44 Chapter 2 Frequency Distributions and Graphs

Make a Grouped Frequency Distribution

(Quantitative Variable)
Example MT2–2
1. Type the data used in Example 2–2 into C1 or open the worksheet from the website.
2. Select Graph>Character Graphs>Histogram. The cursor should be
blinking in the Variables dialog box. If not, click inside the box then double click C1
TEMPERATURES in the list. Only quantitative variables will be shown in this list
since this menu option will only work with that data type.
3. Click in the box for First midpoint: then type in 102.
4. Click in the box for Interval width: then type in 5. You must calculate the class width
(5) and the midpoint of the first class (102) using the instructions in text.
5. Click [OK].

Grouped Frequency Table

The table and a histogram will be displayed in the Session Window. The midpoint of each class is
displayed, not the class limits or boundaries. “TEMPERATURES,” the column name, is truncated
to eight characters for display only.

2–3

Histograms, Frequency After the data have been organized into a frequency distribution, they can be presented
Polygons, and Ogives in graphic forms. The purpose of graphs in statistics is to convey the data to the viewer
Objective 2. Represent data
in pictorial form. It is easier for most people to comprehend the meaning of data pre-
in frequency distributions sented graphically than data presented numerically in tables or frequency distributions.
using histograms, frequency This is especially true if they have little or no statistical knowledge.
polygons, and ogives. Statistical graphs can be used to describe the data set or analyze it. Graphs are
also useful in getting the audience’s attention in a publication or a speaking presenta-
tion. They can be used to discuss an issue, reinforce a critical point, or summarize a
data set. They can also be used to discover a trend or pattern in a situation over a pe-
riod of time.
 Section 2–3 Histograms, Frequency Polygons, and Ogives 45

The three most commonly used graphs in research are

1. The histogram.
2. The frequency polygon.
3. The cumulative frequency graph, or ogive (pronounced o-jive).
An example of each type of graph is shown in Figure 2–1. The data for each graph are
the distribution of the miles that 20 randomly selected runners ran during a given week.
Figure 2–1 y
Examples of Commonly
Used Graphs 5

Frequency 4

1
x
5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5
Class boundaries
(a) Histogram

4
Frequency

1
x
8 13 18 23 28 33 38
Class midpoints
(b) Frequency polygon

20
18
16
Cumulative frequency

14
12
10
8
6
4
2
x
5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5
Class boundaries
(c) Cumulative frequency graph
46 Chapter 2 Frequency Distributions and Graphs

The Histogram
The histogram is a graph that displays the data by using vertical bars of various heights to represent the
frequencies.

Example 2–4 Construct a histogram to represent the data shown below for the record high tempera-
tures for each of the 50 states (see Example 2–2).
Historical Note Class boundaries Frequency
Graphs originated when 99.5–104.5 2
ancient astronomers drew 104.5–109.5 8
the position of the stars in
the heavens. Roman
109.5–114.5 18
surveyors also used 114.5–119.5 13
coordinates to locate 119.5–124.5 7
landmarks on their maps.
124.5–129.5 1
The development of
statistical graphs can be 129.5–134.5 1
traced to William Playfair
(1748–1819), an engineer/
Solution
drafter who used graphs STEP 1 Draw and label the x and y axes. The x axis is always the horizontal axis,
to present economic data
pictorially.
and the y axis is always the vertical axis.
STEP 2 Represent the frequency on the y axis and the class boundaries on the x axis.
STEP 3 Using the frequencies as the heights, draw vertical bars for each class. See
Figure 2–2.
Figure 2–2 y
Histogram for 18
Example 2–4
15
Frequency

12

3
x
0
99.5° 104.5° 109.5° 114.5° 119.5° 124.5° 129.5° 134.5°
Temperature

As the histogram shows, the class with the greatest number of data values (18) is
109.5–114.5, followed by 13 for 114.5–119.5. The graph also has one peak with the data
clustering around it.

The Frequency Polygon Another way to represent the same data set is by using a frequency polygon.

The frequency polygon is a graph that displays the data by using lines that connect points plotted for
the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the
points.
 Section 2–3 Histograms, Frequency Polygons, and Ogives 47

The next example shows the procedure for constructing a frequency polygon.

Example 2–5 Using the frequency distribution given in Example 2–4, construct a frequency polygon.
Solution
STEP 1 Find the midpoints of each class. Recall that midpoints are found by adding
the upper and lower boundaries and dividing by 2.
99.5  104.5 104.5  109.5
 102  107
2 2

And so on. The midpoints are listed next.

Class boundaries Midpoints Frequency
99.5–104.5 102 2
104.5–109.5 107 8
109.5–114.5 112 18
114.5–119.5 117 13
119.5–124.5 122 7
124.5–129.5 127 1
129.5–134.5 132 1

STEP 2 Draw the x and y axes. Label the x axis with the midpoint of each class, and
then use a suitable scale on the y axis for the frequencies.
STEP 3 Using the midpoints for the x values and the frequencies as the y values, plot
the points.
STEP 4 Connect adjacent points with straight lines. Draw a line back to the
x axis at the beginning and end of the graph, at the same distance that
the previous and next midpoints would be located, as shown in
Figure 2–3.
48 Chapter 2 Frequency Distributions and Graphs

Figure 2–3 y
Frequency Polygon for 18
Example 2–5
15

Frequency
12

3
x
0
102° 107° 112° 117° 122° 127° 132°
Temperature

The frequency polygon and the histogram are two different ways to represent
the same data set. The choice of which one to use is left to the discretion of the
researcher.

The Ogive The third type of graph that can be used represents the cumulative frequencies for the
classes. This type of graph is called the cumulative frequency graph or ogive. The cu-
mulative frequency is the sum of the frequencies accumulated up to the upper bound-
ary of a class in the distribution.

The ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

Example 2–6 shows the procedure for constructing an ogive.

Example 2–6 Construct an ogive for the frequency distribution described in Example 2–4.

Solution
STEP 1 Find the cumulative frequency for each class.
Class boundaries Cumulative frequency
99.5–104.5 2
104.5–109.5 10
109.5–114.5 28
114.5–119.5 41
119.5–124.5 48
124.5–129.5 49
129.5–134.5 50

STEP 2 Draw the x and y axes. Label the x axis with the class boundaries. Use
an appropriate scale for the y axis to represent the cumulative frequen-
cies. (Depending on the numbers in the cumulative frequency columns,
scales such as 0, 1, 2, 3, . . . , or 5, 10, 15, 20, . . . , or 1000, 2000,
 Section 2–3 Histograms, Frequency Polygons, and Ogives 49

3000, . . . can be used. Do not label the y axis with the numbers in the
cumulative frequency column.) In this example, a scale of 0, 5, 10, 15, . . .
will be used.
STEP 3 Plot the cumulative frequency at each upper class boundary, as shown in
Figure 2–4. Upper boundaries are used since the cumulative frequencies
represent the number of data values accumulated up to the upper boundary
of each class.

Figure 2–4
y
Plotting the Cumulative
50
Frequency for
45
Example 2–6
40
35
Cumulative
Frequency

30
25
20
15
10
5 x
0
99.5° 104.5° 109.5° 114.5° 119.5° 124.5° 129.5° 134.5°
Temperature

STEP 4 Starting with the first upper class boundary, 104.5, connect adjacent points
with straight lines, as shown in Figure 2–5. Then extend the graph to the
first lower class boundary, 99.5, on the x axis.

Figure 2–5 y
Ogive for Example 2–6 50
45
40
35
Cumulative
Frequency

30
25
20
15
10
5 x
0
99.5° 104.5° 109.5° 114.5° 119.5° 124.5° 129.5° 134.5°
Temperature

Cumulative frequency graphs are used to visually represent how many values are
below a certain upper class boundary. For example, to find out how many record high
temperatures are less than 114.5, locate 114.5 on the x axis, draw a vertical line up un-
til it intersects the graph, and then draw a horizontal line at that point to the y axis. The
y axis value is 28, as shown in Figure 2–6.
50 Chapter 2 Frequency Distributions and Graphs

Figure 2–6 y
Finding a Specific 50
Cumulative Frequency 45
40
35

Cumulative
Frequency
30
28
25
20
15
10
5 x
0
99.5° 104.5° 109.5° 114.5° 119.5° 124.5° 129.5° 134.5°
Temperature

The steps for drawing the three types of graphs are shown in the Procedure Table.

Procedure
Procedure Table
Table

Constructing Statistical Graphs

STEP 1 Draw and label the x and y axes.
STEP 2 Choose a suitable scale for the frequencies or cumulative frequencies and label it
on the y axis.
STEP 3 Represent the class boundaries for the histogram or ogive, or the midpoint for the
frequency polygon, on the x axis.
STEP 4 Plot the points and then draw the bars or lines.

Relative Frequency The histogram, the frequency polygon, and the ogive shown previously were con-
Graphs structed by using frequencies in terms of the raw data. These distributions can be
converted into distributions using proportions instead of raw data as frequencies. These
types of graphs are called relative frequency graphs.
Graphs using relative frequencies instead of frequencies are used when the propor-
tion of data values that fall into a given class is more important than the actual number
of data values that fall into that class. For example, if one wanted to compare the age
distribution of adults in the city of Philadelphia, Pennsylvania, with the age distribution
of adults of Erie, Pennsylvania, one would use relative frequency distributions. The rea-
son is that since the population of Philadelphia is 1,478,002 and the population of Erie
is 105,270, the bars using the actual data values for Philadelphia would be much taller
than those for the same classes for Erie.
To convert a frequency into a proportion or relative frequency, divide the frequency
for each class by the total of the frequencies. The sum of the relative frequencies will al-
ways be 1. These graphs are similar to the ones that use raw data as frequencies, but the
values on the y axis are in terms of proportions. The next example shows the three types
of relative frequency graphs.
 Section 2–3 Histograms, Frequency Polygons, and Ogives 51

Example 2–7 Construct a histogram, frequency polygon, and ogive using relative frequencies for the
distribution (shown here) of the miles 20 randomly selected runners ran during a given
week.
Class Cumulative
boundaries Frequency frequency
5.5–10.5 1 1
10.5–15.5 2 3
15.5–20.5 3 6
20.5–25.5 5 11
25.5–30.5 4 15
30.5–35.5 3 18
35.5–40.5 2 20
20

Solution
STEP 1 Convert each frequency to a proportion or relative frequency by dividing the
frequency for each class by the total number of observations.
For class 5.5–10.5, the relative frequency is 201  0.05.
For class 10.5–15.5, the relative frequency is 202  0.10.
For class 15.5–20.5, the relative frequency is 203  0.15.
And so on.
STEP 2 Using the same procedure, find the relative frequencies for the cumulative
frequency column. The relative frequencies are shown here.
Cumulative
Class Relative relative
boundaries Midpoints frequency frequency
5.5–10.5 8 0.05 0.05
10.5–15.5 13 0.10 0.15
15.5–20.5 18 0.15 0.30
20.5–25.5 23 0.25 0.55
25.5–30.5 28 0.20 0.75
30.5–35.5 33 0.15 0.90
35.5–40.5 38 0.10 1.00
1.00

STEP 3 Draw each graph as shown in Figure 2–7. For the histogram and ogive, use
the class boundaries along the x axis. For the frequency polygon, use the
midpoints on the x axis. The scale on the y axis uses proportions.

When analyzing histograms and frequency polygons, look at the shape of the curve.
For example, does it have one peak or two peaks, or is it relatively flat, or is it
U-shaped? Are the data values spread out on the graph, or are they clustered around the
52 Chapter 2 Frequency Distributions and Graphs

Figure 2–7 y
Graphs for Example 2–7 0.25

0.20

Relative frequency
0.15

0.10

0.05

x
0
5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5
Miles
(a) Histogram

y
0.25

0.20
Relative frequency

0.15

0.10

0.05

x
0
8 13 18 23 28 33 38
Miles
(b) Frequency polygon

y
1.00
Cumulative relative frequency

0.80

0.60

0.40

0.20

x
0
5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5
Miles
(c) Ogive
 Section 2–3 Histograms, Frequency Polygons, and Ogives 53

center? Are there data values in the extreme ends? These may be outliers. (See Section
3–4 for an explanation of outliers.) Are there any gaps in the histogram, or does the fre-
quency polygon touch the x axis somewhere other than the ends? Finally, are the data
clustered at one end or the other, indicating a skewed distribution? (See Section 3–2 for
an explanation of skewness.)
For example, the histogram for the record high temperatures shown in Figure 2–2
(page 46 ) shows a single peaked distribution, with the class 109.5–114.5 containing the
largest number of temperatures. The distribution has no gaps, and there are fewer tem-
peratures in the highest class than in the lowest class.

Exercises
2–19. For 108 randomly selected college applicants, the 2–23. Thirty automobiles were tested for fuel efficiency, in
following frequency distribution for entrance exam scores miles per gallon (mpg). The following frequency
was obtained. Construct a histogram, frequency polygon, distribution was obtained. Construct a histogram, frequency
and ogive for the data. (The data for this exercise will be polygon, and ogive for the data. (The data for this exercise
used for Exercise 2–31.) will be used for Exercise 2–33.)
Class limits Frequency Class boundaries Frequency
90–98 6 7.5–12.5 3
99–107 22 12.5–17.5 5
108–116 43 17.5–22.5 15
117–125 28 22.5–27.5 5
126–134 9 27.5–32.5 2

Applicants who score above 107 need not enroll in a 2–24. Construct a histogram, frequency polygon, and
summer developmental program. In this group, how many ogive for the data in Exercise 2–14 and analyze the results.
students do not have to enroll in the developmental (The data in this exercise will be used for Exercise 2–34.)
program? 2–25. In a class of 35 students, the following grade
2–20. For 75 employees of a large department store, the distribution was found. Construct a histogram, frequency
following distribution for years of service was obtained. polygon, and ogive for the data. (A  4, B  3, C  2,
Construct a histogram, frequency polygon, and ogive for D  1, F  0.) (The data in this exercise will be used for
the data. (The data for this exercise will be used for Exercise 2–35.)
Exercise 2–32.) Grade Frequency
Class limits Frequency 0 3
1–5 21 1 6
6–10 25 2 9
11–15 15 3 12
16–20 0 4 5
21–25 8
A grade of C or better is required for the next level course.
26–30 6
Were the majority of the students able to meet this
requirement?
A majority of the employees have worked for how many
years or less? 2–26. In a study of reaction times of dogs to a specific
stimulus, an animal trainer obtained the following data,
2–21. Construct a histogram, frequency polygon, and
given in seconds. Construct a histogram, frequency
ogive for the data in Exercise 2–11 and analyze the results.
polygon, and ogive for the data and analyze the results.
2–22. Construct a histogram, frequency polygon, and (The histogram in this exercise will be used for Exercises
ogive for the data in Exercise 2–12 and analyze the results. 2–36, 3–16, and 3–68.)
54 Chapter 2 Frequency Distributions and Graphs

Class limits Frequency 2–35. For the data in Exercise 2–25, construct a
histogram, frequency polygon, and ogive, using
2.3–2.9 10
relative frequencies. What proportion of the students
3.0–3.6 12 cannot meet the requirement for enrollment in the
3.7–4.3 6 next course?
4.4–5.0 8
2–36. The animal trainer in Exercise 2–26 selected another
5.1–5.7 4
group of dogs that were much older than the first group and
5.8–6.4 2 measured their reaction times to the same stimulus.
Construct a histogram, frequency polygon, and ogive for
2–27. Construct a histogram, frequency polygon, and the data.
ogive for the data in Exercise 2–15 and analyze the results.
Class limits Frequency
2–28. To determine their lifetimes, 80 randomly selected
batteries were tested. The following frequency distribution 2.3–2.9 1
was obtained. The data values are in hours. Construct a 3.0–3.6 3
histogram, frequency polygon, and ogive for the data and 3.7–4.3 4
analyze the results. 4.4–5.0 16
5.1–5.7 14
Class boundaries Frequency
5.8–6.4 4
63.5–74.5 10
74.5–85.5 15
Analyze the results and compare the histogram for this
85.5–96.5 22 group with the one obtained in Exercise 2–26. Are there
96.5–107.5 17 any differences in the histograms? (The data in this
107.5–118.5 11 exercise will be used for Exercise 3–16 and 3–68.)
118.5–129.5 5
*2–37. Using the following histogram:
a. Construct a frequency distribution; include class limits,
2–29. Construct a histogram, frequency polygon, and class frequencies, midpoints, and cumulative frequencies.
ogive for the data in Exercise 2–16 and analyze the results. b. Construct a frequency polygon.
c. Construct an ogive.
2–30. For the data in Exercise 2–18, construct a histogram
for the home run distances for each player and compare y
them. Are they basically the same, or are there any 7
noticeable differences? Explain your answer. 6
Frequency

2–31. For the data in Exercise 2–19, construct a histogram, 5

frequency polygon, and ogive, using relative frequencies. 4
What proportion of the applicants need to enroll in the 3
summer developmental program? 2
1
2–32. For the data in Exercise 2–20, construct a histogram, x
0
frequency polygon, and ogive, using relative frequencies. 21.5 24.5 27.5 30.5 33.5 36.5 39.5 42.5
What proportion of the employees have been with the store Class boundaries
for more than 20 years?

2–33. For the data in Exercise 2–23, construct a histogram, *2–38. Using the results from Exercise 2–37, answer the
frequency polygon, and ogive, using relative frequencies. following questions.
What proportion of the automobiles had a fuel efficiency of a. How many values are in the class 27.5–30.5?
17.5 miles per gallon or higher? b. How many values fall between 24.5 and 36.5?
c. How many values are below 33.5?
2–34. For the data in Exercise 2–14, construct a histogram,
d. How many values are above 30.5?
frequency polygon, and ogive, using relative frequencies.
 Section 2–3 Histograms, Frequency Polygons, and Ogives 55

Technology Step by Step

Minitab Constructing a Histogram

Step by Step Example MT2–3
1. Enter the data from Example 2–2, the high temperatures for the 50 states.
2. Select Graph>Histogram.
3. In the dialog box double click C1 TEMPERATURES.
4. Click [Options].
a. In Type of Interval: check CutPoint: to use class boundaries instead of
midpoints.
b. In Definition of Intervals: click the button for Midpoint/cutpoint
positions: then type the boundaries MINITAB will use. The format is
Lowest:Highest/Increment.
Histogram Option

5. Click [OK]. The Histogram Dialog Box will be displayed.

6. Click [Edit Attributes]. Select Solid for Fill and Yellow for the Back color.
This will make the bars of the histogram a solid yellow color.
7. Click [OK] twice.
The resulting graph window can be printed, copied to the clipboard or saved.
Histogram of
Temperatures
56 Chapter 2 Frequency Distributions and Graphs

TI-83 Constructing a Histogram

Step by Step In order to display the graphs on the screen, enter the appropriate values in the calculator using the
Window menu. The default values are Xmin  10, Xmax  10, Ymin  10, and Ymax  10.
The Xscl changes the distance between the tick marks on the x axis and can be used to change
the class width for the histogram.
To change the values in the Window:
1. Press WINDOW.
2. Move the cursor to the value that needs to be changed. Then type in the desired value and
press ENTER.
3. Continue until all values are appropriate.
4. Press [2nd] [QUIT] to leave the Window menu.

To plot the histogram:

1. Enter the data in L1.
2. Make sure Window values are appropriate for the histogram.
3. Press [2nd] [STAT PLOT] ENTER.
4. Press ENTER to turn the plot on, if necessary.
5. Move cursor to the Histogram symbol and press ENTER, if necessary.
6. Move cursor to L1 on the Xlist. Press ENTER, if necessary.
7. Move cursor to 1 on Freq. Press ENTER, if necessary.
8. Press GRAPH to display the histogram.
9. To obtain various coordinates, press the TRACE key, followed by  or  keys.

Example TI2–1
Plot a histogram for the following data from Examples 2–2 and 2–4.
112 100 127 120 134 118 105 110 109 112
110 118 117 116 118 122 114 114 105 109
107 112 114 115 118 117 118 122 106 110
116 108 110 121 113 120 119 111 104 111
120 113 120 117 105 110 118 112 114 114
Set the Window values as follows:
Xmin  100
Xmax  135
Ymin  5
Ymax  20

Input Input
 Section 2–3 Histograms, Frequency Polygons, and Ogives 57

Press TRACE and use the arrow keys to determine the number of values in each group.

Output

Excel Constructing a Histogram

Step by Step Example XL2–1
To make a histogram:
1. Press [Ctrl]-N for a new worksheet.
2. Enter the data below in column A, one number per cell.
68 80 69 81 72 100 101 73 102 93
91 92 93 88 82 83 75 75 89 96
103 83 84 85 89

3. Select Tools > Data Analysis.

4. In Data Analysis, select Histogram and click the [OK] button.
5. In the Histogram dialog box, type A1:A25 as the Input Range.
Histogram Dialog Box

6. Select New Worksheet Ply and Chart Output. Click [OK].

Excel presents both a table and a chart on the new worksheet ply. It decides “bins” for the his-
togram itself (here it picked a bin size of seven units), but you can also define your own bin range
on the data worksheet. Here is the histogram with Excel-selected bins.
58 Chapter 2 Frequency Distributions and Graphs

Histogram for
Example XL2–1

2–4 In addition to the histogram, the frequency polygon, and the ogive, several other types
of graphs are often used in statistics. They are the Pareto chart, the time series graph,
Other Types of Graphs and the pie graph. Figure 2–8 shows an example of each type of graph.

Figure 2–8 y
How people get to work
Other Types of Graphs 30

Used in Statistics
25

20
Frequency

15

10

x
0
Auto Bus Trolley Train Walk
(a) Pareto chart

y
Temperature over a 9-hour period Marital status of employees
at Brown’s department store

60°

55° Married
Temperature

50%
50°
Widowed
5%
45° Single
Divorced 18%
27%
40°

x
0
12 1 2 3 4 5 6 7 8 9
Time
(b) Time series graph (c) Pie graph