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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.

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 $\left\{2,6,7,8,8,11\right\}$ . The median here is $7.5$ . So $Q_1=7.5$ .

The "upper half" of the data set is the set $\left\{12,13,14,15,22,23\right\}$ . 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 ( $\text{IQR}=Q_3-Q_1$ ). That is, an outlier is any number less than $Q_1-\left(1.5\times \text{IQR}\right)$ or greater than $Q_3+\left(1.5\times \text{IQR}\right)$ .

There are $15$ values, arranged in increasing order. So, $Q_2$ is the $8^{\text{th}}$ data point, $50$ .

$Q_1$ is the $4^{\text{th}}$ data point, $46$ , and $Q_3$ is the ${12}^{\text{th}}$ data point, $56$ .

The interquartile range $\text{IQR}$ is $Q_3-Q_1$ or $56-47=10$ .

Now we need to find whether there are values less than $Q_1-\left(1.5\times \text{IQR}\right)$ or greater than $Q_3+\left(1.5\times \text{IQR}\right)$ .

$Q_1-\left(1.5\times \text{IQR}\right)=46-15=31$

$Q_3+\left(1.5\times \text{IQR}\right)=56+15=71$

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.