Cumulative Frequency Graphs and Box Plots — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers two ways of summarising grouped data: the cumulative frequency graph, which shows how the values build up across the range, and the box plot, which reduces a whole distribution to five numbers. By the end of this guide you should be able to build a cumulative frequency table, plot the graph correctly, and read the median and quartiles from it.

You should also be able to calculate the interquartile range, draw a box plot from five figures, estimate how many values fall above or below a given amount, and compare two distributions in the way examiners expect.

The organising idea is that cumulative frequency answers "how many so far?". Every feature of the graph follows from that question. The running total only ever goes up, which is why the curve rises from left to right and never falls. The total is reached at the right-hand end. And because the vertical axis counts values rather than measuring them, the median is found by going up the frequency axis to half the total and then across to the curve — not by looking at the middle of the horizontal scale.

Key terms and definitions

Cumulative frequency — a running total of the frequencies up to and including each class.

Upper class boundary — the top value of a class, and the only x-value at which a cumulative frequency point may be plotted.

Median (Q2) — the middle value, read at half the total frequency.

Lower quartile (Q1) — read at a quarter of the total; a quarter of the data lies below it.

Upper quartile (Q3) — read at three-quarters of the total.

Interquartile range (IQR) — Q3 − Q1, describing the spread of the middle half of the data.

Box plot — a diagram of five values: minimum, Q1, median, Q3 and maximum.

Outlier — a value far away from the rest of the data.

Core concepts

Building the cumulative frequency table

Add each class frequency to the running total.

If the frequencies are 4, 7, 12 and 5, the cumulative frequencies are 4, 11, 23 and 28. Each figure includes everything before it, so the final value is always the total number of data values.

That last point is a free check. If the final cumulative frequency does not match the total, an addition has gone wrong.

Plotting the graph

Plot each cumulative frequency against the upper class boundary of its class, then join the points with a smooth curve.

The upper boundary matters because the running total is only complete at the top of the class. For the class 10 < x ≤ 20 with a cumulative frequency of 23, the point is plotted at x = 20, not at the midpoint of 15. Plotting at midpoints is the commonest error here, and it shifts every reading taken afterwards.

The curve should rise from left to right, usually in a stretched S shape. A cumulative frequency graph can never go down, because a running total cannot decrease.

Reading the median and quartiles

Work from the vertical axis. Go up to the right fraction of the total frequency, across to the curve, and down to read the value.

The median is at half the total. The lower quartile is at a quarter. The upper quartile is at three-quarters.

For 40 values: the median is read at 20, the lower quartile at 10 and the upper quartile at 30.

Note that these fractions are of the total frequency, not of the horizontal scale. Reading the median as the middle of the x-axis range is a different and wrong calculation.

The interquartile range

IQR = Q3 − Q1.

It measures the spread of the middle half of the data, ignoring the extreme values at each end. That is exactly why it is useful: a single unusually large value changes the range enormously but leaves the IQR almost untouched.

If Q1 = 10 and Q3 = 25, the IQR is 15.

A smaller IQR means more consistent data, which is the phrase that earns the mark in a comparison question.

Reading "how many" questions

The graph also answers questions about how many values lie above or below a given amount.

To find how many values are under 30, go up from 30 to the curve and across to the cumulative frequency — that reading is the answer directly, since the curve is a running total.

To find how many are above 30, subtract that reading from the total. Forgetting the subtraction is a frequent slip, and the wording "more than" is the signal to make it.

Box plots

A box plot displays five numbers: the minimum, the lower quartile, the median, the upper quartile and the maximum.

The box stretches from Q1 to Q3, so its width is the IQR, and a line inside it marks the median. The whiskers reach out from the box to the minimum and the maximum.

A box plot does not show the mean, and it does not show how many data values there are. It shows position and spread only.

The median line is usually not in the centre of the box. When it sits closer to one end, the data is more tightly packed on that side.

Comparing two distributions

Comparison questions want two statements, one about average and one about spread, and each must be written in the context of the question.

Compare the medians to say which set is higher on average. Compare the IQRs to say which is more consistent.

"The medians are 52 and 47, so the first group scored higher on average, and the IQRs are 12 and 20, so the first group's scores were more consistent" earns both marks. Writing "the first box plot is further right" earns neither, because it describes the diagram instead of the data.

Worked examples

Example 1: Building and checking a table

The frequencies for four classes are 6, 9, 15 and 10. Find the cumulative frequencies and say where each point is plotted for classes 0 < x ≤ 10, 10 < x ≤ 20, 20 < x ≤ 30 and 30 < x ≤ 40.

Running totals: 6, then 6 + 9 = 15, then 15 + 15 = 30, then 30 + 10 = 40.

The points are plotted at the upper boundaries: (10, 6), (20, 15), (30, 30) and (40, 40).

The final cumulative frequency is 40, which matches the total of the frequencies. ✓

Example 2: Reading quartiles

A cumulative frequency graph has been drawn for 60 values. At what cumulative frequencies are the median and the quartiles read, and what is the IQR if Q1 = 14 and Q3 = 32?

The median is read at half of 60, so at 30. The lower quartile is read at a quarter, so at 15, and the upper quartile at three-quarters, so at 45.

The interquartile range is 32 − 14 = 18.

Example 3: A "more than" question

Eighty students sat a test. Reading up from a mark of 55 gives a cumulative frequency of 62. How many students scored more than 55?

The reading of 62 is the number who scored 55 or less, because the curve is a running total.

The number scoring more than 55 is therefore 80 − 62 = 18.

Giving 62 as the answer is the standard error, and re-reading the question for the words "more than" catches it.

Common mistakes and how to avoid them

Plotting at the midpoint. Cumulative frequency points go at the upper class boundary, because the running total is complete only there.

Reading the median off the horizontal axis. Start from half the total frequency on the vertical axis and work across.

Forgetting to subtract in "more than" questions. The curve gives "how many so far", so subtract from the total.

Confusing range with interquartile range. The range uses the two extremes; the IQR uses the quartiles.

Thinking a box plot shows the mean. It shows the median. It also says nothing about how many values there are.

Comparing diagrams rather than data. Say "the median mark was higher", not "the box is further along".

Expecting the median to sit centrally in the box. An off-centre median is normal and describes the shape of the distribution.

Exam technique for "Cumulative Frequency and Box Plots"

Write the total frequency down and then the three reading positions — half, a quarter and three-quarters of it — before touching the graph. That single line gets the readings right and often earns a method mark.

Draw the lines you read along, across from the vertical axis and down to the horizontal. Examiners award marks for a correct reading line even when the value read off is slightly out.

Give the IQR as a subtraction shown in full, so the two quartiles used are visible.

For comparisons, write exactly two sentences: one comparing medians, one comparing IQRs, both mentioning what the data is about.

Check the final cumulative frequency equals the total frequency before going any further.

Read the question for "more than", "less than" or "at least", and subtract from the total where needed.

Quick revision summary

Cumulative frequency is a running total, so it answers "how many so far?". It never decreases, and its final value equals the total number of data values.

Plot each cumulative frequency against the upper class boundary, and join the points with a smooth curve.

Read from the vertical axis: the median at half the total, the lower quartile at a quarter, the upper quartile at three-quarters. For 40 values those are 20, 10 and 30.

IQR = Q3 − Q1, the spread of the middle half, and a smaller IQR means more consistent data. Unlike the range, it is barely affected by one extreme value.

For "how many are more than", read up from the value to the curve and subtract that reading from the total.

A box plot shows five numbers — minimum, Q1, median, Q3, maximum — with the box spanning the IQR. It does not show the mean or the number of values.

To compare two distributions, give one statement about the medians and one about the IQRs, both in context.