Numerical Measures of Central Tendency and Variability

The interquartile range (IQR) is the range of the middle \(\mathrm{50\%}\) of the scores in a distribution. It is computed as follows:

For Quiz 1, the \(\mathrm{75th}\) percentile is \(\mathrm{8}\) and the \(\mathrm{25th}\) percentile is \(\mathrm{6}\). The interquartile range is therefore \(\mathrm{2}\). For Quiz 2, which has greater spread, the \(\mathrm{75th}\) percentile is \(\mathrm{9}\), the \(\mathrm{25th}\) percentile is \(\mathrm{5}\), and the interquartile range is \(\mathrm{4}\). Recall that in the discussion of box plots, the \(\mathrm{75th}\) percentile was called the upper hinge and the \(\mathrm{25th}\) percentile was called the lower hinge. Using this terminology, the interquartile range is referred to as the H-spread.

A related measure of variability is called the semi-interquartile range. The semi-interquartile range is defined simply as the interquartile range divided by \(\mathrm{2}\). If a distribution is symmetric, the median plus or minus the semi-interquartile range contains half the scores in the distribution.