Cumulative Frequency Diagrams: Median, Quartiles and IQR for GCSE Higher

Cumulative frequency is one of the most reliably examined topics on Higher tier Statistics — and one of the easiest to pick up full marks on, because the method almost never changes. Complete the table, plot the points, join them up, then read values off. The only students who lose marks here are the ones who plot in the wrong place or misread the scale.

Let’s work through the whole thing properly.

What “cumulative” actually means

A regular frequency table tells you how many are in each group. A cumulative frequency table tells you how many are in that group and every group before it — a running total.

Here’s a set of test marks:

Test mark	Frequency	Test mark	Cumulative frequency
0 < m ≤ 20	6	0 < m ≤ 20	6
20 < m ≤ 40	22	0 < m ≤ 40	28
40 < m ≤ 60	36	0 < m ≤ 60	64
60 < m ≤ 80	45	0 < m ≤ 80	109
80 < m ≤ 100	11	0 < m ≤ 100	120

Two things to notice:

1. The first entry is always identical in both tables — there’s nothing before it to add on.
2. Every interval in the cumulative column starts at 0. The second row isn’t “20 to 40”, it’s “0 to 40”, because it now includes everyone from the first group too.

The quick way to fill it in is to add each new frequency onto the previous running total: 6, then 6 + 22 = 28, then 28 + 36 = 64, then 64 + 45 = 109, then 109 + 11 = 120.

Check: the final cumulative frequency must equal the sum of all the original frequencies. Here that’s 120. If it doesn’t match, you’ve made an addition slip — find it before you start plotting, or every point after it will be wrong.

Drawing the diagram

Four steps, in this order:

1. Scale the axes. The variable (test mark) goes on the horizontal axis, cumulative frequency always on the vertical. Your vertical axis must reach at least 120 here.
2. Plot at the END of each interval. This is the rule examiners are really testing. The entry “0 to 20, cumulative frequency 6” means six people scored up to 20 marks, so it goes at (20, 6). Then (40, 28), (60, 64), (80, 109), (100, 120).
3. Add the extra point at the start. Nobody scored below 0, so the cumulative frequency at 0 is 0 — plot (0, 0). Careful: the first interval doesn’t always begin at zero. If your table starts at 10, you plot a cumulative frequency of 0 at 10.
4. Join them up, left to right, with straight lines or a smooth curve.

Tip: Both straight lines and a smooth curve earn full marks. Straight lines are easier to draw and much easier to read off accurately, so unless your teacher insists otherwise, use a ruler and keep it simple.

The single biggest error here is plotting at the midpoint of each interval. That’s the method for a frequency polygon — a completely different diagram. Cumulative frequency goes at the upper bound, every time.

Reading the median and quartiles

Once the curve is drawn, every follow-up question uses the same technique: go up the vertical axis, across to the curve, then down.

What you want	Go up to	Then
Lower quartile (LQ)	n ÷ 4	Across to curve, down, read off
Median	n ÷ 2	Across to curve, down, read off
Upper quartile (UQ)	3n ÷ 4	Across to curve, down, read off
Interquartile range	—	UQ − LQ

Here, n is the total cumulative frequency — found by reading across from the highest point on the graph, not by looking at the top of the axis.

Take a diagram of the number of minutes students were late to school, where the highest point reads across to 40. So n = 40:

• Median: half of 40 is 20. Across from 20, down to the axis → 6.5 minutes.
• Lower quartile: a quarter of 40 is 10. Across from 10, down → 2 minutes.
• Upper quartile: three quarters of 40 is 30. Across from 30, down → 25 minutes.
• Interquartile range: 25 − 2 = 23 minutes.

Watch the scale. In a second example about restaurant waiting times, the vertical axis was numbered up to 100 — but the curve topped out at 80. Plenty of students grabbed 100 as the total and halved it to 50. The total is whatever the curve reaches, which was 80, so the median sits at 40 on the vertical axis. Read the top of the curve, not the top of the axis.

For that restaurant data (n = 80): median at 40 → 46 minutes; LQ at 20 → 34 minutes; UQ at 60 → 55 minutes; IQR = 55 − 34 = 21 minutes.

If those terms feel shaky, it’s worth a detour through Mean, Median, Mode and Range before you go further — cumulative frequency assumes you’re already fluent with what an average and a spread actually mean.

Working backwards: from a value to a frequency

The other half of the exam question flips the direction. You’re given a value on the horizontal axis and asked how many are above or below it. Now you go up first, then across.

Students more than 5 minutes late get a detention. What percentage received one?

1. Find 5 on the horizontal axis, go up to the curve, read across → cumulative frequency of 18.
2. That 18 is the number below 5 minutes — so they’re safe.
3. Detentions go to everyone above: 40 − 18 = 22 students.
4. As a percentage: 22 ÷ 40 × 100 = 55%.

The trap in step 2 is real. A cumulative frequency graph always reads “how many are less than this”. If the question asks for “more than”, you must subtract from the total.

One diner is chosen at random. Find the probability they waited between 50 and 70 minutes.

1. At 50 minutes: cumulative frequency 49.
2. At 70 minutes: cumulative frequency 77.
3. Between them: 77 − 49 = 28 people.
4. As a probability: 28/80.

Same idea as an “and” region on a two-way table — subtract the smaller running total from the larger. If probability notation is what’s slowing you down rather than the graph, Probability Tree Diagrams covers the fraction work in more depth.

The five mistakes that cost marks

1. Plotting at midpoints instead of upper bounds.
2. Reading the total off the top of the axis rather than the top of the curve.
3. Forgetting the (0, 0) point — or forcing it in when the first interval doesn’t start at 0.
4. Giving the number above a value when the graph gives you the number below.
5. Doing LQ − UQ instead of UQ − LQ and getting a negative interquartile range. A spread can never be negative.

Remember: every value you read off a cumulative frequency diagram is an estimate. Examiners often want the word “estimate” in your answer, and they’ll accept a tolerance either side of the exact reading — so don’t panic if you get 46 and the mark scheme says 45–47.

What comes next

Cumulative frequency is the gateway to the rest of Higher tier data handling. Once you can pull a median and quartiles off a curve, you can build a box plot from them directly — the five-number summary (minimum, LQ, median, UQ, maximum) is exactly what a cumulative frequency diagram hands you. Our guide to Box and Whisker Plots picks up precisely where this one stops, and exam questions frequently ask you to do both in sequence: draw the curve in part (a), then the box plot in part (b).

Practise this on your phone

Cumulative frequency is a technique, not a fact — you can’t revise it by reading. You need to complete tables and read off values until the “up, across, down” motion is automatic.

GCSE Statistics covers the whole Statistics and Probability strand: surveys, sampling, charts, averages, spread and probability theory — including cumulative frequency tables, quartiles and the interquartile range, with worked solutions so you can see exactly where a reading went wrong. It’s the fastest way to drill the ten-question-a-day habit that makes this topic feel routine by exam day.

If you’re revising the whole subject rather than one strand, GCSE Maths is the complete bundle — Number, Algebra, Geometry and Statistics in a single app, with 2900+ questions across the four subject apps. That matters more than it sounds for Higher tier Statistics, because cumulative frequency questions rarely arrive alone: they get bundled with percentages, probability fractions and comparison-in-context questions that need clean Number skills underneath.

Fifteen minutes a day, working through diagram questions until you stop hesitating over which axis to start from. That’s all this topic takes.