25.1 Quartiles, interquartile range and box plots — Learn

Prerequisites: order data, find a median, and calculate a range.

Objectives: find quartiles and the interquartile range, draw a box plot, and interpret these measures.

A quartile divides ordered data into four equal groups. The lower quartile $Q_1$ is the median of the lower half. The upper quartile $Q_3$ is the median of the upper half. The interquartile range (IQR) measures the spread of the middle half: $\mathrm{IQR}=Q_3-Q_1$.

Example 1. Find the median, quartiles and IQR for the chapter travel times.

There are 20 values, so the median is the mean of positions 10 and 11:

The lower half has positions 1 to 10, so $Q_1=(17+18)/2=17.5$.

The upper half has positions 11 to 20, so $Q_3=(26+27)/2=26.5$.

Example 2. Draw a box plot from the five-number summary $12,17.5,21.5,26.5,42$.

Draw a number line using a consistent scale. Mark the minimum and maximum with whiskers, and draw a box from $Q_1$ to $Q_3$. Mark the median inside the box.

Predict: predict how the box changes if the upper quartile moves from $26.5$ to $30$ while the other values stay fixed. Write your prediction down first.

Test: use the control to test your prediction.

Explain: which part represents the middle half of the data? Write your explanation before reading on.

Watch out. Do not calculate the IQR as $Q_1-Q_3$. The upper quartile is larger, so subtract $Q_1$ from $Q_3$; check that the spread is positive.

Progress check

For the travel times, the middle half runs from 17.5 to 26.5 minutes. Does this match your first prediction of a narrow or wide interval? Give a reason.

You should now be able to

• find the median and quartiles of an ordered list;
• calculate and interpret the interquartile range;
• draw and read a box plot from a five-number summary.