by Denise Harvey and David M. Lane
Chapter 1: Levels of Measurement
The Board of Trustees at a university commissioned a top management-consulting
ﬁrm to address the admission processes for academic and athletic programs. The
consulting ﬁrm wrote a report discussing the trade-off between maintaining
academic and athletic excellence. One of their key ﬁndings was:
The standard for an athlete’s admission, as reﬂected in SAT
scores alone, is lower than the standard for non-athletes by as
much as 20 percent, with the weight of this difference being
carried by the so-called “revenue sports” of football and
basketball. Athletes are also admitted through a different
process than the one used to admit non-athlete students.
What do you think?
Based on what you have learned in this chapter about measurement scales, does it
make sense to compare SAT scores using percentages? Why or why not?
Think about this before continuing:
As you may know, the SAT has an arbitrarily-determined lower
limit on test scores of 200. Therefore, SAT is measured on
either an ordinal scale or, at most, an interval scale. However, it
is clearly not measured on a ratio scale. Therefore, it is not
meaningful to report SAT score differences in terms of
percentages. For example, consider the effect of subtracting 200
from every student's score so that the lowest possible score is 0.
How would that affect the difference as expressed in
All material presented in Chapter: “Introduction”
1. A teacher wishes to know whether the males in his/her class have more
conservative attitudes than the females. A questionnaire is distributed assessing
attitudes and the males and the females are compared. Is this an example of
descriptive or inferential statistics?
2. A cognitive psychologist is interested in comparing two ways of presenting
stimuli on sub- sequent memory. Twelve subjects are presented with each method
and a memory test is given. What would be the roles of descriptive and
inferential statistics in the analysis of these data?
3. If you are told only that you scored in the 80th percentile, do you know from
that description exactly how it was calculated? Explain.
4. A study is conducted to determine whether people learn better with spaced or
massed practice. Subjects volunteer from an introductory psychology class. At
the beginning of the semester 12 subjects volunteer and are assigned to the
massed-practice condition. At the end of the semester 12 subjects volunteer and
are assigned to the spaced-practice condition. This experiment involves two
kinds of non-random sampling: (1) Subjects are not randomly sampled from
some speciﬁed population and (2) subjects are not randomly assigned to
conditions. Which of the problems relates to the generality of the results? Which
of the problems relates to the validity of the results? Which problem is more
5. Give an example of an independent and a dependent variable.
6. Categorize the following variables as being qualitative or quantitative:
Rating of the quality of a movie on a 7-point scale
Country you were born in
Time to respond to a question
7. Specify the level of measurement used for the items in Question 6.
8. Which of the following are linear transformations?
Converting from meters to kilometers
Squaring each side to ﬁnd the area
Converting from ounces to pounds
Taking the square root of each person's height.
Multiplying all numbers by 2 and then adding 5
Converting temperature from Fahrenheit to Centigrade
9. The formula for ﬁnding each student’s test grade (g) from his or her raw score
(s) on a test is as follows: g = 16 + 3s
Is this a linear transformation?
If a student got a raw score of 20, what is his test grade?
10. For the numbers 1, 2, 4, 16, compute the following:
11. Which of the frequency polygons has a large positive skew? Which has a large
12. What is more likely to have a skewed distribution: time to solve an anagram
problem (where the letters of a word or phrase are rearranged into another
word or phrase like “dear” and “read” or “funeral” and “real fun”) or scores on
a vocabulary test?
Questions from Case Studies
Angry Moods (AM) case study
13. (AM) Which variables are the participant variables? (They act as independent
variables in this study.)
14. (AM) What are the dependent variables?
15. (AM) Is Anger-Out a quantitative or qualitative variable?
Teacher Ratings (TR) case study
16. (TR) What is the independent variable in this study?
ADHD Treatment (AT) case study
17. (AT) What is the independent variable of this experiment? How many levels
does it have?
18. (AT) What is the dependent variable? On what scale (nominal, ordinal, interval,
ratio) was it measured?
2. Graphing Distributions
1. Stem and Leaf Displays
3. Frequency Polygons
4. Box Plots
5. Bar Charts
6. Line Graphs
7. Dot Plots
Graphing data is the ﬁrst and often most important step in data analysis. In this day
of computers, researchers all too often see only the results of complex computer
analyses without ever taking a close look at the data themselves. This is all the
more unfortunate because computers can create many types of graphs quickly and
This chapter covers some classic types of graphs such bar charts that were
invented by William Playfair in the 18th century as well as graphs such as box
plots invented by John Tukey in the 20th century.
Graphing Qualitative Variables
by David M. Lane
1. Create a frequency table
2. Determine when pie charts are valuable and when they are not
3. Create and interpret bar charts
4. Identify common graphical mistakes
When Apple Computer introduced the iMac computer in August 1998, the
company wanted to learn whether the iMac was expanding Apple’s market share.
Was the iMac just attracting previous Macintosh owners? Or was it purchased by
newcomers to the computer market and by previous Windows users who were
switching over? To ﬁnd out, 500 iMac customers were interviewed. Each customer
was categorized as a previous Macintosh owner, a previous Windows owner, or a
new computer purchaser.
This section examines graphical methods for displaying the results of the
interviews. We’ll learn some general lessons about how to graph data that fall into
a small number of categories. A later section will consider how to graph numerical
data in which each observation is represented by a number in some range. The key
point about the qualitative data that occupy us in the present section is that they do
not come with a pre-established ordering (the way numbers are ordered). For
example, there is no natural sense in which the category of previous Windows
users comes before or after the category of previous Macintosh users. This
situation may be contrasted with quantitative data, such as a person’s weight.
People of one weight are naturally ordered with respect to people of a different
All of the graphical methods shown in this section are derived from frequency
tables. Table 1 shows a frequency table for the results of the iMac study; it shows
the frequencies of the various response categories. It also shows the relative
frequencies, which are the proportion of responses in each category. For example,
the relative frequency for “none” of 0.17 = 85/500.
Table 1. Frequency Table for the iMac Data.
The pie chart in Figure 1 shows the results of the iMac study. In a pie chart, each
category is represented by a slice of the pie. The area of the slice is proportional to
the percentage of responses in the category. This is simply the relative frequency
multiplied by 100. Although most iMac purchasers were Macintosh owners, Apple
was encouraged by the 12% of purchasers who were former Windows users, and
by the 17% of purchasers who were buying a computer for the ﬁrst time.
Figure 1. Pie chart of iMac purchases illustrating frequencies of previous
Pie charts are effective for displaying the relative frequencies of a small number of
categories. They are not recommended, however, when you have a large number of
categories. Pie charts can also be confusing when they are used to compare the
outcomes of two different surveys or experiments. In an inﬂuential book on the use
of graphs, Edward Tufte asserted “The only worse design than a pie chart is several
Here is another important point about pie charts. If they are based on a small
number of observations, it can be misleading to label the pie slices with
percentages. For example, if just 5 people had been interviewed by Apple
Computers, and 3 were former Windows users, it would be misleading to display a
pie chart with the Windows slice showing 60%. With so few people interviewed,
such a large percentage of Windows users might easily have occurred since chance
can cause large errors with small samples. In this case, it is better to alert the user
of the pie chart to the actual numbers involved. The slices should therefore be
labeled with the actual frequencies observed (e.g., 3) instead of with percentages.
Bar charts can also be used to represent frequencies of different categories. A bar
chart of the iMac purchases is shown in Figure 2. Frequencies are shown on the Y-
axis and the type of computer previously owned is shown on the X-axis. Typically,
the Y-axis shows the number of observations in each category rather than the
percentage of observations in each category as is typical in pie charts.
Number of Buyers
Figure 2. Bar chart of iMac purchases as a function of previous computer
Often we need to compare the results of different surveys, or of different
conditions within the same overall survey. In this case, we are comparing the
“distributions” of responses between the surveys or conditions. Bar charts are often
excellent for illustrating differences between two distributions. Figure 3 shows the
number of people playing card games at the Yahoo web site on a Sunday and on a
Wednesday in the spring of 2001. We see that there were more players overall on
Wednesday compared to Sunday. The number of people playing Pinochle was
nonetheless the same on these two days. In contrast, there were about twice as
many people playing hearts on Wednesday as on Sunday. Facts like these emerge
clearly from a well-designed bar chart.
Figure 3. A bar chart of the number of people playing different card games
on Sunday and Wednesday.
The bars in Figure 3 are oriented horizontally rather than vertically. The horizontal
format is useful when you have many categories because there is more room for
the category labels. We’ll have more to say about bar charts when we consider
numerical quantities later in this chapter.
Some graphical mistakes to avoid
Don’t get fancy! People sometimes add features to graphs that don’t help to convey
their information. For example, 3-dimensional bar charts such as the one shown in
Figure 4 are usually not as effective as their two-dimensional counterparts.
Documents you may be interested
Documents you may be interested