6 1.5 Frequency & Frequency Tables
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.
The following table 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 |
In this research, 3 students studied for 2 hours. 5 students studies for 3 hours.
A frequency is the number of times a value of the data occurs. According to the table, 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.
Relative frequency = [latex]\frac{\text{frequency of the class}}{\text{total}}[/latex]
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 the table below.
Cumulative relative frequency = sum of previous relative frequencies + current class frequency
Example 1
| DATA VALUE | FREQUENCY | RELATIVE
FREQUENCY |
CUMULATIVE RELATIVE
FREQUENCY |
|---|---|---|---|
| 2 | 3 | [latex]\frac{3}{20}[/latex] or 0.15 | 0.15 |
| 3 | 5 | [latex]\frac{5}{20}[/latex] or 0.25 | 0.15 + 0.25 = 0.40 |
| 4 | 3 | [latex]\frac{3}{20}[/latex] or 0.15 | 0.40 + 0.15 = 0.55 |
| 5 | 6 | [latex]\frac{6}{20}[/latex] or 0.30 | 0.55 + 0.30 = 0.85 |
| 6 | 2 | [latex]\frac{2}{20}[/latex] or 0.10 | 0.85 + 0.10 = 0.95 |
| 7 | 1 | [latex]\frac{1}{20}[/latex] 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.
Example 2
We sample the height of 100 soccer players. The result is shown below.
| Height (inches) | Frequency |
| 59.95 – 61.95 | 5 |
| 61.95 – 63.95 | 3 |
| 63.95 – 65.95 | 15 |
| 65.95 – 67.95 | 40 |
| 67.95 – 69.95 | 17 |
| 69.95 – 71.95 | 12 |
| 71.95 – 73.95 | 7 |
| 73.95 – 75.95 | 1 |
| Total = 100 |
Find:
a. the relative frequency for each class.
Show Answer
| Height (Inches) | Frequency | Relative Frequency | Cumulative Relative Frequency |
| 59.95 – 61.95 | 5 | [latex]\frac{5}{100}[/latex] or 0.05 | 0.05 |
| 61.95 – 63.95 | 3 | [latex]\frac{3}{100}[/latex] or 0.03 | 0.05 + 0.03 = 0.08 |
| 63.95 – 65.95 | 15 | [latex]\frac{15}{100}[/latex] or 0.15 | 0.08 + 0.15 = 0.23 |
| 65.95 – 67.95 | 40 | [latex]\frac{4}{100}[/latex] or 0.04 | 0.23 + 0.40 = 0.63 |
| 67.95 – 69.95 | 17 | [latex]\frac{17}{100}[/latex] or 0.17 | 0.63 + 0.17 = 0.80 |
| 69.95 – 71.95 | 12 | [latex]\frac{12}{100}[/latex] or 0.12 | 0.80 + 0.12 = 0.92 |
| 71.95 – 73.95 | 7 | [latex]\frac{7}{100}[/latex] or 0.07 | 0.92 + 0.07 = 0.99 |
| 73.95 – 75.95 | 1 | [latex]\frac{1}{100}[/latex] or 0.01 | 0.99 + 0.01 = 1.00 |
| Total = 100 | Total = 1 |
b. the percentage for height that is less than 63.95 inches.
Show Answer
[latex]\frac{5+3}{100}[/latex] = 0.08 = 8%
c. the percentage for height that is between 69.95 inches and 73.95 inches.
Show Answer
[latex]\frac{12}{100}[/latex] + [latex]\frac{9}{100}[/latex] = 0.12 + 0.07 = 0.19
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 3
The table shows the amount, in inches, of annual rainfall in a sample of towns.
| Rainfall (inches) | Frequency |
| 2.95 – 4.97 | 6 |
| 4.97 – 6.99 | 7 |
| 6.99 – 9.01 | 15 |
| 9.01 – 11.03 | 8 |
| 11.03 – 13.05 | 9 |
| 13.05 – 15.07 | 5 |
Find
- the relative frequency and cumulative relative frequency for each class.
Show Answer
Total = sum of all frequencies = 6 + 7 + 15 + 8 + 9 + 5 = 50
Rainfall (inches) Frequency Relative frequency Cumulative relative frequency 2.95 – 4.97 6 [latex]\frac{6}{50}[/latex] = 0.12 0.12 4.97 – 6.99 7 [latex]\frac{7}{50}[/latex] = 0.14 0.12 + 0.14 = 0.26 6.99 – 9.01 15 [latex]\frac{15}{50}[/latex] = 0.30 0.26 + 0.30 = 0.56 9.01 – 11.03 8 [latex]\frac{8}{50}[/latex] = 0.16 0.56 + 0.16 = 0.72 11.03 – 13.05 9 [latex]\frac{9}{50}[/latex] = 0.18 0.72 + 0.18 = 0.90 13.05 – 15.07 5 [latex]\frac{5}{50}[/latex] = 0.10 0.90 + 0.10 = 1.00 - the percentage of rainfall that is less than 9.01 inches.
Show Answer
The percentage of rainfall that is less than 9.01 inches = 0.12 + 0.14 + 0.30 = 0.56
- the percentage of heights that fall between 61.95 and 65.95 inches.
Show Answer
The percentage of heights that fall between 6.99 inches and 11.03 inches = [latex]\frac{15}{50}[/latex] + [latex]\frac{8}{50}[/latex] = 0.26
Try It
The table 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 | 823,356 |
- What is the frequency of deaths measured from 2006 through 2009?
Show Answer
97,118
- What percentage of deaths occurred after 2009?
Show Answer
41.6%
- What is the relative frequency of deaths that occurred in 2003 or earlier?
Show Answer
[latex]\frac{67,092}{823,356}[/latex] = 0.081
- What is the percentage of deaths that occurred in 2004?
Show Answer
27.8%
- What kind of data are the numbers of deaths?
Show Answer
Quantitative discrete
- 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?
Show Answer
Quantitative continuous
Example 4
The table contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 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 |
- What is the frequency of deaths measured from 2000 through 2004?
Show Answer
37,526 + 37,862 + 38,491 + 38,477 + 38,444 = 190,800
- What percentage of deaths occurred after 2006?
Show Answer
[latex]\frac{37,435 + 34,172 + 30,862 + 30,296 + 29,757}{653,782}[/latex] or 24.9%
- What is the relative frequency of deaths that occurred in 2000 or before?
Show Answer
[latex]\frac{260,086}{653,782}[/latex] or 39.8%
- What is the percentage of deaths that occurred in 2011?
Show Answer
[latex]\frac{29,757}{653,782}[/latex] or 4.6%
- What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.
Show Answer
75.1% of all fatal traffic crashes for the period from 1994 to 2011 happened from 1994 to 2006.