Resistant Statistic vs Outlier

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.

An outlier is a value that stands apart from the rest of the data, and the AP course gives two rules for deciding when apart counts. The first flags any value more than $1.5 \times \text{IQR}$ above the third quartile $Q_3$ or below the first quartile $Q_1$. The second flags any value more than 2 standard deviations from the mean. Neither rule is the definition of an outlier. They are criteria, and they do not always agree.

Take 12, 13, 19, 19, 20, 23, 25, 27, 35, 40. The quartiles are $Q_1 = 19$ and $Q_3 = 27$, so the IQR is 8 and the upper fence sits at $27 + 1.5(8) = 39$. The value 40 clears it, so the IQR rule calls 40 an outlier. Now the other rule: the mean is 23.3 and $s = 8.91$, so two standard deviations reach $23.3 + 2(8.91) = 41.12$, and 40 does not get there. Same data, two rules, opposite answers. Say which rule you used.

"There is an outlier, so it must be a mistake and I should drop it." An outlier is a statement about position inside this one data set and nothing more. A 40 from a broken sensor is an error; a 40 from an unusually productive week is the most informative point you have. Removing a flagged value is a decision about the study that needs a reason you can write down, not an arithmetic step.

The 2 standard deviation rule quietly fails on small samples. No observation among $n$ values can sit more than $\frac{n-1}{\sqrt{n}}$ sample standard deviations from the mean, because the value doing the stretching also inflates $s$. With $n = 5$ that ceiling is 1.79, so the rule flags nothing at all however extreme one of the five values is. The 1.5 IQR rule has no such blind spot.

Outliers sit in topic 1.7 of the Fall 2026 course, which is where both rules are listed, and topic 1.6 names outliers, gaps, and clusters as the unusual features every description of a distribution has to mention in context.