Frequency
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3.
Table 1.12 lists the different data values in ascending order and their frequencies.
DATA VALUE 
FREQUENCY 

2 
3 
3 
5 
4 
3 
5 
6 
6 
2 
7 
1 
Table 1.12 Frequency Table of Student Work Hours
A frequency is the number of times a value of the data occurs. According to Table 1.12, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample—in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
DATA VALUE 
FREQUENCY 
RELATIVE FREQUENCY 

2 
3 
$\frac{3}{20}$ or .15 
3 
5 
$\frac{5}{20}$ or .25 
4 
3 
$\frac{3}{20}$ or .15 
5 
6 
$\frac{6}{20}$ or .30 
6 
2 
$\frac{2}{20}$ or .10 
7 
1 
$\frac{1}{20}$ or .05 
Table 1.13 Frequency Table of Student Work Hours with Relative Frequencies
The sum of the values in the relative frequency column of Table 1.13 is $\frac{20}{20}$ , or 1.
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.14.
In the first row, the cumulative frequency is simply .15 because it is the only one. In the second row, the relative frequency was .25, so adding that to .15, we get a relative frequency of .40. Continue adding the relative frequencies in each row to get the rest of the column.
DATA VALUE 
FREQUENCY 
RELATIVE
FREQUENCY

CUMULATIVE RELATIVE
FREQUENCY


2 
3 
$\frac{3}{20}$ or .15 
.15 
3 
5 
$\frac{5}{20}$ or .25 
.15 + .25 = .40 
4 
3 
$\frac{3}{20}$ or .15 
.40 + .15 = .55 
5 
6 
$\frac{6}{20}$ or .30 
.55 + .30 = .85 
6 
2 
$\frac{2}{20}$ or .10 
.85 + .10 = .95 
7 
1 
$\frac{1}{20}$ or .05 
.95 + .05 = 1.00 
Table 1.14 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
NOTE
Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.
Table 1.15 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
HEIGHTS
(INCHES)

FREQUENCY 
RELATIVE
FREQUENCY

CUMULATIVE
RELATIVE
FREQUENCY


59.95–61.95 
5 
$\frac{5}{100}$ = .05 
.05 
61.95–63.95 
3 
$\frac{3}{100}$ = .03 
.05 + .03 = .08 
63.95–65.95 
15 
$\frac{15}{100}$ = .15 
.08 + .15 = .23 
65.95–67.95 
40 
$\frac{40}{100}$ = .40 
.23 + .40 = .63 
67.95–69.95 
17 
$\frac{17}{100}$ = .17 
.63 + .17 = .80 
69.95–71.95 
12 
$\frac{12}{100}$ = .12 
.80 + .12 = .92 
71.95–73.95 
7 
$\frac{7}{100}$ = .07 
.92 + .07 = .99 
73.95–75.95 
1 
$\frac{1}{100}$ = .01 
.99 + .01 = 1.00 

Total = 100 
Total = 1.00 

Table 1.15 Frequency Table of Soccer Player Height
The data in this table have been grouped into the following intervals:
 59.95–61.95 inches
 61.95–63.95 inches
 63.95–65.95 inches
 65.95–67.95 inches
 67.95–69.95 inches
 69.95–71.95 inches
 71.95–73.95 inches
 73.95–75.95 inches
Note
This example is used again in Descriptive Statistics, where the method used to compute the intervals will be explained.
In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.
Example 1.15
From Table 1.15, find the percentage of heights that are less than 65.95 inches.
Solution 1.15
If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are 5 + 3 + 15 = 23 players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then $\frac{23}{100}$ or 23 percent. This percentage is the cumulative relative frequency entry in the third row.
Try It 1.15
Table 1.16 shows the amount, in inches, of annual rainfall in a sample of towns.
Rainfall (Inches) 
Frequency 
Relative Frequency 
Cumulative Relative Frequency 

2.95–4.97 
6 
$\frac{6}{50}$ = .12 
.12 
4.97–6.99 
7 
$\frac{7}{50}$ = .14 
.12 + .14 = .26 
6.99–9.01 
15 
$\frac{15}{50}$ = .30 
.26 + .30 = .56 
9.01–11.03 
8 
$\frac{8}{50}$ = .16 
.56 + .16 = .72 
11.03–13.05 
9 
$\frac{9}{50}$ = .18 
.72 + .18 = .90 
13.05–15.07 
5 
$\frac{5}{50}$ = .10 
.90 + .10 = 1.00 

Total = 50 
Total = 1.00 

Table 1.16
From Table 1.16, find the percentage of rainfall that is less than 9.01 inches.
Example 1.16
From Table 1.15, find the percentage of heights that fall between 61.95 and 65.95 inches.
Solution 1.16
Add the relative frequencies in the second and third rows: .03 + .15 = .18 or 18 percent.
Try It 1.16
From Table 1.16, find the percentage of rainfall that is between 6.99 and 13.05 inches.
Example 1.17
Use the heights of the 100 male semiprofessional soccer players in Table 1.15. Fill in the blanks and check your answers.
 The percentage of heights that are from 67.95–71.95 inches is ________.
 The percentage of heights that are from 67.95–73.95 inches is ________.
 The percentage of heights that are more than 65.95 inches is ________.
 The number of players in the sample who are between 61.95 and 71.95 inches tall is ________.
 What kind of data are the heights?
 Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.
Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
Solution 1.17
 29 percent
 36 percent
 77 percent
 87
 quantitative continuous
 get rosters from each team and choose a simple random sample from each
Try It 1.17
From Table 1.16, find the number of towns that have rainfall between 2.95 and 9.01 inches.
Collaborative Exercise
In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:
 What percentage of the students in your class have no siblings?
 What percentage of the students have from one to three siblings?
 What percentage of the students have fewer than three siblings?
Example 1.18
Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows:
2, 5, 7, 3, 2, 10, 18, 15, 20, 7, 10, 18, 5, 12, 13, 12, 4, 5, 10. Table 1.17 was produced.
DATA 
FREQUENCY 
RELATIVE
FREQUENCY

CUMULATIVE
RELATIVE
FREQUENCY


3 
3 
$\frac{3}{19}$ 
.1579 
4 
1 
$\frac{1}{19}$ 
.2105 
5 
3 
$\frac{3}{19}$ 
.1579 
7 
2 
$\frac{2}{19}$ 
.2632 
10 
3 
$\frac{4}{19}$ 
.4737 
12 
2 
$\frac{2}{19}$ 
.7895 
13 
1 
$\frac{1}{19}$ 
.8421 
15 
1 
$\frac{1}{19}$ 
.8948 
18 
1 
$\frac{1}{19}$ 
.9474 
20 
1 
$\frac{1}{19}$ 
1.0000 
Table 1.17 Frequency of Commuting Distances
 Is the table correct? If it is not correct, what is wrong?
 True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
 What fraction of the people surveyed commute five or seven miles?
 What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?
Solution 1.18
 No. The frequency column sums to 18, not 19. Not all cumulative relative frequencies are correct.
 False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read 1052, 01579, 02105, 03684, 04737, 06316, 07368, 07895, 08421, 09474, 1.0000.
 $\frac{5}{19}$
 $\frac{7}{19}$, $\frac{12}{19}$, $\frac{7}{19}$
Try It 1.18
Table 1.16 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?
Example 1.19
Table 1.18 contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.
Year 
Total Number of Deaths 

2000 
231 
2001 
21,357 
2002 
11,685 
2003 
33,819 
2004 
228,802 
2005 
88,003 
2006 
6,605 
2007 
712 
2008 
88,011 
2009 
1,790 
2010 
320,120 
2011 
21,953 
2012 
768 
Total 
823,856 
Table 1.18
Answer the following questions:
 What is the frequency of deaths measured from 2006 through 2009?
 What percentage of deaths occurred after 2009?
 What is the relative frequency of deaths that occurred in 2003 or earlier?
 What is the percentage of deaths that occurred in 2004?
 What kind of data are the numbers of deaths?
 The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?
Solution 1.19
 97,118 (11.8 percent)
 41.6 percent
 67,092/823,356 or 0.081 or 8.1 percent
 27.8 percent
 quantitative discrete
 quantitative continuous
Try It 1.19
Table 1.19 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994–2011.
Year 
Total Number of Crashes 
Year 
Total Number of Crashes 

1994 
36,254 
2004 
38,444 
1995 
37,241 
2005 
39,252 
1996 
37,494 
2006 
38,648 
1997 
37,324 
2007 
37,435 
1998 
37,107 
2008 
34,172 
1999 
37,140 
2009 
30,862 
2000 
37,526 
2010 
30,296 
2001 
37,862 
2011 
29,757 
2002 
38,491 
Total 
653,782 
2003 
38,477 


Table 1.19
Answer the following questions:
 What is the frequency of deaths measured from 2000 through 2004?
 What percentage of deaths occurred after 2006?
 What is the relative frequency of deaths that occurred in 2000 or before?
 What is the percentage of deaths that occurred in 2011?
 What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.