Cumulative relative frequency refers to the proportion of data values that are less than or equal to a certain value. It is usually expressed in the form of a percentage.
For example, suppose that in a test of 100 marks the cumulative frequency for 60 marks is 85%. This means that 85% of the students have obtained less than 60 marks in the exam.
The cumulative relative frequency can be represented easily in the form of a cumulative relative frequency table.
How do you find the cumulative relative frequency?
- Represent the data in the form of a frequency distribution table.
- Find the cumulative frequencies by summing up the frequencies of the current and the preceding class intervals.
- Divide the cumulative frequencies by the total frequency. This gives us the cumulative relative frequency.
- If needed, we convert the cumulative relative frequencies obtained above into percentages by multiplying the proportion by 100.
Example:
Consider the following data given in the form of a frequency table.
| Class Intervals | Frequency |
| 0-10 | 10 |
| 10-20 | 15 |
| 20-30 | 7 |
| 30-40 | 18 |
We find out the cumulative relative frequency table as follows.
Step 1: We find out the cumulative frequency by adding up the frequencies of the current and previous class intervals.
| Class Intervals | Frequency | Cumulative Frequency |
| 0-10 | 10 | 10 |
| 10-20 | 15 | 10+15 = 25 |
| 20-30 | 7 | 10+15+7 = 32 |
| 30-40 | 18 | 10+15+7+18 = 50 |
| TOTAL = 50 |
Step 2: We find out the relative cumulative frequency by dividing the cumulative frequency by the total frequency.
| Class Intervals | Frequency | Cumulative Frequency | Relative Cumulative Frequency |
| 0-10 | 10 | 10 | 10/50 = 0.2 |
| 10-20 | 15 | 25 | 25/50 = 0.5 |
| 20-30 | 7 | 32 | 32/50 = 0.64 |
| 30-40 | 18 | 50 | 50/50 = 1 |
| Total=50 |
Step 3: We convert the relative cumulative frequency into percentages by multiplying it by 100
| Class Intervals | Frequency | Cumulative Frequency | Relative Cumulative Frequency |
| 0-10 | 10 | 10 | 0.2 or 20% |
| 10-20 | 15 | 25 | 0.5 or 50% |
| 20-30 | 7 | 32 | 0.64 or 64% |
| 30-40 | 18 | 50 | 1 or 100% |
| Total=50 |
Remarks:
1. The cumulative relative frequency for the last class must always be 1 because we are dividing the total frequency by itself to calculate the final cumulative relative frequency.
2. A cumulative relative frequency distribution shows us the percentage of data values that lie below an upper class interval.
3. A cumulative relative frequency graph or a cumulative frequency histogram is drawn in the usual way except for the fact that we plot the cumulative relative frequencies on the Y axis.
