What do they measure?For each of the following summary statistics, decide (i) whether it could be used to measure center or variability and (ii) whether it is resistant.

$$\begin{gathered} \left(a\right)\frac{Q_1+Q_3}{2} \\ \left(b\right)\frac{\mathrm{Max}-\mathrm{Min}}{2} \end{gathered}$$

The given information is:

$\frac{Q_1+Q_3}{2}\mathrm{and}\frac{\mathrm{Max}-\mathrm{Min}}{2}$

(i)

The average of the first two and third quartiles are:

$\frac{Q_1+Q_3}{2}$

This variable would be equal to the median in a symmetric distribution because the median is located exactly in the middle of the first and third quartiles.

The variable $\frac{Q_1+Q_3}{2}$is likewise a measure of centre because the median is a measure of centre.

ii)

First quartile and third quartile are resistant to outliers, thus the variable $\frac{Q_1+Q_3}{2}$will be as well.

$\frac{\mathrm{Max}-\mathrm{Min}}{2}$

Because the range is the difference between the maximum and minimum, the given variable is half the range.

The given variable is likewise a measure of spread because the range is a measure of spread.

Outliers are not tolerated by the range, therefore the given variable is not resistant.