You can click the link Section 1.3 to access the online version of this section of the textbook. Below is an edited version of section 1.3.

1.3 Frequency, Frequency Tables, and Levels of Measurement

Frequency and Frequency Tables

Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.

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 1.9 lists the different data values in ascending order and their frequencies.

Table 1.9
DATA VALUE	FREQUENCY
2	3
3	5
4	3
5	6
6	2
7	1

A frequency is the number of times a value of the data occurs. According to Table 1.9, 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.

Table 1.10
DATA VALUE	FREQUENCY	RELATIVE FREQUENCY
2	3	$\frac{3}{20}$ or 0.15
3	5	$\frac{5}{20}$ or 0.25
4	3	$\frac{3}{20}$ or 0.15
5	6	$\frac{6}{20}$ or 0.30
6	2	$\frac{2}{20}$ or 0.10
7	1	$\frac{1}{20}$ or 0.05

The sum of the values in the relative frequency column of Table 1.10 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 1.11.

Table 1.11
DATA VALUE	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
2	3	$\frac{3}{20}$ or 0.15	0.15
3	5	$\frac{5}{20}$ or 0.25	0.15 + 0.25 = 0.40
4	3	$\frac{3}{20}$ or 0.15	0.40 + 0.15 = 0.55
5	6	$\frac{6}{20}$ or 0.30	0.55 + 0.30 = 0.85
6	2	$\frac{2}{20}$ or 0.10	0.85 + 0.10 = 0.95
7	1	$\frac{1}{20}$ or 0.05	0.95 + 0.05 = 1.00

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

Table 1.12 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

Table 1.12
HEIGHTS (INCHES)	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
59.95–61.95	5	$\frac{5}{100}$ = 0.05	0.05
61.95–63.95	3	$\frac{3}{100}$ = 0.03	0.05 + 0.03 = 0.08
63.95–65.95	15	$\frac{15}{100}$ = 0.15	0.08 + 0.15 = 0.23
65.95–67.95	40	$\frac{40}{100}$ = 0.40	0.23 + 0.40 = 0.63
67.95–69.95	17	$\frac{17}{100}$ = 0.17	0.63 + 0.17 = 0.80
69.95–71.95	12	$\frac{12}{100}$ = 0.12	0.80 + 0.12 = 0.92
71.95–73.95	7	$\frac{7}{100}$ = 0.07	0.92 + 0.07 = 0.99
73.95–75.95	1	$\frac{1}{100}$ = 0.01	0.99 + 0.01 = 1.00
	Total = 100	Total = 1.00

The data in this table have been grouped into the following intervals:

59.95 to 61.95 inches
61.95 to 63.95 inches
63.95 to 65.95 inches
65.95 to 67.95 inches
67.95 to 69.95 inches
69.95 to 71.95 inches
71.95 to 73.95 inches
73.95 to 75.95 inches

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 1.14

From Table 1.12, find the percentage of heights that are less than 65.95 inches.

Try It 1.14

Table 1.13 shows the amount, in inches, of annual rainfall in a sample of towns.

Table 1.13
Rainfall (Inches)	Frequency	Relative Frequency	Cumulative Relative Frequency
2.95–4.97	6	$\frac{6}{50}$ = 0.12	0.12
4.97–6.99	7	$\frac{7}{50}$ = 0.14	0.12 + 0.14 = 0.26
6.99–9.01	15	$\frac{15}{50}$ = 0.30	0.26 + 0.30 = 0.56
9.01–11.03	8	$\frac{8}{50}$ = 0.16	0.56 + 0.16 = 0.72
11.03–13.05	9	$\frac{9}{50}$ = 0.18	0.72 + 0.18 = 0.90
13.05–15.07	5	$\frac{5}{50}$ = 0.10	0.90 + 0.10 = 1.00
	Total = 50	Total = 1.00

From Table 1.13, find the percentage of rainfall that is less than 9.01 inches.

TRY IT 1.14

Table 1.13 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	$\frac{6}{50}$ = 0.12	0.12
4.97–6.99	7	$\frac{7}{50}$ = 0.14	0.12 + 0.14 = 0.26
6.99–9.01	15	$\frac{15}{50}$ = 0.30	0.26 + 0.30 = 0.56
9.01–11.03	8	$\frac{8}{50}$ = 0.16	0.56 + 0.16 = 0.72
11.03–13.05	9	$\frac{9}{50}$ = 0.18	0.72 + 0.18 = 0.90
13.05–15.07	5	$\frac{5}{50}$ = 0.10	0.90 + 0.10 = 1.00
	Total = 50	Total = 1.00

From Table 1.13, find the percentage of rainfall that is less than 9.01 inches.

Example 1.15

From Table 1.12, find the percentage of heights that fall between 61.95 and 65.95 inches.

Try It 1.15

From Table 1.13, find the percentage of rainfall that is between 6.99 and 13.05 inches.

Example 1.16

Use the heights of the 100 male semiprofessional soccer players in Table 1.12. Fill in the blanks and check your answers.

The percentage of heights that are from 67.95 to 71.95 inches is: ____.
The percentage of heights that are from 67.95 to 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.

Try It 1.16

From Table 1.13, 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 1.17

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 1.14 was produced:

Table 1.14
DATA	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
3	3	$\frac{3}{19}$	0.1579
4	1	$\frac{1}{19}$	0.2105
5	3	$\frac{3}{19}$	0.1579
7	2	$\frac{2}{19}$	0.2632
10	3	$\frac{4}{19}$	0.4737
12	2	$\frac{2}{19}$	0.7895
13	1	$\frac{1}{19}$	0.8421
15	1	$\frac{1}{19}$	0.8948
18	1	$\frac{1}{19}$	0.9474
20	1	$\frac{1}{19}$	1.0000

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)?

Try It 1.17

Table 1.13 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 1.18

Table 1.15 contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.

Table 1.15
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

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?

Try It 1.18

Table 1.16 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.

Table 1.16
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

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.

Solutions

Solution 1.14

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%. This percentage is the cumulative relative frequency entry in the third row.

Solution 1.15

Add the relative frequencies in the second and third rows: 0.03 + 0.15 = 0.18 or 18%.

Solution 1.16

29%
36%
77%
87
quantitative continuous
get rosters from each team and choose a simple random sample from each

Solution 1.17

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: 0.1052, 0.1579, 0.2105, 0.3684, 0.4737, 0.6316, 0.7368, 0.7895, 0.8421, 0.9474, 1.0000.
$\frac{5}{19}$
$\frac{7}{19}$ , $\frac{12}{19}$ , $\frac{7}{19}$

Solution 1.18

97,118 (11.8%)
41.6%
67,092/823,356 or 0.081 or 8.1 %
27.8%
Quantitative discrete
Quantitative continuous

MATH 103-Statistics

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Section 1.3

1.3 Frequency, Frequency Tables, and Levels of Measurement

Frequency and Frequency Tables

Frequency

Example 1.14

TRY IT 1.14

From Table 1.13, find the percentage of rainfall that is less than 9.01 inches.

Example 1.15

Example 1.16

Collaborative Exercise

Example 1.17

Example 1.18

Solutions

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