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.
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.
DATA VALUE  FREQUENCY  RELATIVE FREQUENCY 

2  3  320 or 0.15 
3  5  520 or 0.25 
4  3  320 or 0.15 
5  6  620 or 0.30 
6  2  220 or 0.10 
7  1  120 or 0.05 
The sum of the values in the relative frequency column of Table 1.10 is 2020 , 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.
DATA VALUE  FREQUENCY  RELATIVE FREQUENCY 
CUMULATIVE RELATIVE FREQUENCY 

2  3  320 or 0.15  0.15 
3  5  520 or 0.25  0.15 + 0.25 = 0.40 
4  3  320 or 0.15  0.40 + 0.15 = 0.55 
5  6  620 or 0.30  0.55 + 0.30 = 0.85 
6  2  220 or 0.10  0.85 + 0.10 = 0.95 
7  1  120 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.
HEIGHTS (INCHES) 
FREQUENCY  RELATIVE FREQUENCY 
CUMULATIVE RELATIVE FREQUENCY 

59.95–61.95  5  5100 = 0.05  0.05 
61.95–63.95  3  3100 = 0.03  0.05 + 0.03 = 0.08 
63.95–65.95  15  15100 = 0.15  0.08 + 0.15 = 0.23 
65.95–67.95  40  40100 = 0.40  0.23 + 0.40 = 0.63 
67.95–69.95  17  17100 = 0.17  0.63 + 0.17 = 0.80 
69.95–71.95  12  12100 = 0.12  0.80 + 0.12 = 0.92 
71.95–73.95  7  7100 = 0.07  0.92 + 0.07 = 0.99 
73.95–75.95  1  1100 = 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.
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  650 = 0.12  0.12 
4.97–6.99  7  750 = 0.14  0.12 + 0.14 = 0.26 
6.99–9.01  15  1550 = 0.30  0.26 + 0.30 = 0.56 
9.01–11.03  8  850 = 0.16  0.56 + 0.16 = 0.72 
11.03–13.05  9  950 = 0.18  0.72 + 0.18 = 0.90 
13.05–15.07  5  550 = 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  6/ 50 = 0.12  0.12 
4.97–6.99  7  7/ 50 = 0.14  0.12 + 0.14 = 0.26 
6.99–9.01  15  15/50 = 0.30  0.26 + 0.30 = 0.56 
9.01–11.03  8  8/50 = 0.16  0.56 + 0.16 = 0.72 
11.03–13.05  9  9/50 = 0.18  0.72 + 0.18 = 0.90 
13.05–15.07  5  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.
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.
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:

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:
DATA  FREQUENCY  RELATIVE FREQUENCY 
CUMULATIVE RELATIVE FREQUENCY 

3  3  319  0.1579 
4  1  119  0.2105 
5  3  319  0.1579 
7  2  219  0.2632 
10  3  419  0.4737 
12  2  219  0.7895 
13  1  119  0.8421 
15  1  119  0.8948 
18  1  119  0.9474 
20  1  119  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)?
Example 1.18
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?
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
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 23100 or 23%. This percentage is the cumulative relative frequency entry in the third row.
Add the relative frequencies in the second and third rows: 0.03 + 0.15 = 0.18 or 18%.
 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.
 519
 719 , 1219 , 719
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
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