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%.

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 × IQR) or greater than Q_{3} +
(1.5 × 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_{1} 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 × IQR) or greater than Q_{3} +
(1.5 × IQR).

Q_{1} −
(1.5 × IQR) = 46 − 15 = 31

Q_{3} +
(1.5 × 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.