Mathematics

What Is a Box Plot?

Every part of a box plot labelled, why a wide box means spread and never amount, and the two-sentence comparison that mark schemes actually award marks for.

Box plot

A box plot is a diagram of five values from a data set — the minimum, lower quartile, median, upper quartile and maximum — drawn against a single scale.

Also called: Box-and-whisker diagram, Box and whisker plot
Where students meet it: Higher tier GCSE Mathematics, usually Grade 10, once quartiles and cumulative frequency have been covered.

The short answer
A box plot draws five numbers on one scale: minimum, lower quartile, median, upper quartile and maximum. The box spans the middle half of the data and each whisker covers a quarter, so a wider section means more spread, never more data.

An example

Minimum 21, LQ 48, median 62, UQ 71, maximum 94

The left whisker runs from 21 to 48, the box from 48 to 71 with a line drawn at 62, and the right whisker from 71 to 94. The interquartile range is 71 − 48 = 23 and the range is 94 − 21 = 73. The right whisker is much longer than the left, so the top quarter of the marks is stretched out while the bottom quarter is comparatively bunched.

Five numbers, and a quarter of the data between each

Every mark on a box plot is one of five values, read against the scale underneath. Nothing else on the diagram carries information.

The five values divide the data into four quarters, and each quarter holds 25 per cent of the values however long or short its section looks. That is the whole idea, and it is what a box plot is for: not showing individual results, but showing where the data sits and how far it is spread, in a shape you can put directly above another one.

End of the left whisker — the minimum
Left edge of the box — the lower quartile, with a quarter of the data below it
Line inside the box — the median, with half the data on each side
Right edge of the box — the upper quartile, with a quarter of the data above it
End of the right whisker — the maximum, so the whole diagram spans the range

Width is spread, never amount

The commonest misreading is that a wide box means more data. It means the opposite of concentrated: the middle half of the values is stretched across a wider range of the scale. There are exactly as many values in a narrow box as in a wide one.

Read the shape and you get the skew for free. A long right whisker with a short left one says the top quarter of the values is strung out, which is a distribution skewed to the right. A median sitting off-centre inside its own box says the same thing about the middle half. And a box plot never tells you the sample size — two plots drawn side by side may come from groups of 20 and 2,000.

Comparing two box plots the way a mark scheme wants

The question is nearly always "compare the two distributions", and the marks split cleanly. One comparison of average, one comparison of spread, and both written in the context of the data rather than in the language of the diagram.

Say Class A has a median of 62 with an interquartile range of 23, and Class B a median of 55 with an interquartile range of 34. The answer is: Class A's median mark is higher, so on average Class A scored better; and Class A's interquartile range is smaller, so Class A's marks were more consistent. Writing "Class A's box is further to the right" describes the picture rather than the students and earns nothing.

Proportion of the data lying inside the box, whatever width the box happens to be: 50%Proportion of the data lying inside the box, whatever width the box happens to be

Common questions

What do the whiskers on a box plot show?

The bottom quarter and the top quarter of the data. The left whisker runs from the minimum to the lower quartile and the right whisker from the upper quartile to the maximum, so between them they cover half the values while the box covers the other half.

Can you find the mean from a box plot?

No. A box plot shows the median and the quartiles, none of which is the mean, and it does not show the individual values needed to calculate one. If a comparison question asks about averages, the median is the average you have — and it is the sensible one to use when the data is skewed.

Does a wider box mean more data?

No, it means more spread. The box always contains the middle 50 per cent of the values, so widening it stretches the same half of the data across a larger range. A box plot carries no information about the number of values it was drawn from.

What does the interquartile range look like on a box plot?

It is the width of the box: the upper quartile minus the lower quartile. Because it ignores the top and bottom quarters, it is unaffected by a single extreme value, which is why it is preferred to the range when comparing how consistent two sets of data are.

How do I compare two box plots?

Make exactly two statements, both in context. Compare the medians to say which group did better on average, and compare the interquartile ranges to say which was more consistent. Describing the position of the boxes on the page, rather than what that means for the data, does not score.

Sources

GCSE mathematics: subject content and assessment objectives — Department for Education

Last updated August 21, 2026