Cumulative frequency is one of those topics that looks complicated on paper but follows a very clear, repeatable method. Once you know the steps, you can pick up 6–8 marks every time it comes up — and it does come up, usually in Paper 2 or Paper 3 on AQA 8300, Edexcel 1MA1, and OCR J560. It often links to box plots, so mastering both together is time well spent.
What You Need to Know First
- How to read a grouped frequency table
- How to plot coordinates on a graph accurately
- What the median, lower quartile (LQ) and upper quartile (UQ) mean
- How to calculate the interquartile range (IQR): IQR = UQ − LQ
- What a box plot looks like and its five key values
Drawing a Cumulative Frequency Graph
Cumulative frequency means the running total of frequencies as you move through the data. You always plot cumulative frequency against the upper class boundary of each group — never the midpoint.
Step 1: Add a cumulative frequency column to your table by adding up frequencies as you go.
Step 2: Write down the coordinates. Each point is (upper class boundary, cumulative frequency).
Step 3: Plot the points and join them with a smooth curve. The curve always starts at (lowest value, 0) — it must go through the origin of your data, not the graph's origin.
Step 4: Label your axes. The y-axis is cumulative frequency; the x-axis is the variable being measured.
Worked Example 1
The table below shows the times (in minutes) 80 students took to complete a test.
| Time (minutes) | Frequency |
|---|---|
| 0 < t ≤ 20 | 5 |
| 20 < t ≤ 40 | 18 |
| 40 < t ≤ 60 | 32 |
| 60 < t ≤ 80 | 20 |
| 80 < t ≤ 100 | 5 |
Step 1 — Build the cumulative frequency column:
| Time (minutes) | Frequency | Cumulative Frequency |
|---|---|---|
| 0 < t ≤ 20 | 5 | 5 |
| 20 < t ≤ 40 | 18 | 23 |
| 40 < t ≤ 60 | 32 | 55 |
| 60 < t ≤ 80 | 20 | 75 |
| 80 < t ≤ 100 | 5 | 80 |
Step 2 — Write the coordinates using upper class boundaries:
(20, 5), (40, 23), (60, 55), (80, 75), (100, 80)
Also plot the starting point: (0, 0) — because 0 students had finished in 0 minutes.
Step 3 — Plot and join with a smooth S-shaped curve. The curve should pass through all six points.
Finding the Median and Quartiles
Once you have your cumulative frequency graph, you can read off the median, lower quartile and upper quartile using horizontal and vertical lines.
For a dataset of n values:
- Median = value at n ÷ 2
- Lower Quartile (LQ) = value at n ÷ 4
- Upper Quartile (UQ) = value at n ÷ 4 × 3 (or 3n ÷ 4)
- IQR = UQ − LQ
Worked Example 2
Using the graph from the example above (n = 80 students):
Median: Read across from 80 ÷ 2 = 40 on the cumulative frequency axis → draw a horizontal line to the curve → drop down to the x-axis → Median ≈ 53 minutes
Lower Quartile: Read across from 80 ÷ 4 = 20 → → LQ ≈ 38 minutes
Upper Quartile: Read across from 3 × 80 ÷ 4 = 60 → → UQ ≈ 66 minutes
IQR = 66 − 38 = 28 minutes
These five values — minimum (0), LQ (38), median (53), UQ (66), maximum (100) — are exactly what you need to draw a box plot.
To draw the box plot:
- Draw a number line covering the full range
- Draw a box from LQ to UQ
- Draw a vertical line inside the box at the median
- Draw whiskers out to the minimum and maximum values
Common Mistakes Students Make
- Plotting against the midpoint instead of the upper class boundary. The point (40, 23) uses the upper boundary of 20 < t ≤ 40, which is 40 — not the midpoint 30. This is one of the most common errors.
- Forgetting to plot the starting point (0, 0). Without this, your curve won't start in the right place and the shape will be wrong.
- Joining points with straight lines instead of a smooth curve. Cumulative frequency graphs should always be a smooth S-shaped curve. Straight lines between points lose the presentation mark.
- Reading the wrong value from the axis. When finding the median, go across from the cumulative frequency axis first, then down to the x-axis — not the other way around. Students who go up from the x-axis are guessing.
- Using n instead of n ÷ 2 for the median. The median is at the n/2 position, not at n. For 80 values, go across from 40, not 80.
Exam Tips for Cumulative Frequency
- Always label your axes — you'll lose a mark if they're missing. The y-axis must say "Cumulative frequency" and the x-axis must show the variable and units.
- Use a ruler for the horizontal and vertical construction lines when reading off the median and quartiles. Examiners expect to see these lines drawn on the graph — they are your working, and some mark schemes award a method mark specifically for them.
- The final cumulative frequency must equal the total frequency. Check your last value matches n before you start plotting — it's a quick error-check that takes two seconds.
- AQA and Edexcel both award method marks for a correct cumulative frequency table even if the graph is slightly inaccurate. Fill the table in first, carefully, before touching the graph.
- For box plots, read values off your curve to 1 decimal place if the graph allows it. Being precise shows the examiner you used the graph properly and earns full marks where another student might lose an accuracy mark.
- If the question says "use your graph," you must read the answer from the graph — not calculate it from the table. Examiners check that construction lines are drawn.
Practise Now
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