Box Plots: Reading the Five-Number Summary

The five-number summary

Five values describe a whole data set’s position and spread:

Value	Meaning
minimum	the smallest value
$Q_1$	a quarter of the data lies below here
median	half lies below here
$Q_3$	three quarters lie below here
maximum	the largest value

A box plot draws all five on a number line.

Take these seven scores: $4, 7, 9, 12, 14, 15, 21$.

$$\text{min } 4 \quad Q_1 = 7 \quad \text{median } 12 \quad Q_3 = 15 \quad \text{max } 21$$

What each part means

The box runs from $Q_1$ to $Q_3$. Its width is the interquartile range:

$$\text{IQR} = 15 - 7 = 8$$

Half of all the data sits inside that box.

The line inside the box is the median, not the mean. It is drawn heaviest because where it falls is the first thing worth looking at.

The whiskers stretch from the box out to the smallest and largest values.

Every section holds a quarter of the data

This is the part that reads wrongly at first glance.

Section	Share of the data
left whisker	$25\%$
left half of box	$25\%$
right half of box	$25\%$
right whisker	$25\%$

All four hold the same number of values, however different their widths look.

So a long whisker does not mean more data out there — it means those values are spread out. A short section means that quarter of the data is packed tightly.

Why the shape of the plot tells you about skew

If the data were symmetric, the median would sit in the middle of the box and the whiskers would match. When it does not, the plot is telling you something.

The median sits left of centre and the right whisker is long. Both say the same thing: the upper values are more spread out than the lower ones, so the distribution is skewed right.

Reading it in quarters makes the reason plain. The bottom quarter is squeezed into a narrow band, and the top quarter is stretched across a wide one — same number of values, very different room.

That is worth knowing because skew decides which average to report. In a right-skewed set the mean is pulled up above the median, so the median is the honest summary — and a box plot shows you the skew before you have calculated anything.

Comparing two groups

Box plots are at their most useful in pairs, because five numbers each is a fair comparison that fits in one picture.

Two comparisons come straight off the pictures:

• Class B scored higher. Its median sits to the right of Class A’s.
• Class B was more consistent. Its box is narrower, so its middle half is packed into a smaller range.

Both plots share one axis, which is what makes the comparison fair. Two box plots drawn on different scales cannot be compared by eye at all.

What a box plot cannot tell you

It shows five numbers, so anything not among them is gone:

• How many observations there are. Twenty values and two thousand can draw the identical plot.
• The individual values. You cannot recover the data from the picture.
• Gaps or a second peak. A set clustered at both ends with nothing in the middle can look exactly like an evenly spread one.

For those, a dot plot or histogram is the right display.

Worked examples

Common mistakes

Practice problems