## Numerical Measures of Central Tendency and Variability

Read these sections and complete the questions at the end of each section. First, we will define central tendency and introduce mean, median, and mode. We will then elaborate on median and mean and discusses their strengths and weaknesses in measuring central tendency. Finally, we'll address variability, range, interquartile range, variance, and the standard deviation.

### Measures of Variability

#### Interquartile Range

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

$\mathrm{IQR} = \mathrm{75th} \, \; \text{percentile} - \mathrm{25th} \; \text{percentile}$

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.