Construct and interpret box plots

Detailed Explanation: Construct and interpret box plots

A box plot shows a data set using five values:

• Minimum
• Lower quartile, $Q_1$
• Median
• Upper quartile, $Q_3$
• Maximum

The box shows the middle $50\%$ of the data, and the lines extending from the box are called whiskers.

Worked example

Construct and interpret a box plot for:

The data is already in order.

1. Find the median

There are $8$ values, so the median is the average of the middle two values:

2. Find the lower quartile

The lower half is:

Its median is:

3. Find the upper quartile

The upper half is:

Its median is:

4. Record the five-number summary

5. Draw the box plot

Draw a labelled, evenly scaled number line. Mark $5$, $7.5$, $9.5$, $13$, and $20$.

• Draw a box from $7.5$ to $13$.
• Draw a line inside the box at $9.5$ for the median.
• Draw whiskers from $7.5$ to $5$ and from $13$ to $20.

The box plot should have this general structure:

The positions must be placed according to the scale. For example, the distance from $13$ to $20$ should be longer than the distance from $5$ to $7.5$ because $7$ is greater than $2.5$.

Interpret the box plot

The range is the distance from the minimum to the maximum:

The interquartile range is the spread of the middle half of the data:

The median is $9.5$, so a typical centre of the data is about $9.5$. The middle half of the values lies from $7.5$ to $13$.

The right whisker, from $13$ to $20$, is longer than the left whisker, from $5$ to $7.5$. This suggests the data may be skewed to the right. However, the box plot does not show the individual data values, so the original list cannot generally be recovered from it.

Do not call $20$ an outlier unless a particular outlier rule has been given. Using the common $1.5\times\text{IQR}$ rule, the upper fence is

so $20$ would not be identified as an outlier by that rule.