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    Section 1.3

     

    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 3 320 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.

    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.

    Table 1.12
    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.

    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 6 650 = 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.

    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.

    1. The percentage of heights that are from 67.95 to 71.95 inches is: ____.
    2. The percentage of heights that are from 67.95 to 73.95 inches is: ____.
    3. The percentage of heights that are more than 65.95 inches is: ____.
    4. The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.
    5. What kind of data are the heights?
    6. 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:

    1. What percentage of the students in your class have no siblings?
    2. What percentage of the students have from one to three siblings?
    3. 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 3 319 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
    1. Is the table correct? If it is not correct, what is wrong?
    2. 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.
    3. What fraction of the people surveyed commute five or seven miles?
    4. 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.

    1. What is the frequency of deaths measured from 2006 through 2009?
    2. What percentage of deaths occurred after 2009?
    3. What is the relative frequency of deaths that occurred in 2003 or earlier?
    4. What is the percentage of deaths that occurred in 2004?
    5. What kind of data are the numbers of deaths?
    6. 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.

    1. What is the frequency of deaths measured from 2000 through 2004?
    2. What percentage of deaths occurred after 2006?
    3. What is the relative frequency of deaths that occurred in 2000 or before?
    4. What is the percentage of deaths that occurred in 2011?
    5. What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.

     


    Solutions

    Solution 1.18

    1. 97,118 (11.8%)
    2. 41.6%
    3. 67,092/823,356 or 0.081 or 8.1 %
    4. 27.8%
    5. Quantitative discrete
    6. Quantitative continuous

     

     

     

     

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