Resistant Statistic vs Median

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.

Sort the values from smallest to largest and the median is whatever sits at position $\frac{n+1}{2}$ (n plus 1, over 2), where $n$ is how many values there are. When $n$ is odd that position lands on an actual observation. When $n$ is even it lands between two of them and the median is their average, so the median of an even-sized set need not be a number that appears in the data at all. It is commonly written $M$.

Take the seven quiz scores 12, 15, 15, 18, 22, 24, 91. They are already sorted and $n = 7$, so the median is the value in position 4: $M = 18$. The mean of the same seven scores is 28.14. Now change the 91 to 27. The median is still 18, while the mean drops to 19. That is resistance: the median only counts how many values sit on each side of it and ignores how far away they are, so moving one extreme value cannot shift it by more than one position.

With an even count the averaging step kicks in. For 2, 6, 8, 10 the median is $(6 + 8) / 2 = 7$, a number that appears nowhere in the list.

The most common error is reporting the position instead of the value. For those seven scores the answer is 18, not 4; position 4 only tells you where to look. Two related slips are taking the middle of the list before sorting it, and averaging the smallest and largest values, which gives the midrange rather than the median.

The median is also the 50th percentile and the center of the five-number summary. The quartiles are the medians of the halves on either side of it, with the median itself left out of each half under the TI-84 convention this site uses: for the seven scores, 12, 15, 15 gives $Q_1 = 15$ and 22, 24, 91 gives $Q_3 = 24$. Comparing the two centers is also a quick read on shape: here the mean of 28.14 sits far above the median of 18, which usually signals either a right skew or a single high value dragging the mean up. In this set it is the second of those, since the 1.5 IQR rule puts the upper fence at $24 + 1.5(9) = 37.5$ and flags 91 as an outlier.