Lesson 12 of 15 · free to read
Construct frequency, relative frequency, and cumulative relative frequency tables from raw data and interpret them.
You already know how to classify data as qualitative or quantitative. But when you have a list of numbers, how do you see the pattern? A frequency table turns raw data into a clear summary.
The idea
Frequency tables count how often each value occurs; relative and cumulative relative frequencies turn those counts into proportions that describe the whole data set.
Frequency table of student work hours (n = 20)
| DATA VALUE | FREQUENCY |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 3 |
| 5 | 6 |
| 6 | 2 |
| 7 | 1 |
Each row pairs a data value with how many students reported that many hours. The frequencies add to 20, the total number of students.
Definition
relative frequency
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.
The fraction or percentage of all data points that have that value.
Frequency table with relative frequencies
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
| 2 | 3 | or 0.15 |
| 3 | 5 | or 0.25 |
| 4 | 3 | or 0.15 |
| 5 | 6 | or 0.30 |
| 6 | 2 | or 0.10 |
| 7 | 1 | or 0.05 |
Each relative frequency is the frequency divided by 20. The relative frequencies sum to 20/20 = 1.
Definition
cumulative relative frequency
The accumulation of the previous relative frequencies; to find it, add all the previous relative frequencies to the relative frequency for the current row.
The running total of relative frequencies up to and including the current value.
Frequency table with relative and cumulative relative frequencies
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
| 2 | 3 | or 0.15 | 0.15 |
| 3 | 5 | or 0.25 | 0.15 + 0.25 = 0.40 |
| 4 | 3 | or 0.15 | 0.40 + 0.15 = 0.55 |
| 5 | 6 | or 0.30 | 0.55 + 0.30 = 0.85 |
| 6 | 2 | or 0.10 | 0.85 + 0.10 = 0.95 |
| 7 | 1 | or 0.05 | 0.95 + 0.05 = 1.00 |
Each cumulative entry adds the current row's relative frequency to the previous total. The last entry is 1.00, meaning all data has been accumulated.
Because of rounding, the relative frequency column may not always sum to exactly one, and the last entry in the cumulative relative frequency column may not be exactly one. However, they each should be close to one.
When data are continuous, like heights, we group them into intervals. The table below shows the heights, in inches, of a sample of 100 semiprofessional soccer players, grouped into intervals such as 59.95–61.95 inches.
Frequency table of soccer player heights (n = 100)
| HEIGHTS (INCHES) | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
| 59.95–61.95 | 5 | 0.05 | |
| 61.95–63.95 | 3 | 0.05 + 0.03 = 0.08 | |
| 63.95–65.95 | 15 | 0.08 + 0.15 = 0.23 | |
| 65.95–67.95 | 40 | 0.23 + 0.40 = 0.63 | |
| 67.95–69.95 | 17 | 0.63 + 0.17 = 0.80 | |
| 69.95–71.95 | 12 | 0.80 + 0.12 = 0.92 | |
| 71.95–73.95 | 7 | 0.92 + 0.07 = 0.99 | |
| 73.95–75.95 | 1 | 0.99 + 0.01 = 1.00 | |
| Total = 100 | Total = 1.00 |
Each row is an interval of heights. The frequency column counts players in that interval; the cumulative column builds to 1.00 by the last row.
Worked example
Use the cumulative relative frequency column.
Identify the rows with heights less than 65.95 inches: the first three intervals (59.95–61.95, 61.95–63.95, 63.95–65.95).
These intervals cover all heights below 65.95 inches.
Another example: nineteen people were asked how many miles they commute to work each day. The data are: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. A frequency table was produced, but it contains errors.
Frequency table of commuting distances (n = 19)
| DATA | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
| 3 | 3 | 0.1579 | |
| 4 | 1 | 0.2105 | |
| 5 | 3 | 0.1579 | |
| 7 | 2 | 0.2632 | |
| 10 | 3 | 0.4737 | |
| 12 | 2 | 0.7895 | |
| 13 | 1 | 0.8421 | |
| 15 | 1 | 0.8948 | |
| 18 | 1 | 0.9474 | |
| 20 | 1 | 1.0000 |
Check the frequency column: it sums to 18, not 19. Also, the value 2 is missing, and several cumulative relative frequencies are wrong.
The table is incorrect: the frequency column sums to 18, not 19, because the value 2 (which occurs twice) was left out. The cumulative relative frequencies are also wrong. The correct cumulative 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.
Frequency tables also work for discrete data like years of service. The table below shows the number of years of service for 70 federal employees.
Frequency table of years of service for 70 federal employees
| Number of Years of Service | Number of Federal Employees |
|---|---|
| 24 | 2 |
| 25 | 1 |
| 26 | 3 |
| 27 | 0 |
| 28 | 4 |
| 29 | 6 |
| 30 | 11 |
| 31 | 12 |
| 32 | 7 |
| 33 | 8 |
| 34 | 6 |
| 35 | 10 |
Each row is a whole number of years. The frequencies add to 70. Notice the zero frequency for 27 years.
Worked example
Use the definition of relative frequency.
Read the frequency for 30 years: 11.
The table shows 11 employees have 30 years of service.
Practice with the same table:
Recap
Say each point to yourself, then tap it to check.
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Adapted from Introductory Statistics 2e, © 2023 OpenStax, Rice University
Barbara Illowsky and Susan Dean
Licensed under CC BY 4.0. This course is an adaptation, not the original work.
Access for free at openstax.org/details/books/introductory-statistics-2e