Cumulative Frequency
Cumulative frequency is used to show running totals in grouped data. It is especially useful for estimating the median, lower quartile, upper quartile and interquartile range from a graph.
Cumulative frequency means adding frequencies as you move down a table.
It answers questions like: How many values are less than or equal to this amount?
How to complete a cumulative frequency table
To complete a cumulative frequency table, keep a running total of the frequency column.
Example 1: Complete a cumulative frequency table
The table shows test scores for a class.
0 < score ≤ 10: frequency 3
10 < score ≤ 20: frequency 7
20 < score ≤ 30: frequency 12
30 < score ≤ 40: frequency 8
Now add the frequencies as a running total:
3
3 + 7 = 10
10 + 12 = 22
22 + 8 = 30
The cumulative frequencies are 3, 10, 22 and 30.
How to draw a cumulative frequency graph
For grouped data, plot the cumulative frequency against the upper class boundary.
Use the upper boundary on the horizontal axis.
Use the cumulative frequency on the vertical axis.
Join the points with a smooth increasing curve.
Example 2: Plotting points
Using the table from Example 1, plot these points:
(10, 3)
(20, 10)
(30, 22)
(40, 30)
The first coordinate is the upper class boundary. The second coordinate is the cumulative frequency.
Finding the median from cumulative frequency
To estimate the median, first find half of the total frequency. Then use the cumulative frequency graph to read the matching value.
Example 3: Estimate the median
The total frequency is 30.
30 ÷ 2 = 15
On the cumulative frequency graph, find 15 on the vertical axis, draw across to the curve, then draw down to the horizontal axis.
The value you read from the horizontal axis is the estimated median.
Quartiles and interquartile range
Quartiles split the data into four parts. The lower quartile is one quarter of the way through the data. The upper quartile is three quarters of the way through the data.
Lower quartile position = total frequency ÷ 4
Median position = total frequency ÷ 2
Upper quartile position = 3 × total frequency ÷ 4
Interquartile range = upper quartile − lower quartile
Example 4: Quartile positions
A cumulative frequency graph has a total frequency of 80.
Lower quartile position = 80 ÷ 4 = 20
Median position = 80 ÷ 2 = 40
Upper quartile position = 3 × 80 ÷ 4 = 60
You would read the values at cumulative frequencies 20, 40 and 60 from the graph.
A common mistake is plotting frequency instead of cumulative frequency. The graph should always increase or stay level; it should not go down.
In GCSE exams, use a ruler carefully when reading from a cumulative frequency graph. Small reading errors are usually accepted, but your method must be clear.
Video explanation
A short Worthing Maths Tutor video explanation for cumulative frequency can be embedded here later to improve student engagement and time on page.
Practice questions
- Frequencies are 4, 6, 9 and 11. Write down the cumulative frequencies.
- A grouped table has upper boundaries 10, 20, 30 and 40 with cumulative frequencies 5, 14, 25 and 32. What points should be plotted?
- A cumulative frequency graph has total frequency 60. What cumulative frequency should you use to estimate the median?
- A cumulative frequency graph has total frequency 100. What cumulative frequencies should you use for the lower and upper quartiles?
- The lower quartile is 18 and the upper quartile is 43. Find the interquartile range.
Answers
- 4, 10, 19, 30
- (10, 5), (20, 14), (30, 25), (40, 32)
- 30
- Lower quartile: 25, upper quartile: 75
- 25
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