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# Cumulative Frequency
- URL: https://www.esheets.io/cumulative-frequency/
- Published: 2026-10-06T18:59:57.000Z
- Updated: 2026-10-06T18:59:57.000Z
- Author: Richard Linnington
- Tags: Maths, Statistics

Cumulative frequency is a running total used with grouped data. It helps you estimate how many observations lie below a value and locate the middle and spread of the data.

This Higher GCSE worksheet moves from completing and reversing tables to plotting cumulative-frequency graphs at upper class boundaries. Then practise estimating medians, quartiles, interquartile ranges and frequencies, and reading backwards from a cumulative count to a data value. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Ten questions, ten marks: complete a cumulative-frequency table, recover ordinary frequencies, then plot two graphs. The second plotting question also requires you to construct the cumulative table.

Read a median, estimate quartiles and an interquartile range, find frequencies below and above thresholds, and reverse-read a value. Finally, read cumulative totals from a graph, then use successive differences to recover the class frequencies and tackle a graph whose axis extends above the actual total.

#### What you’ll practise

- Running totals and successive differences.
- Plotting cumulative frequencies at upper class boundaries; a lower-boundary zero start may also be included.
- Reading medians, quartiles, interquartile ranges and frequencies.
- Using the curve endpoint to identify the total frequency.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

# Cumulative frequency

Complete the tables, plot the cumulative-frequency graphs and use the graphs to estimate medians, quartiles and frequencies. Each whole question is worth one mark.

## Topic guide

### Frequency and cumulative frequency

Ordinary frequency counts observations in one class. Cumulative frequency is the running total up to an upper class boundary: add the current frequency to all preceding frequencies.

For task times in seconds, let the classes be 10 < t ≤ 20, 20 < t ≤ 30, 30 < t ≤ 40, 40 < t ≤ 50 and 50 < t ≤ 60.

__Worked example: building running totals__
| Time (seconds) | Frequency | Cumulative frequency |
| -------------- | --------- | -------------------- |
| 10 < t ≤ 20    | 6         | 6                    |
| 20 < t ≤ 30    | 14        | 20                   |
| 30 < t ≤ 40    | 20        | 40                   |
| 40 < t ≤ 50    | 14        | 54                   |
| 50 < t ≤ 60    | 6         | 60                   |

For example, the third total is 6 + 14 + 20 = 40\. The final cumulative frequency, 60, is the number of observations.

### Reversing a cumulative table

Take successive differences, starting from zero. For cumulative totals 7, 19, 36, 52, 60, the ordinary frequencies are 7 − 0 = 7, 19 − 7 = 12, 36 − 19 = 17, 52 − 36 = 16 and 60 − 52 = 8\. Check that these frequencies add to 60.

### Plotting the graph

Put the measured variable on the horizontal axis and cumulative frequency on the vertical axis. For the task-time table, plot (20, 6), (30, 20), (40, 40), (50, 54) and (60, 60). You may also include the lower-boundary zero start (10, 0). It is optional, and is not the origin (0, 0): the first class begins at 10 seconds.

Use each **upper class boundary**, not the midpoint or the lower boundary. Join the points with a sensible smooth increasing cumulative-frequency curve (an ogive). Straight joined segments may be tolerated in exams, but this worksheet models the smooth curve convention. Its drawing questions join your points automatically; the positions of your points are marked.

### Median, quartiles and interquartile range

First read the total n from the curve endpoint. The median position is n/2, the lower quartile position is n/4 and the upper quartile position is 3n/4\. These are levels on the cumulative-frequency axis, not the data values you give as answers. Read horizontally from each level to the curve, then vertically down to the data axis.

For the task-time example, n = 60\. The positions are 15, 30 and 45\. Reading the smooth model curve gives graph estimates of a lower quartile of about 26.9 seconds, a median of 35 seconds and an upper quartile of about 43.1 seconds. Using the unrounded curve readings, the interquartile range is UQ − LQ ≈ 16.1 seconds (about 16 seconds from a graph).

### Below, above and reverse reading

To estimate how many observations are below a value, read up from that value to the curve, then across to the cumulative-frequency axis. At 40 seconds in the example, the cumulative frequency is 40\. For the number above 40 seconds, subtract from the total: 60 − 40 = 20 people.

For “The fastest 20 people took under how many seconds?”, start at cumulative frequency 20, read across to the curve, then down: 30 seconds. For “The slowest 6 people took longer than how many seconds?”, start at 60 − 6 = 54: the reading is 50 seconds. These are estimates from grouped data.

### Checks that prevent common mistakes

- Use running totals for plotting, rather than individual class frequencies. Include every preceding class.
- Plot cumulative frequencies at the upper class boundaries. If you include a zero-frequency start, place it at the lower class boundary, not automatically at x = 0.
- Read n from the final point. An axis ending at 80 can still contain only 60 observations: the median level is then 30, not 40.
- Keep the quartile levels in order: n/4, n/2, 3n/4\. Subtract data values, not cumulative counts, for the interquartile range.
- For “above”, subtract the cumulative count from n. For reverse reading, start on the cumulative-frequency axis.
- To rebuild ordinary frequencies from a graph, read cumulative totals at consecutive class boundaries and subtract the earlier total from the later one.