Box plots & range

Two classes take the same test and end up with the same mean. Still the results can look completely different: in one class almost all the scores sit close together, in the other they run from very weak to very strong. A single mean does not reveal that. The box plot, in contrast, shows in one small picture where the middle lies and how far the values spread apart.

The range

The simplest measure of spread is the range. You compute it as the largest value minus the smallest value. If the values reach from 4 to 19, the range is 19 − 4 = 15. It tells you how wide a span the data cover in total, and it carries the same unit as the data themselves, points or degrees for example.

Median and quartiles

The median splits the sorted data into two halves. The quartiles split each half once more: the lower quartile is the median of the lower half, the upper quartile the median of the upper half. With an odd number of values the median itself stays out of this, it belongs to neither half. Between the two quartiles lie the middle 50 percent of all values.

The box plot

The box plot brings exactly these numbers together as a picture. The box reaches from the lower to the upper quartile, a line inside it marks the median. From the box two lines run out, the whiskers, up to the smallest and the largest value. Minimum, lower quartile, median, upper quartile and maximum are the five key values of a box plot. From them you read off the middle, the middle half and the range at a glance.

Reading the spread

The width of the box reveals the spread of the middle values. A wide box means the middle 50 percent lie far apart. A narrow box means the values crowd around the median. Two data sets with the same median can thus have quite different boxes.