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