Resistant statistic

A statistic is resistant when moving a few observations arbitrarily far barely changes its value. Resistance belongs to the statistic, not to the data: the same numbers can carry a resistant summary and a nonresistant one at once. The median, the quartiles, and the IQR are resistant. The mean $\bar{x}$ (x-bar), the standard deviation $s$, and the range are not.

Take 21, 23, 24, 26, 28, 30, 31, 33, 36. The mean is $252/9 = 28$, the median is the fifth value, also 28, and $s = 4.95$. Change the 36 to 96. The median stays at 28, $Q_1$ at 23.5 and $Q_3$ at 32, so the IQR is still 8.5, while the mean climbs to $312/9 = 34.67$ and $s$ to 23.33, almost five times over. Only the summaries that read magnitudes moved.

"The median is resistant, so it is the better measure of center" does not follow. Resistance is stability, not accuracy: a resistant summary buys it by refusing to look at how far the extreme values actually are. On a roughly symmetric distribution with no outliers, the mean uses more of the information and is what t procedures are built on. It is a reason to switch summaries when the shape calls for it, not a ranking.

Resistant is not a spare word for robust. Resistant describes a statistic: how much its value moves when a few observations are extreme. Robust describes a procedure: whether it still performs as advertised when a condition is only approximately met, the sense in which t procedures tolerate mild non-normality. AP does not keep them apart: topic 1.7 writes "a resistant (or robust) measure of center", so on the exam the two words point at one idea.

Resistance has a limit. A value already beyond a quartile can move further out without touching $Q_1$ or $Q_3$, but one pulled from inside a half does move it: in 1, 2, 3, 4, 5, 6, 7, 8 the third quartile is 6.5, and changing the 5 to 100 raises it to 7.5. Once more than a quarter of the data sits far out on one side, that quartile moves too.