to senior year, the percent who say "no" also decreases from
freshman to senior year. This holds true because the data is
summarized as percents within each school year.7.3 Overview
Part II:
Chapters 12and 13 expand on the statistical methods that are
possible with categorical variables. In order to fully appreciate
what methods are now possible, we must first look at the overall
picture of statistical methods.
Figure 7.5. Breakdown of Statistical Methods
Statistics is a collection of methods (procedures) for extracting
information from data. Overall, these procedures are classified
as either descriptive methods or inferential methods.
· Descriptive methods are procedures that are used to describe a
sample. These procedures can either be graphs or numerical
summaries. We have seen examples of descriptive methods. The
choice of the graph and numerical summary varies based on the
type and number of variables that are being described.

· Inferential methods are procedures that are used to make
conclusions about a population. So far, we have only seen one
example of an inferential method. In Lesson 4, we learned
about an interval that be can be used to estimate the population
percent. This (confidence) interval is a type of inferential
method. In this lesson, we will learn more about inferential
methods.7.4 Example 7.3. Two Different Categorical Variables
Suppose a researcher conducted a study to determine if there is
a gender effect when comparing individuals who frequently
order a vegetarian meal when eating out.Table 7.4 numerically
summarizes the results for 1180 people who were surveyed
about this topic.
Table 7.4. Numerical Summary of Results from Survey
Do you frequently order a vegetarian meal when eating out?
Gender
Yes
No
Total
Female
195
240
435
Male
260
485
745
Total
455
725
1180
1. What is your gender? Female Male
2. Do you frequently order a vegetarian meal when eating out?
Yes No
In statistics, Table 7.4 is also called a contingency

table because it summarizes the data for two variables.
Specifically, this table is called a 2 × 2 contingency tableor just
a 2 × 2 tablebecause both variables (questions) have two
choices. Therefore, we have 2 rows and 2 columns in the
table. Chapter 13 explores issues about 2 × 2 tables.
Below are possible questions that could be answered using
the contingency table found in Table 7.4.
1. What percent of the sample is female?
Answer: 435/1180 = .37 or 37%
2. Among males, what percent said "yes" to the question about
frequently ordering a vegetarian meal when eating out?
Answer: 260/745 = .35 or 35% (Note: An example of a
conditional percent.)
Table 7.5. Conditional Percents by Gender on Data from Table
7.4
Gender
Yes
No
Total
Female
195/435 = .45(45%)
240/435 = .55(55%)
435/435 = 1.0 (100%)
Male
260/745 = .35(35%)
485/745 = .65(65%)
745/745 = 1.0 (100%)
Total
455
725
1180
The conditional percents are more valuable because they allow
us to compare results of the two genders ignoring that fact that

the two sample sizes are different. In this instance, the results
suggest that females are more likely than males to frequently
order a vegetarian meal when eating out, because the percent of
females that said "yes" is 45% while the percent of males that
said "yes" is 35%.
These conditional percents can also be displayed on a cluster
bar graph as shown in Figure 7.6.
Figure 7.6. Likelihood of Frequently Ordering a Vegetarian
Meal when Eating Out by Gender
However, even though this is an important finding, a
comparison of these two sample percents is only a descriptive
result and not an inferential conclusion about the two
underlying populations. In order to make inferential conclusions
about the two genders, we need to first calculate two 95%
intervals, also known as 95% confidence intervals.7.5 95%
Confidence Intervals
Earlier, we learned how to calculate a 95% confidence interval
to estimate the population percent:
95% Confidence Interval Formula: Sample Percent ± (Margin of
Error)
Table 7.6 shows the calculation of two 95% confidence
intervals that estimate the population percent who said "yes"
about frequently ordering a vegetarian meal when eating out.
Table 7.6. The 95% Confidence Intervals to Estimate Population
Percent who said "Yes"
Gender
Sample Percent that said "yes"
Sample Size (n)
Margin of Error (M.E.)
M.E. = 1/√n
95% Confidence Interval To Estimate Population Percent That
Said "Yes"
Female
45%
435

M.E. = 5%
45% ± 5% = (40 to 50)%
Male
35%
745
M.E. = 4%
35% ± 4% = (31 to 39)%
As you examine the two calculated confidence intervals (C.I.s)
found in Table 7.6, you should notice that these two confidence
intervals have no common values. Figure 7.7 shows that the
two calculated confidence intervals do not overlap. Because
these two confidence intervals do not overlap or have any
common values, we can conclude, at 95% confidence, that there
is a difference in the two genders with regard to percent whom
say "yes" about frequently ordering vegetarian meals when
eating out.
Figure 7.7. 95% Confidence Intervals from Table 7.6Decision
Rule used with two 95% Confidence Intervals to Make
Conclusions
· If the two confidence intervals do not overlap, we can
conclude that there is a difference in the two population
percents at 95% confidence.
· If the two confidence intervals do overlap, we cannot conclude
that there is a difference in two population percents, at 95%
confidence.7.6 Second Inferential Method (Assessing Statistical
Significance)
Hopefully, while listening to the news, you have at least once
heard someone report a finding that was "statistically
significant." Chapter 13 will allow us to learn what is behind
this statement of being "statistically significant"when
considering data in 2 × 2 tables.
Example 7.3. (Continued)
Recall the data that was used in Example 7.3 and displayed
in Table 7.4.
Table 7.7. Data from Table 7.4

Gender
Yes
No
Total
Female
195
240
435
Male
260
485
745
Total
455
725
1180
The research question of interest can be worded one of two
ways: The two possible wordings include:
Wording 1: Is there a statistically significant difference between
the percent who said "yes" when considering gender?
Wording 2: Is there a statistically significant relationship
between gender and the likelihood of saying "yes"?
Because this data is summarized as a 2 × 2 table, each wording
is equally acceptable. Two categorical variables that are
measured on the same individuals are related (associated) if
some choices of one variable tend to occur more often with
some choices of the second variable. Both wordings include the
phrase "statistically significant." Below is the proper definition
of the term: "statistically significant".
A statistically significantrelationship or difference is one that is
large enough to be unlikely to have occurred in the sample if
there is no relationship or difference in the population.7.7 The
Chi-Squared Statistic

Note: The term Chi-Square and Chi-Squared refer to the same
statistic. Both terms are used in textbooks.
A "statistically significant" relationship between two
categorical values is determined from a quantity called the chi-
squared statistic. This chi-squared statistic is a single number
that quantifies the amount of disparity between the actual
(observed) counts that are found in the 2 × 2 table and the
counts that would be expected if there were no relationship in
the population. The first step in determining the chi-squared
statistic is to calculate the expected count for each cell in the 2
× 2 table. Below is the proper definition of an expected count.
An expected count is a hypothetical count that would occur if in
fact there is no relationship between the two variables
Computer software was used to calculate both the expected
counts and chi-squared statistic as shown in Figure 7.8.
Figure 7.8. Chi-Squared Results
Although you will not be expected to calculate an expected
count or a chi-squared statistic, a more explicit idea of how
expected counts are calculated may help you to understand what
the chi-squared statistic is measuring.
In Figure 7.8, note that we have (455 / 1180) people who say
"yes" to ordering a vegetarian meal when eating out. If there is
no relationship between gender and ordering a vegetarian meal,
then we would expect the same proportion of the 435 females to
order vegetarian as in the overall sample. Therefore, we expect
((455 / 1180) × (435)) = 168 females to order a vegetarian meal
if there is no relationship between the two categorical variables.
We continue in this manner until we have calculated all of the
expected counts. The computer results found in Figure 7.8show
an expected count that is lower than the actual (observed)
count in two of the four cells of the table. Theexpected count is
lower than the observed count for females who say "yes" and
for males who say "no". Therefore, more females than
expected are ordering a vegetarian meal when eating out and
fewer males than expected are ordering a vegetarian meal when
eating out.

The following are interpretations of the numbers found in the
first cell of the 2 × 2 table.Interpretations
· The survey found that 195 females actually said "yes." (Note:
this is an interpretation of an actual (observed) count).
· One would expect 168 females to say "yes" if there is no
relationship between gender and likelihood of frequently
ordering a vegetarian meal when eating out. (Note this is an
interpretation of an expected count.)
As you compare the observed count of 195 with the expected
count of 168, you do notice that there is a difference of 27
between the two counts. Because of this finding, we do have
some support for a relationship between the two variables
because the observed count is not consistent with the expected
count that assumes the two variables are not related. The results
from the other three cells show similar disparities between the
observed and expected counts. The difference between the
observed and expected counts must be large enough to suggest a
relationship. A difference of only a few counts would not be
sufficient because such a small difference could result by
chance alone.
The chi-squared statistic is a single number that quantifies the
amount of disparity between the actual (observed) counts and
the expected counts for all the cells of the table combined. With
the chi-squared statistic the following holds true.
For any size contingency table:
· If the chi-squared statistic = 0, there is no relationship
between the two variables. This means that for every cell in the
table, the actual count will equal the expected count. This is the
smallest value that a chi-squared statistic can assume.
For a 2 × 2 contingency table:
· If the chi-squared statistic ≥ 3.84 (a value called the "critical
value"), there is support for a statistically significant
relationship between the two variables. There is no upper
boundary for the chi-squared statistic. For contingency tables
larger than 2 rows and 2 columns, the "critical value" is larger
than 3.84.

From Figure 7.8, we find that with our example, the chi-squared
statistic is 11.43. Since 11.43 > 3.84 there is support for a
statistically significant relationship between the two
variables.7.8 The P-value
With the advent of computer software, we now have another
way to determine whether or not a relationship between two
categorical variables is statistically significant. This is good
news because most people, other than statisticians, have no clue
why 3.84 is the magic boundary for a statistically significant
relationship between the variables in a 2 × 2 contingency table.
This is especially helpful for contingency tables that are larger
than 2 × 2. You do not have to determine the "critical value"
for the chi-squared statistic for every size table. The p-value of
the chi-squared statistic will give you all the information you
need to determine statistical significance. The p-value is
an inferential method.
Remember that a statistically significant relationship is one that
is large enough to be unlikely to have occurred in the sample if
there’s no relationship in the population. A p-value is a
probability that measures how likely it is to observe the
relationship or one even stronger if there’s really no
relationship in the population. Two properties about a p-
value are:
· possible values for the p-value are 0 to 1.0 because it is a
probability
· calculation is based on the value of the chi-squared statistic
If you look at the computer output from Figure 7.8. you find
that the p-value is .001. Remember this probability is based on
the fact that our chi-squared statistic is 11.43.
· Interpretation of our p-value: The likelihood of getting our
chi-squared statistic of 11.43 or any value more extreme, if in
fact there is no relationship in the population, is .001.
Since it is highly unlikely (.001 = .1%) that we would get our
chi-squared statistic of 11.43 or any chi-square statistic larger
than 11.43, if there is really no relationship in the population,
we can conclude that our results are inconsistent with the

position that there is no relationship in the population. So we
can conclude that there is a statistically significant relationship
in the population.
Since the p-value is confusing to some, we will revisit what the
chi-squared statistic is measuring. The chi-squared statistic is a
measure of the magnitude of the difference between what we
observe (observed counts) and what we would expect to observe
if there is no relationship between the variables (expected
counts). Statisticians have calculated the probability of having
a chi-squared statistic of a certain value or larger value.
Statisticians call this the right-tail probability since values get
larger as you move to the right on a number line. The right-
tail probability is what the p-value indicates when used with the
chi-squared statistic. Small p-values are associated with large
chi-squared values and large chi-squared values mean that the
difference between the observed and expected counts is too
large to be by chance alone and that there must be a relationship
between the two variables.
To further illustrate how unlikely a chi-squared value that is
11.43 or larger is, look at the histogram in Figure 7.9below. The
histogram shows the frequency (out of 10,000) of chi-squared
values (for a 2 × 2 table) when there is no relationship in the
population. Most of the chi-squared values are 0 and the
histogram is right-skewed. Values above 11.43 are almost non-
existent which is demonstrated by our p-value of .001.
Figure 7.9 Histogram of the Chi-Squared Statistic for a 2 × 2
Table
The histogram will change shape for tables larger than 2 rows
and 2 columns, but the shape will still be right-skewed. So,
larger chi-square values will always be highly unlikely and have
lowp-values. A histogram for the chi-squared statistic for a 2 ×
8 table is shown in Figure 7.10 below.
Figure 7.10 Histogram of the Chi-Squared Statistic for a 2 × 8
Table Decision Rule used with P-Value to Make Conclusions

· If the p-value ≤ .05, we can conclude that there is a
statistically significant relationship between the variables
· If the p-value > .05 we cannot conclude that there is a
statistically significant relationship between the variables
Figure 7.11. Right-tail p-value probability for Chi-squared
distribution
Figure 7.11 shows the relationship between the chi-squared
value (χ2) and the right-tail p-value probability.
Table 7.8 shows that we obtained the same conclusion with both
inferential methods.
Table 7.8. Overall Inferential Conclusions with Example 7.3
Inferential Procedure
Inferential Result
Inferential Conclusion
95% Confidence Intervals (C.I.)
C.I.’s for Population Percent who said "yes"
95% C.I. for females: (40 to 50)%
95% C.I. for males: (31 to 39)%
Since the two C.I.’s do not overlap, we can conclude that there
is a statistically significant difference in the two genders with
regard to the percent who said "yes," at 95% confidence
Significance Test (P-value)
P-value= .001 from chi-squared statistic
Since the P-value ≤ .05 we can conclude that there is a
statistically significant relationship between gender and
likelihood of saying "yes"
Below is an overall summary of what has been so far discussed
in the three examples.
Figure 7.12. Categorical Variables Breakdown
Sometimes one data set can be viewed either as (one sample of
more than one categorical variable) or (two different categorical
variables.) This is certainly possible with both Example
7.2and Example 7.3. To illustrate the point, let’s
examine Example 7.3.

With Example 7.3, the first inferential procedure involved
comparing two 95% Confidence Intervals. This seems to work
best if you assume that there are really two samples (females
and males) of the one categorical variable of interest: "whether
they frequently order a vegetarian meal". In contrast, the
second inferential procedure involved using the p-value from
the chi-squared statistic. In this instance, it makes more sense to
assume that there are two different categorical variables
because stated conclusions include the word "relationship".
Either approach is acceptable. So it’s up to the researcher to
decide which wording bests fits the proposed research
question.7.9 Other Numbers That Can Describe 2 × 2 Tables
Sometimes data that is collected in a 2 × 2 table has an outcome
that is undesirable.
Because of this, measures other than the chi-squared statistic
may be more informative.
These measures, which are found in Chapter 12, include the
following:
1. Risk
2. Relative Risk
3. Increased Risk
Each of the measures is a number that is used to evaluate
chance. Below are the formulas that are used to obtain these
measures. (Note: you will not be asked to calculate any of these
measures.)
Risk = (number with trait/total)
Relative Risk = Risk1/Risk2 (Note: Always put smaller risk on
the bottom)
Increased Risk = (Relative Risk - 1.0) × 100%
Example 7.4. Risk, Relative Risk, and Increased Risk
A recent study examined the incidence of injuries for male and
female high school athletes. The response variable was whether
the athlete has experienced an injury during the school year or
not. Suppose the data were as found in Table 7.9.
Table 7.9. Experiencing Injury by Gender

Experienced Injury?
Gender
Yes
No
Total
Female
150
350
500
Male
100
400
500
Total
250
750
1000
In this example, the undesirable trait (outcome) is experiencing
injury. So the calculated risk of injury for each gender is:
For Females: Risk = (number with trait)/total = 150/500 = .3
(30%)
For Males: Risk =(number with trait)/total = 100/500 = .2 (20%)
Riskis just another name for a probability or proportion. Risks
can also be converted to percents. A risk is a type of conditional
percent. In this example, we find that the risk for injury is
higher for females than males. However, sometimes researchers
prefer to report the two risks as a single quantity. Two
possibilities are the relative risk and the increased risk.
Relative Risk = Riskfemales/Riskmales = .3/.2 = 30%/20% =
1.5
Relative Risk Interpretation: A female athlete is 1.5 times more
likely to experience injury than a male athlete during the school
year.
Increased Risk = (Relative Risk - 1.0) × 100% = (1.5 – 1.0) ×
100% = 50%

Increased Risk Interpretation: The risk for injury during the
school year is 50% higher for female athletes than for male
athletes.
Relative risks and increased risks are reported in the news all
the time. However, these measures are only descriptive and
cannot be used to make inferential conclusions.
Example 7.5. Clarification of Risk, Relative Risk, and Increased
Risk
Which of the following three choices show the largest
magnitude of difference in the two risks when comparing
"percent that quit smoking" for users and non-users of the
patch? The results are based on sample of 50 smokers who used
the patch and a sample of 50 smokers who did not use the patch.
Choice A: 50% of users quit whereas 25% of non-users quit
Choice B: 10% of users quit whereas 5% of non-users quit
Choice C: 2% of users quit whereas 1% of non-users quit
I suspect that most of you correctly selected Choice A.
However, what is interesting is the fact that all three choices
have the same relative risk and increased risk as shown in Table
7.10.
Table 7.10. Results for the Three Choices
Choice
Relative Risk
Increased Risk
Chi-Square Statistic & P-value
A
50%/25% = 2.0
(2.0-1) × 100% = 100%
Chi-Square = 17.14 & P-value = .000*
B
10%/5% = 2.0
(2.0-1) × 100% = 100%
Chi-Square = 1.81 & P-value = .179
C
2%/1% = 2.0
(2.0-1) × 100% = 100%

Chi-Square = .338 & P-value = .561
*Statistically Significant
From Table 7.10, the following interpretations can be made.
· Relative Risk Interpretation: A person is 2 times more likely
to quit smoking when using the patch.
· Increased Risk Interpretation: There is a 100% increase in the
chance of quitting smoking when using the patch.
Seeing an increased risk of 100% does make one believe that
there is a substantial difference in the two groups, which, in this
case, suggests that using the nicotine patch does make a huge
difference when trying to quit smoking. The smallest value that
an increased risk can assume is 0%, which indicates that the
risk is the same for the two groups.
In contrast, when compared to the increased risk, the relative
risk may seem more subtle and less impressive. The smallest
value that a relative risk can assume is 1.0, which indicates that
the risk is the same for the two groups. So while a relative risk
of 2.0 certainly suggests that using the nicotine patch does make
some difference, the finding is not as dramatic as the increased
risk.
The critical point is that measures such as relative risk and
increased risk are just one numerical result and by themselves
cannot tell the entire story. You should also read Section
12.3 in your textbook to see how a missing baseline risk can
also lead to misinterpreting the relative risk and the increased
risk.
Choice A has the largest magnitude of effect that leads to not
only the smallest p-value, but also the only p-value that is
statistically significant. So with Choice A one can claim that
there is a statistically significant relationship between quitting
smoking and whether or not a person uses the patch. This
finding supports that Choice A has the largest magnitude of
difference in the two risks.Lesson 7 Practice Questions

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