Robustness vs Resistant Statistic

Robustness is a property of a procedure, and always robustness against a named violation. Every interval and test comes with conditions, real data never satisfy them exactly, and robustness answers how much that costs. "The $t$ test is robust" is an incomplete sentence: robust against mild non-normality, yes; against a sample that was not random, not at all.

The word can be measured rather than asserted. A nominal 95% interval promises to capture the parameter in 95% of repeated samples, so break a condition, simulate, and count. Drawing from a strongly right-skewed population, one-sample t-intervals actually cover about 88% at $n = 5$, about 93% at $n = 30$, and about 94% at $n = 100$ across 200,000 samples. Strong skew costs something at every size, and the cost shrinks as $n$ grows. For a test the same check reads the true false-alarm rate against $\alpha$ (alpha).

"My data are robust" is the sentence to catch. Data are never robust; procedures are. Watch the neighboring word too. In wider statistical usage, resistant describes a statistic whose value barely moves when a few observations sit far out, which is why the median is resistant and the mean is not, while robust describes a procedure that still performs as advertised when a condition holds only approximately. AP topic 1.7 does not draw that line: it treats the two as interchangeable labels for measures of center and variability, calling the median and IQR resistant or robust and the mean, range, and standard deviation nonresistant or non-robust. Read each word from what the sentence is about.

Robustness has a hard edge at bias. Keep a perfectly normal population and sample by a method that never reaches its bottom 20%: coverage of a nominal 95% interval falls from about 80% at $n = 10$ to about 0% at $n = 100$. It gets worse as the sample grows, because more data pin down the wrong number more tightly. Bias moves the center, and no sample size repairs that. Robustness is also never a licence to skip a condition check, which is scored separately.

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.