Math Core

A box plot (also called a box-and-whisker plot) shows the center and spread of a data set in one simple picture. It doesn't show every value like a dot plot does. Instead it shows five key numbers, which makes it great for comparing groups.

The five-number summary

A box plot is built from five values you already know how to find.

Definition

Five-number summary

The five-number summary of a data set is

the minimum,
the first quartile, $Q_1$ ,
the median,
the third quartile, $Q_3$ ,
the maximum.

These five numbers split the data into four parts, and each part holds about one quarter (25%) of the values.

Drawing a box plot

How to draw a box plot

Find the five-number summary.
Draw a number line that covers the minimum to the maximum.
Draw a box from $Q_1$ to $Q_3$ . Draw a line inside the box at the median.
Draw whiskers: a line from the box out to the minimum, and a line from the box out to the maximum.

Worked example: Plant heights

A class measured 11 bean plants, in centimeters:

4, \ 6, \ 7, \ 7, \ 9, \ 10, \ 12, \ 13, \ 14, \ 16, \ 19

Make a box plot.

Five-number summary. The data is already in order.

Minimum: 4. Maximum: 19.
Median: the 6th value, 10.
Lower half (leave out the median): $4, 6, 7, 7, 9$ , so $Q_1 = 7$ .
Upper half: $12, 13, 14, 16, 19$ , so $Q_3 = 14$ .

Draw it. The box goes from 7 to 14 with a line at 10. The whiskers reach out to 4 and 19.

Bean plant heights (cm)

Reading a box plot

Every section of a box plot holds about 25% of the data: the left whisker, the left part of the box, the right part of the box, and the right whisker. So:

The whole box holds the middle 50% of the data. Its length is the IQR.
The distance from the end of one whisker to the other is the range.
About 50% of the values are above the median and 50% are below it.

Worked example: Reading minutes

The box plot shows how many minutes a group of students spent reading on Saturday.

Minutes spent reading on Saturday

Find the median, the range and the IQR.
About what percent of the students read for more than 30 minutes?
About what percent read between 30 and 50 minutes?

Solutions.

The five-number summary is 20, 30, 45, 50, 80. The median is the line inside the box: 45 minutes. The range is $80 - 20 = 60$ minutes. The IQR is $50 - 30 = 20$ minutes.
30 is $Q_1$ . About 25% of the data is below $Q_1$ , so about $100\% - 25\% = 75\%$ is above it.
30 to 50 is the whole box, from $Q_1$ to $Q_3$ . That's about 50%.

Common mistake

A longer section of a box plot does not mean it has more data. In the reading plot, the right whisker (50 to 80) is much longer than the right part of the box (45 to 50), but each holds about 25% of the students. A long section means those values are more spread out, not that there are more of them.

Comparing two box plots

Box plots drawn above the same number line make it easy to compare two groups. Compare the medians to see which group is typically higher, and compare the IQRs to see which group varies more.

Worked example: Two classes, one quiz

The box plots show the scores of two classes on the same quiz.

Quiz scores

Center: Class B's median (80) is higher than Class A's (75), so Class B typically scored a bit higher.
Spread: Class A's IQR is $85 - 65 = 20$ . Class B's IQR is $85 - 75 = 10$ . Class B's scores were more consistent.
The top 25% of both classes scored 85 or more, since both have $Q_3 = 85$ .

Tip

If you can't tell the exact values from a box plot, read the five-number summary off the number line first and write it down. Then every question becomes a quick subtraction or a "25%, 50%, 75%" question.