Learning OUTCOMES
- Calculate and use relative frequencies and cumulative relative frequencies to answer questions about a distribution
Recall: Converting Fractions to Decimals
To convert a fraction into a decimal, divide the numerator (the number above the division symbol) by the denominator (the number below the division symbol). In probability, the numerator represents the number of events, and the denominator represents the number of possible outcomes.
Example: [latex]\frac{3}{20} = 3 \div 20 = 0.15[/latex]
Frequency
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: [latex]5[/latex], [latex]6[/latex], [latex]3[/latex], [latex]3[/latex], [latex]2[/latex], [latex]4[/latex], [latex]7[/latex], [latex]5[/latex], [latex]2[/latex], [latex]3[/latex], [latex]5[/latex], [latex]6[/latex], [latex]5[/latex], [latex]4[/latex], [latex]4[/latex], [latex]3[/latex], [latex]5[/latex], [latex]2[/latex], [latex]5[/latex], [latex]3[/latex].
The following table lists the different data values in ascending order and their frequencies.
| DATA VALUE | FREQUENCY |
|---|---|
| [latex]2[/latex] | [latex]3[/latex] |
| [latex]3[/latex] | [latex]5[/latex] |
| [latex]4[/latex] | [latex]3[/latex] |
| [latex]5[/latex] | [latex]6[/latex] |
| [latex]6[/latex] | [latex]2[/latex] |
| [latex]7[/latex] | [latex]1[/latex] |
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, [latex]20[/latex], 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, [latex]20[/latex]. Relative frequencies can be written as fractions, percents, or decimals.
Recall: Calculating A Probability
To calculate the probability of an event occurring, you find the number of times the event occurs and divide that by the total number of possible outcomes.
[latex]\mathrm{Probability \ of \ an \ event} = \frac{\mathrm{number \ of \ events}}{\mathrm{number \ of \ outcomes}}[/latex]
Example: Rolling an odd number on a 6-sided die. There are 3 odd numbers (1, 3, 5) out of the 6 total numbers on the die.
[latex]\mathrm{Probability \ of \ rolling \ an \ odd \ number} = \frac{3}{6}[/latex]
Notice, the probability is not a simplified fraction, this is commonly seen in statistics how to find the probability of an event.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| [latex]2[/latex] | [latex]3[/latex] | [latex]\displaystyle\frac{3}{20}[/latex] or [latex]0.15[/latex] |
| [latex]3[/latex] | [latex]5[/latex] | [latex]\displaystyle\frac{5}{20}[/latex] or [latex]0.25[/latex] |
| [latex]4[/latex] | [latex]3[/latex] | [latex]\displaystyle\frac{3}{20}[/latex] or [latex]0.15[/latex] |
| [latex]5[/latex] | [latex]6[/latex] | [latex]\displaystyle\frac{6}{20}[/latex] or [latex]0.30[/latex] |
| [latex]6[/latex] | [latex]2[/latex] | [latex]\displaystyle\frac{2}{20}[/latex] or [latex]0.10[/latex] |
| [latex]7[/latex] | [latex]1[/latex] | [latex]\displaystyle\frac{1}{20}[/latex] or [latex]0.05[/latex] |
The sum of the values in the relative frequency column of the previous table is [latex]\frac{20}{20}[/latex], or [latex]1[/latex].
Recall: Converting Fractions to Decimals
To add fractions with like denominators, like the fractions we will have when calculating the relative frequency, you add the numerators and leave the denominators the same.
Fraction Addition
If [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then
[latex]\Large\frac{a}{c}\normalsize+\Large\frac{b}{c}\normalsize=\Large\frac{a+b}{c}[/latex]
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.
You will be adding different numbers of events but the number of outcomes (denominator) in a situation does not change.
Recall: Adding or Subtracting Decimals
- Write the numbers vertically so the decimal points line up.
- Use zeros as place holders, as needed.
- Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
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.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| [latex]2[/latex] | [latex]3[/latex] | [latex]\displaystyle\frac{3}{20}[/latex] or [latex]0.15[/latex] | [latex]0.15[/latex] |
| [latex]3[/latex] | [latex]5[/latex] | [latex]\displaystyle\frac{5}{20}[/latex] or [latex]0.25[/latex] | [latex]0.15 + 0.25 = 0.40[/latex] |
| [latex]4[/latex] | [latex]3[/latex] | [latex]\displaystyle\frac{3}{20}[/latex] or [latex]0.15[/latex] | [latex]0.40 + 0.15 = 0.55[/latex] |
| [latex]5[/latex] | [latex]6[/latex] | [latex]\displaystyle\frac{6}{20}[/latex] or [latex]0.30[/latex] | [latex]0.55 + 0.30 = 0.85[/latex] |
| [latex]6[/latex] | [latex]2[/latex] | [latex]\displaystyle\frac{2}{20}[/latex] or [latex]0.10[/latex] | [latex]0.85 + 0.10 = 0.95[/latex] |
| [latex]7[/latex] | [latex]1[/latex] | [latex]\displaystyle\frac{1}{20}[/latex] or [latex]0.05[/latex] | [latex]0.95 + 0.05 = 1.00[/latex] |
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 1.12 below 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 | [latex]\frac{5}{100} = 0.05 [/latex] | 0.05 |
| 61.95–63.95 | 3 | [latex]\frac{3}{100}= 0.03[/latex] | 0.05 + 0.03 = 0.08 |
| 63.95–65.95 | 15 | [latex]\frac{15}{100}= 0.15[/latex] | 0.08 + 0.15 = 0.23 |
| 65.95–67.95 | 40 | [latex]\frac{40}{100} = 0.40[/latex] | 0.23 + 0.40 = 0.63 |
| 67.95–69.95 | 17 | [latex]\frac{17}{100}= 0.17[/latex] | 0.63 + 0.17 = 0.80 |
| 69.95–71.95 | 12 | [latex]\frac{12}{100}= 0.12[/latex] | 0.80 + 0.12 = 0.92 |
| 71.95–73.95 | 7 | [latex]\frac{7}{100}= 0.07[/latex] | 0.92 + 0.07 = 0.99 |
| 73.95–75.95 | 1 | [latex]\frac{1}{100}= 0.01[/latex] | 0.99 + 0.01 = 1.00 |
| Total = 100 | Total = 1.00 |
Table 1.12
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
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 5 players whose heights fall within the interval 59.95-61.95 inches, 3 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, 7 players whose heights fall within the interval 71.95-73.95, and 1 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
From Table 1.12, find the percentage of heights that are less than 65.95 inches.
Try It
Table 1.13 shows the amount, in inches, of annual rainfall in a sample of towns.
| Rainfall (Inches) | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| [latex]2.95–4.97[/latex] | [latex]6[/latex] | [latex] \frac{6}{50} = 0.12 [/latex] | [latex]0.12[/latex] |
| [latex]4.97–6.99[/latex] | [latex]7[/latex] | [latex] \frac{7}{50} = 0.14[/latex] | [latex]0.12 + 0.14 = 0.26[/latex] |
| [latex]6.99–9.01[/latex] | [latex]15[/latex] | [latex] \frac{15}{50} = 0.30[/latex] | [latex]0.26 + 0.30 = 0.56[/latex] |
| [latex]9.01–11.03[/latex] | [latex]8[/latex] | [latex] \frac{8}{50} =0.16 [/latex] | [latex]0.56 + 0.16 = 0.72[/latex] |
| [latex]11.03–13.05[/latex] | [latex]9[/latex] | [latex] \frac{9}{50} = 0.18[/latex] | [latex]0.72 + 0.18 = 0.90[/latex] |
| [latex]13.05–15.07[/latex] | [latex]5[/latex] | [latex] \frac{5}{50} = 0.10 [/latex] | [latex]0.90 + 0.10 = 1.00[/latex] |
| Total [latex]= 50[/latex] | Total [latex]= 1.00[/latex] |
Example
From Table 1.12, find the percentage of heights that fall between 61.95 and 65.95 inches.
Try It
From Table 1.13, find the percentage of rainfall that is between [latex]6.99[/latex] and [latex]13.05[/latex] inches.
Example
Use the heights of the [latex]100[/latex] male semiprofessional soccer players in Table 1.12. Fill in the blanks and check your answers.
- The percentage of heights that are from [latex]67.95[/latex] to [latex]71.95[/latex] inches is: ____.
- The percentage of heights that are from [latex]67.95[/latex] to [latex]73.95[/latex] inches is: ____.
- The percentage of heights that are more than [latex]65.95[/latex] inches is: ____.
- The number of players in the sample who are between [latex]61.95[/latex] and [latex]71.95[/latex] 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
From Table 1.13, find the number of towns that have rainfall between [latex]2.95[/latex] and [latex]9.01[/latex] inches.
Activity
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
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 | [latex]\frac{3}{19}[/latex] | 0.1579 |
| 4 | 1 | [latex]\frac{1}{19}[/latex] | 0.2105 |
| 5 | 3 | [latex]\frac{3}{19}[/latex] | 0.1579 |
| 7 | 2 | [latex]\frac{2}{19}[/latex] | 0.2632 |
| 10 | 3 | [latex]\frac{4}{19}[/latex] | 0.4737 |
| 12 | 2 | [latex]\frac{2}{19}[/latex] | 0.7895 |
| 13 | 1 | [latex]\frac{1}{19}[/latex] | 0.8421 |
| 15 | 1 | [latex]\frac{1}{19}[/latex] | 0.8948 |
| 18 | 1 | [latex]\frac{1}{19}[/latex] | 0.9474 |
| 20 | 1 | [latex]\frac{1}{19}[/latex] | 1.0000 |
Table 1.14 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)?
Try It
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
Table 1.15 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 1.15
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
Table 1.16 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 |
Table 1.16
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.
