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Box-and-Whisker Plots

To understand box-and-whisker plots, you have to understand medians and quartiles of a data set.

The median is the middle number of a set of data, or the average of the two middle numbers (if there are an even number of data points).

The median ( $Q_{2}$ ) divides the data set into two parts, the upper set and the lower set. The lower quartile ( $Q_{1}$ ) is the median of the lower half, and the upper quartile ( $Q_{3}$ ) is the median of the upper half.

Example:

Find $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ for the following data set, and draw a box-and-whisker plot.

${2, 6, 7, 8, 8, 11, 12, 13, 14, 15, 22, 23}$

There are $12$ data points. The middle two are $11$ and $12$ . So the median, $Q_{2}$ , is $11.5$ .

The "lower half" of the data set is the set ${2, 6, 7, 8, 8, 11}$ . The median here is $7.5$ . So $Q_{1} = 7.5$ .

The "upper half" of the data set is the set ${12, 13, 14, 15, 22, 23}$ . The median here is $14.5$ . So $Q_{3} = 14.5$ .

A box-and-whisker plot displays the values $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ , along with the extreme values of the data set ( $2$ and $23$ , in this case):

A box & whisker plot shows a "box" with left edge at $Q_{1}$ , right edge at $Q_{3}$ , the "middle" of the box at $Q_{2}$ (the median) and the maximum and minimum as "whiskers".

Note that the plot divides the data into $4$ equal parts. The left whisker represents the bottom $25 %$ of the data, the left half of the box represents the second $25 %$ , the right half of the box represents the third $25 %$ , and the right whisker represents the top $25 %$ .

Outliers

If a data value is very far away from the quartiles (either much less than $Q_{1}$ or much greater than $Q_{3}$ ), it is sometimes designated an outlier . Instead of being shown using the whiskers of the box-and-whisker plot, outliers are usually shown as separately plotted points.

The standard definition for an outlier is a number which is less than $Q_{1}$ or greater than $Q_{3}$ by more than $1.5$ times the interquartile range ( $IQR = Q_{3} - Q_{1}$ ). That is, an outlier is any number less than $Q_{1} - (1.5 \times IQR)$ or greater than $Q_{3} + (1.5 \times IQR)$ .

Example:

Find $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ for the following data set. Identify any outliers, and draw a box-and-whisker plot.

${5, 40, 42, 46, 48, 49, 50, 50, 52, 53, 55, 56, 58, 75, 102}$

There are $15$ values, arranged in increasing order. So, $Q_{2}$ is the $8^{th}$ data point, $50$ .

$Q_{1}$ is the $4^{th}$ data point, $46$ , and $Q_{3}$ is the $12^{th}$ data point, $56$ .

The interquartile range $IQR$ is $Q_{3} - Q_{1}$ or $56 - 47 = 10$ .

Now we need to find whether there are values less than $Q_{1} - (1.5 \times IQR)$ or greater than $Q_{3} + (1.5 \times IQR)$ .

$Q_{1} - (1.5 \times IQR) = 46 - 15 = 31$

$Q_{3} + (1.5 \times IQR) = 56 + 15 = 71$

Since $5$ is less than $31$ and $75$ and $102$ are greater than $71$ , there are $3$ outliers.

The box-and-whisker plot is as shown. Note that $40$ and $58$ are shown as the ends of the whiskers, with the outliers plotted separately.