Cumulative frequency is found by adding the frequencies of each class interval to the sum of the frequencies of all the class intervals preceding it in a frequency distribution table. The final value will always be the total number of observations.
Here's a step-by-step explanation:
Steps to Calculate Cumulative Frequency:
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Create a Frequency Distribution Table: Begin with your data organized into class intervals (or categories) and their corresponding frequencies.
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Add a Cumulative Frequency Column: Add a new column to your frequency distribution table labeled "Cumulative Frequency."
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Calculate the First Cumulative Frequency: The first cumulative frequency is the same as the frequency of the first class interval. Place this value in the cumulative frequency column for the first class interval.
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Calculate Subsequent Cumulative Frequencies: For each subsequent class interval, add the frequency of that interval to the previous cumulative frequency.
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Repeat: Continue this process until you have calculated the cumulative frequency for all class intervals.
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Verification: The last cumulative frequency should be equal to the total number of observations in your dataset. This serves as a check that you have calculated correctly.
Example:
Let's say we have the following frequency distribution of ages:
| Age (Years) | Frequency |
|---|---|
| 10-20 | 5 |
| 21-30 | 12 |
| 31-40 | 8 |
| 41-50 | 3 |
To find the cumulative frequency, we would follow these steps:
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Start with the Frequency Distribution: As above.
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Add Cumulative Frequency Column:
| Age (Years) | Frequency | Cumulative Frequency |
|---|---|---|
| 10-20 | 5 | |
| 21-30 | 12 | |
| 31-40 | 8 | |
| 41-50 | 3 |
- First Cumulative Frequency: The cumulative frequency for the 10-20 age group is 5.
| Age (Years) | Frequency | Cumulative Frequency |
|---|---|---|
| 10-20 | 5 | 5 |
| 21-30 | 12 | |
| 31-40 | 8 | |
| 41-50 | 3 |
- Second Cumulative Frequency: The cumulative frequency for the 21-30 age group is 5 + 12 = 17.
| Age (Years) | Frequency | Cumulative Frequency |
|---|---|---|
| 10-20 | 5 | 5 |
| 21-30 | 12 | 17 |
| 31-40 | 8 | |
| 41-50 | 3 |
- Continue: Continue the process. The cumulative frequency for the 31-40 age group is 17 + 8 = 25. The cumulative frequency for the 41-50 age group is 25 + 3 = 28.
| Age (Years) | Frequency | Cumulative Frequency |
|---|---|---|
| 10-20 | 5 | 5 |
| 21-30 | 12 | 17 |
| 31-40 | 8 | 25 |
| 41-50 | 3 | 28 |
- Verification: The total number of observations is 5+12+8+3=28, which is the same as the final cumulative frequency.
Why is Cumulative Frequency Useful?
Cumulative frequency is useful for understanding the proportion of data that falls below a certain value, for finding percentiles, and for creating ogives (cumulative frequency graphs). It provides a quick way to see how many data points fall within or below a given range.
