Chapter 2: Graphical Descriptive Techniques
This chapter discussed graphical descriptive methods used to summarize and describe sets of data.
At the completion of this chapter, you are expected to know the following:
1. How to recognize whether the type of data under consideration is quantitative, qualitative, or
2. How to summarize a set of quantitative data by means of a frequency distribution, histogram,
relative frequency polygon, and stem and leaf display.
3. How to summarize a set of qualitative data by means of a pie chart and bar chart.
2.2 Types of Data
This section introduced the two m ain t ypes of dat a t hat are referred t o t hroughout t he t ext:
quantitative (numerical) data and qualitative (categori cal) data. The appropriate graphical method to be
used in presenting data depends, in part, on the type of data under consideration. Later in the text, when
statistical inference is covered, the data type will help to identify the appropriate statistical technique to
be used in solving a problem. In a few situations, it will be necessary to recognize whether or not a set of
nonquantitative data can be ordered. If the categories for a set of nonquantitative data can be ordered or
ranked, we have a third type of data, called ranked data.
At the completion of this section, you should be able to identify whether the type of data under con-
sideration is quantitative, qualitative, or ranked.
Question: How do I identify quantitative data?
Answer: Quantitative data are real numbers. They are not numbers arbitrarily assigned
to represent qualitative data. An experi ment that produces qualitative data al-
ways asks for verbal, nonnumerical responses (e.g., yes and no; defective and
nondefective; Catholic, Protestant, and other).
For each of the following examples of data, determine whether the data type is quantitative, qualita-
tive, or ranked.
a) the weekly level of the prime interest rate during the past year
b) the make of car driven by each of a sample of executives
c) the number of contacts made by each of a companys salespersons during a week
4 d) the rating (excellent, good, fa ir, or poor) given to a particular television program by each of a
sample of viewers
e) the number of shares traded on the New York Stock Exchange each week throughout 1987
a) Quantitative, if th e interest rate lev el is ex pressed as a percentage. If the level is simply ob-
served as being high, moderate, or low, then the data type is qualitative.
b ) Qualitative
d) Ranked, because the categories can be ordered
2.1 Describe the difference between quantitative data and qualitative data.
2.2 For each of the following exam ples of data , determine whether the data are quantitative,
qualitative, or ranked.
a) the month of the highest sales for each firm in a sample
b) the department in which each of a sample of university professors teaches
c) the weekly closing price of gold throughout a year
d) the size of soft drink (large, medium, or sm all) ordered by a sam ple of cust omers in a
e) the number of barrels of crude oil imported monthly by the United States
2.3 Identify the type of data observed for each of the following variables.
a) the number of students in a statistics class
b) the student evaluations of the professor (1 = poor, 5 = excellent)
c) the political preferences of voters
d) the states in the United States of America
e) the size of a condominium (in square feet)
5 2.3 Graphical Techniques for Quantitative Data
This section introduced the basic methods of descriptive statistics used for organi zing a set of nu-
merical data in tabular form and present ing it graphically. Summarizing data in this way requires that
you first group t he data into classes. Judgment is required concerning the number and t he size of t he
classes t o be used. The i mportant poi nt t o bear i n m ind when making this judgment is that the
presentation of t he grouped dat a shoul d enabl e the user t o qui ckly grasp t he general shape of the
distribution of the data.
At the completion of this section, you should be able to summarize a set of numerical data in the fol-
1. Organize the data into a frequency distribution.
2. Construct a histogram.
3. Construct a frequency polygon.
4. Construct the relative frequency counterparts of points 1, 2, and 3.
5. Construct an ogivethe graph of a cumulative relative frequency distribution.
6. Construct a stem and leaf display.
Question: How do I choose the number of cl asses and t he width of t he classes to be
used in constructing a frequency distribution?
Answer: Although this choice is arbitrary and no hard and fast rules can be given, here
are a few useful guidelines:
1. The classes must be nonoverlapping, so that each measurement falls into
exactly one class. Therefore, choose the classes so that no measurement falls
on a class boundary.
2. Choose the number of classes to be used as a num ber between 5 and 20,
with smaller numbers of classes being chosen for smaller data sets.
3. The approximate width of each class is given by the following:
Largest value Smallest value
Approximate class width =
Number of classes
Choose the actual class width to be a val ue close to the approximate width
that is convenient to work with. Avoid awkward fractional values.
6 Example 2.2
The weights in pounds of a group of workers are as follows:
173 165 171 175 188
183 177 160 151 169
162 179 145 171 175
168 158 186 182 162
154 180 164 166 157
a) Construct a stem and leaf display for these data.
b) Construct a frequency distribution for these data.
a) The first step in constructing a stem and leaf display is to decide how to split each observation
(weight) into two parts: a stem and a leaf. For this example, we will define the first two digits of
an observation to be its stem and the third digit to be its leaf. Th us, the first two weights are
split into a stem and a leaf as follows:
Weight Stem Leaf
173 17 3
183 18 3
Scanning the remaining weights, we find that there are five possible stems (14, 15, 16, 17 and
18), which we list in a column from smallest to largest, as sh own below. Next, we co nsider
each observation in turn and place its leaf in the same row as its stem, to the right of the vertical
line. The resulting stem and l eaf display shown bel ow has grouped t he 25 wei ghts into five
categories. The second row of the display, corresponding to the stem 15, has four leaves: 4, 8, 1
and 7. The four weights represented in the second row are therefore 154, 158, 151, and 157.
15 4 8 1 7
16 2 8 5 0 4 6 9 2
17 3 7 9 1 5 1 5
18 3 0 6 2 8