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 () divides the data set into two parts, the upper set and the lower set. The lower quartile () is the median of the lower half, and the upper quartile () is the median of the upper half.
Find , , and for the following data set, and draw a box-and-whisker plot.
There are data points. The middle two are and . So the median, , is .
The "lower half" of the data set is the set . The median here is . So .
The "upper half" of the data set is the set . The median here is . So .
A box-and-whisker plot displays the values , , and , along with the extreme values of the data set ( and , in this case):
A box & whisker plot shows a "box" with left edge at , right edge at , the "middle" of the box at (the median) and the maximum and minimum as "whiskers".
Note that the plot divides the data into equal parts. The left whisker represents the bottom of the data, the left half of the box represents the second , the right half of the box represents the third , and the right whisker represents the top .
If a data value is very far away from the quartiles (either much less than or much greater than ), 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 or greater than by more than times the interquartile range (). That is, an outlier is any number less than or greater than .
Find , , and for the following data set. Identify any outliers, and draw a box-and-whisker plot.
There are values, arranged in increasing order. So, is the data point, .
is the data point, , and is the data point, .
The interquartile range is or .
Now we need to find whether there are values less than or greater than .
Since is less than and and are greater than , there are outliers.
The box-and-whisker plot is as shown. Note that and are shown as the ends of the whiskers, with the outliers plotted separately.