Frequency distributions and percentage distributions show the number and percentage, respectively, of observations that fall in each class of a variable.
Sometimes the administrator needs to know how many observations (or what percentage of observations) fall below or above a certain standard. For example, the fi re chief of Metro, Texas, is quite concerned about how long it takes fi re crews to arrive at the scene of a fi re. Th e Metro Morning News has run several stories about fi res, in which it claimed the Metro fi re department was slow in
Table 4.4 Tons of Garbage Collected by Sanitary Engineer Teams, Week of June 8, 2011
Percentage of Work Crews Tons of Garbage Normal Moore
50–60 16 13
60–70 24 22
70–80 30 30
80–90 20 22
90–100 10 13
100 100
N 5 100 N 5 165
responding. Because the Metro fi re department automatically records the time of fi re calls on its computer and also records the dispatched fi re truck’s report that it has arrived at the fi re, the response times to all fi res are available. An analyst has made a frequency distribution of these response times (Table 4.5). Th e Metro fi re chief considers 5 minutes to be an excellent response time, 10 minutes to be an acceptable response time, 15 minutes to be an unsatisfactory response time, and 20 minutes to be unacceptable. As a result, the fi re chief wants to know the percentage of fi re calls answered in under 5 minutes, under 10 minutes, under 15 minutes, and under 20 minutes. To provide the fi re chief with the informa- tion she wants, the analyst must construct a cumulative percentage distribution.
Th e fi rst step in developing a cumulative percentage distribution is to pre- pare a running total of responses to fi re calls. To the right of the “Number of Calls”
column in Table 4.5, you will fi nd a blank column labeled “Running Total.” In this column, we will calculate the total number of responses made that were less than each interval’s upper limit. For example, how many fi res were responded to in less than 1 minute? From the table, we can see seven fi res had response times of under a minute. Enter the number 7 for the fi rst class in the “Running Total”
column. How many fi re responses were under 2 minutes? Th ere were 21, with 7 under 1 minute, plus 14 between 1 and 2 minutes. Enter 21 as the value for the second class. Using this logic, fi ll in the rest of the values in Table 4.5.
Table 4.5 Response Times of the Metro Fire Department, 2011 Response Time
(Minutes) Number of Calls
Running Total
Cumulative Percentage
0–1 7
1–2 14
2–3 32
3–4 37
4–5 48
5–6 53
6–7 66
7–8 73
8–9 42
9–10 40
10–11 36
11–12 23
12–13 14
13–14 7
14–15 2
15–20 6
500
The second step is to construct a cumulative percentage column. This step is performed by dividing each frequency in the “Running Total” column by the total frequency (in this case, 500). In the fourth column of Table 4.5, “Cumulative Percentage,” enter the following numbers. Th e fi rst entry should be 1.4 (7 4 500); the second entry should be 4.2 (21 4 500); the third entry should be 10.6 (53 4 500). Fill in the remaining values for this column.
You now have a cumulative frequency distribution, and a cumulative percentage distribution, for the fire chief. The distribution is a bit awkward, however, because it has so many categories. Th e next step would be to collapse the cumulative percentage distribution into fewer categories. Because the fi re chief is concerned with response times of 5, 10, 15, and 20 minutes, these times would be the best categories. Table 4.6 should result from your calculations. From this table, what can you tell the chief about fi re department response times in Metro?
How good are the department’s response times?
Graphical Presentations
Often a public or nonprofi t administrator wants to present information visually so that leaders, citizens, and staff can get a general feel for a problem without reading a table. Two methods of visual presentation will be described here: the frequency polygon and the histogram.
Let us say that the Normal city manager, as part of her budget justifi cation, wants to show the city council the number of complaints that the city animal control offi ce receives about barking dogs. An assistant has prepared the frequency distribution shown in Table 4.7.
To construct a frequency polygon, follow these steps.
Step 1: On a sheet of graph paper, write the name of the variable across the bot- tom and the frequency along the side. Here, the variable is the number of complaints about dogs, and the frequency is the number of weeks.
See Figure 4.1.
Table 4.6 Response Times of the Metro Fire Department, 2011 Response Time Percentage (Cumulative) of
Response Times
Under 5 minutes 27.6
Under 10 minutes 82.4
Under 15 minutes 98.8
Under 20 minutes 100.00
N 5 500
Step 2: Calculate the midpoint for each class interval. Add the two boundar- ies for each class and divide by 2. For the fi rst class, the midpoint is (5 1 9) 4 2, or 7. Th e midpoints for the other classes are 12, 17, 22, 27, 32.
Step 3: On the horizontal dimension or axis of the graph, fi nd the fi rst class midpoint (7). Directly above this point, mark the frequency for this class (also 7) with a dot. Repeat this procedure for the fi ve other classes.
Your graph should look like the one in Figure 4.2.
Table 4.7 Complaints per Week about Barking Dogs, 2011
Number of Complaints Number of Weeks
5–9 7
10–14 6
15–19 15
20–24 17
25–29 5
30–34 2
52
Figure 4.1 First Step in Constructing a Frequency Polygon
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2011
Step 4: Calculate the midpoint for the class below the lowest class observed (this class would be 0–4 complaints) and for the class above the highest class observed (35–39). Plot these midpoints with a frequency of 0 on your graph (i.e., on the horizontal axis).
Step 5: Draw a line connecting the points in sequence. Your first fre- quency polygon should look like the one in Figure 4.3. (Note that whereas the frequency polygon presents a useful visual representa- tion of the data, the line segments do not correspond to actual data points.)
One nice aspect of frequency polygons is that the analyst can draw more than one on the same graph. For example, suppose that the Normal city manager wants to show how complaints about barking dogs have changed over time. The city manager gives you the data shown in Table 4.8. In Figure 4.4, graph frequency polygons for both years on the same graph. What does the graph tell you about barking dog complaints in 2011 as opposed to those in 2010?
Note: Whenever two or more frequency polygons are drawn on the same set of axes, each polygon should be drawn in a diff erent color or with a diff erent type of line (such as solid, broken, bold) so the reader can tell them apart. Be sure to label each line. Figure 6.1 in Chapter 6 illustrates using diff erent types of lines on the same set of axes.
Figure 4.2 Third Step in Constructing a Frequency Polygon
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2011
A histogram is a bar graph for a variable that takes on many values (such as income or gross national product [GNP]). Th e term bar chart is sometimes used when a variable can take only a very limited set of values (for example, a variable assessing an opinion that calls for the responses “agree,” “undecided,”
or “disagree”). Our intention is not to multiply terms (or confusion), but some statistical package programs loaded onto computers, such as the Statistical Package for the Social Sciences (SPSS), do make this distinction. You may need to use SPSS in class or on the job.
Figure 4.3 The Frequency Polygon
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2011
Table 4.8 Complaints per Week about Barking Dogs, 2010 and 2011
Number of Weeks Number of Complaints 2010 2011
5–9 8 7
10–14 12 6
15–19 14 15
20–24 10 17
25–29 6 5
30–34 2 2
52 52
To construct a histogram of barking dog complaints in Normal for 2011, complete the following steps.
Steps 1–3: Follow the same procedures given for constructing frequency polygons in Steps 1, 2, and 3 above. Following this procedure should yield the graph shown in Figure 4.5.
Step 4: Using the points on the graph, fi rst draw a horizontal line from the lower to the upper class boundary for each class. Th en draw in the vertical lines along the class boundaries from these horizontal lines to the horizontal axis of the graph. Each class is now represented by a bar.
Step 5: Shade in the bars you have drawn in Step 4. Your graph should appear as shown in Figure 4.6.
You should use histograms rather than frequency polygons whenever you want to emphasize the distinctiveness of each class. As you can see by looking at the graphs, the frequency polygon tends to smooth out class diff erences. Frequency polygons should be used whenever you want to emphasize a smooth trend or when two or more graphs are placed on a single chart, table, or pair of coor- dinate axes.
Figure 4.4 Frequency Polygons for Number of Complaints about Dogs, 2010 and 2011
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2010 and 2011
Figure 4.5 Third Step in Constructing a Histogram
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2011
Figure 4.6 The Histogram
20
15
10
5
0
5 10 15 20 25 30 35 40
Number of Weeks
Number of Complaints about Barking Dogs, 2011
Cumulative frequency distributions can also be graphed. For example, the cumulative distribution for the Metro fi re department response times shown in Table 4.6 can be made into a frequency polygon. For each response time in the table (under 5 minutes, under 10 minutes, and so on), simply plot the corre- sponding percentage, and connect the consecutive points. Your graph should look like the one in Figure 4.7. For comparative purposes, make a second frequency polygon in Figure 4.7 for the city of Atlantis fi re department; the response times are presented in Table 4.9. Be sure to label the two lines for clarity.
A frequency polygon for a cumulative distribution is called an ogive.
Compare the ogives in Figure 4.7. Which fi re department appears to respond more quickly to fi res? Why do you think so?
Figure 4.7 Cumulative Frequency Polygons
100
20 40 60 80
0 5 10 15 20
Number of Minutes
Cumulative Percentage of Response Times
Table 4.9 Response Times of Atlantis Fire Department, 2011 Response Time
Percentage (Cumulative) of Response Times
Under 5 minutes 21.2
Under 10 minutes 63.9
Under 15 minutes 86.4
Under 20 minutes 100.0
Chapter Summary
Descriptive statistics summarize a body of raw data so that the data can be more easily understood. Frequency distributions, percentage distributions, and cumula- tive frequency distributions are three ways to condense raw data into a table that is easier to read and interpret. A frequency distribution displays the number of times each value, or range of values, of a variable occurs. Th e frequency distribution shows classes appropriate for the variable under study and the number of data points falling into each class. A percentage distribution shows the percentage of total data points that fall into each class. A cumulative frequency (or percentage) distribution displays the number (or percentage) of observations that fall above or below a certain class.
To add visual appeal and increase interpretability, graphical presentations of data are used. Graphical techniques discussed in this chapter include the fre- quency polygon, the histogram and bar chart, and the ogive. Th e frequency poly- gon is a plot of the frequency distribution information (class versus frequency), with the plotted points connected in sequence by line segments. Th e histogram is a bar graph of a frequency distribution; each class is represented by a horizontal bar, and its frequency corresponds to the height of the bar from the horizontal axis. Th e term bar chart is sometimes used in place of histogram when the variable can take on only a very limited set of values. An ogive is a frequency polygon for a cumulative frequency distribution.
Problems
4.1 You are the research assistant to the administrator of a small bureau in the federal government. Your boss has received some criticism that the bureau does not respond promptly to congressional requests. Th e only information you have is the day the agency received the request and the day the agency mailed the response. From those fi gures, you have calculated the number of days the agency took to respond.
Days Necessary to Respond to Congressional Requests
9 1 6 10 8 12 9 14 15 7
19 8 21 10 50 37 9 4 28 44
9 18 8 39 7 1 4 15 7 28
47 9 6 7 24 10 41 7 9 29
6 4 12 7 9 15 39 24 9 2
20 31 18 9 33 8 6 3 7 16
20 26 9 9 16 5 3 12 36 11
8 6 28 35 8 10 11 20 3 10
16 8 12 4 6 9 10 10 9 16
4 14 11 8 5 8 11 9 7 6
11 9 7 8 10 9 11
Do the following:
(a) Prepare the frequency distribution.
(b) Present the distribution graphically.
(c) Prepare a cumulative frequency distribution.
(d) Present the cumulative distribution graphically.
(e) Write a paragraph explaining what you have found.
4.2 Allan Wiese, the mayor of Orva, South Dakota, feels that the productivity of meter butlers has declined in the past year. Mayor Wiese’s research assistant provides him with the accompanying data. Convert the frequency distributions to comparable distributions. What can you tell Mayor Wiese about the produc- tivity of his meter butlers?
Number of Butlers Parking Tickets Issued
per Meter Butler May 2010 May 2011
21–30 5 6
31–40 7 9
41–50 9 12
51–60 5 7
61–70 3 1
29 35
4.3 Scotty Allen, the civil service director for Maxwell, New York, compiles the accompanying frequency distribution of scores on the Maxwell civil service exam. Construct a cumulative frequency distribution and a cumulative frequency polygon for Mr. Allen.
Exam Score Number of Applicants
61–65 20
66–70 13
71–75 47
76–80 56
81–85 33
86–90 27
91–95 41
96–100 34
4.4 Th e incumbent governor of a large state is campaigning on the platform that he eliminated a great many large, “do-nothing” bureaucracies. As the research assis- tant for the challenger, you are asked to present the accompanying data (numbers are the size of bureaus eliminated under the incumbent and under his predeces- sor) graphically in the manner most favorable for the challenger.
Incumbent Predecessor
6 16 15
14 5 28
7 3 48
3 7 104
24 19 37
6 21 56
3 12 15
1 4 6
2 3 3
21 6 27
41 1 39
4.5 Refer to Problem 4.4. Construct a frequency distribution, and present it to refl ect favorably on the incumbent.
4.6 Th e city clerk has received numerous complaints over the past year that couples applying for a marriage license have to wait too long to receive one. Although the clerk is skeptical (couples applying for a license are usually young and impatient), she pulls a representative sample of marriage licenses issued in the past year.
Because a machine stamps each license application with the time the application is received and the time it is issued, she can tell how long the young (and old) lovers had to wait for the marriage license. Th e clerk considers service received in less than 10 minutes good and service received in less than 15 minutes acceptable.
Her tabulation of the license data shows the following:
Minutes Waited for
Marriage License Number of Couples
Less than 5 28
5–9 36
10–14 60
15–19 82
20–24 44
25–29 39
Prepare the percentage distribution for the marriage license data and the appropriate graphical displays. Write a short memorandum explaining the results and addressing the issue of whether couples have to wait too long for marriage licenses.
4.7 Th e city clerk from Problem 4.6 is intrigued by the fi ndings of her survey of marriage licenses issued in the past year (data analysis often has this effect).
Accordingly, she decides to pull another representative sample of marriage licenses, this time from 2 years ago. She is interested in determining whether service to the public from her unit has improved or declined over the past 2 years. As before, the clerk considers service received in less than 10 minutes good and service received in less than 15 minutes acceptable. Her tabulation of the sample of marriage licenses issued 2 years ago shows the following:
Minutes Waited for
Marriage License Number of Couples
Less than 5 112
5–9 87
10–14 31
15–19 27
20–24 29
25–29 3
Prepare the percentage distribution for the marriage license data and the appropriate graphical displays. Write a short memorandum explaining the results and addressing the question of whether service to the public from her unit has improved or declined over the past 2 years.
4.8 Because of cutbacks in agency funding, the United Way of Megopolis has had to forgo routine maintenance of its computer terminals for the past 5 years.
(Th e equipment is made by the Indestructible Computer Company.) Th e head of the agency is concerned that the agency will face a major equipment crisis this year, because the recommended maintenance schedule for the terminals is once every 3 years. Over the past 5 years, the agency has been able to purchase new terminals. In an eff ort to obtain more funding from the state legislature, the agency chief compiles the following data. Th e data show the time since the last routine maintenance of the terminal or, if the terminal was purchased in the last 2 years, the time since the terminal was purchased.
Years since Last
Maintenance Number of Terminals
1 or less 103
2 187
3 97
4 56
5 37
6 12
7 or more 5
Prepare the percentage distribution for the terminal maintenance data and the appropriate graphical displays. Write a short memorandum both explaining the results and trying to convince the state legislature to provide funding for routine maintenance of computer terminals.
4.9 Assume that you are a staff analyst working for the head of the United Way of Megopolis in Problem 4.8. Write a short memorandum both explaining the results of the data tabulation in Problem 4.8 and trying to convince the agency head that the equipment “crisis” at the agency is overblown.
4.10 Th e local humane society is concerned about available space for impounded ani- mals. Th e agency keeps careful count of the number of animals it shelters each day. To determine the load on the agency, its head, Anna Trueheart, selects a
representative sample of days from the last 2 years and records the number of animals impounded on each day. Her data appear as follows:
65 49 84 72 43 91
57 46 77 69 90 64
85 67 52 44 95 79
48 63 55 96 75 48
88 81 93 67 58 72
51 49 96 79 73 80
65 54 86 98 42 63
92 71 79 84 59 45
Prepare the frequency and percentage distributions for the animal impound- ment data and the appropriate graphical displays for Ms. Trueheart. Write a short memorandum explaining both the results and the demands on the humane soci- ety to shelter animals.
4.11 Th e director of the state Department of Public Works wants to upgrade the de- partment’s automobile fl eet; she claims that the fl eet is too old. Th e governor appoints a staff analyst to investigate the issue. Th e analyst compiles data on both the age and the odometer readings (mileage) of the department’s automobile fl eet. Her data appear as follows:
Age (in years) Number of Automobiles
Less than 2 16
2–4 24
4–6 41
6–8 57
8–10 64
10 or more 39
Mileage Number of Automobiles
Less than 10,000 76
10,000–20,000 63
20,000–30,000 51
30,000–40,000 32
40,000–50,000 12
50,000 or more 7
Prepare percentage distributions for the age and mileage data and the appropriate graphical displays. Write a short memorandum to the governor both explaining the results and making a recommendation regarding whether the department’s automobile fl eet should be upgraded.
5
T
he most commonly used descriptive statistics are measures of central ten- dency. As you can guess from this title, measures of central tendency at- tempt to locate the mid-most or center point in a group of data. For example, what was the average starting salary of the students who graduated from the MPA program last year? On average over the past 5 years, how many MPA students accepted job off ers from nonprofi t organizations? On last week’s midterm exam, what was the mid-most score (i.e., the score that divided the observations in half )? On average, how many employees of the Mechanicsburg city govern- ment report job-related accidents every month? What was the mid-most amount of monetary donations received by the Agency for Civic Renewal such that half the donations received were larger and half were smaller? Th e measures of central tendency give a shorthand indication of the main trend in the data.A measure of central tendency is a number or score or data value that represents the average in a group of data. Th ree diff erent types of averages are calculated and used most often in public and nonprofi t management. Th e fi rst is the mean, which is the arithmetic average of the observations; the second is the median, which is the observation that falls exactly in the middle of the group; and the third is the mode, or the data value that occurs with greatest frequency. Th is chapter shows how to calculate the three measures of central tendency for both ungrouped and grouped data (data that have been assembled into a frequency distribution) and discusses the use and interpretation of these measures. It concludes with a discussion of the relationship between the measures of central tendency and the diff erent levels of measurement—interval, ordinal, and nominal—you learned about in Chapter 2.