What is a resistant statistic? Median vs mean

By Jude Wallis · Updated July 31, 2026

A resistant statistic barely changes when one data value moves far out. The median, the quartiles, and the IQR are resistant. The mean, standard deviation, range, correlation, and least-squares regression line are not. Report the median and IQR for skewed data or data with outliers.

AP Statistics: Unit 1 (topics 1.6 Descriptions for One Quantitative Variable Distributions, 1.7 Summary Statistics for One Quantitative Variable, 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable). The Fall 2026 course names the median and IQR as resistant and the mean, range, and standard deviation as nonresistant in Unit 1 topic 1.7, and the idea carries across topics 1.6 to 1.8 on describing distributions, summary statistics, and boxplots.

What resistant means

A statistic is resistant (also called robust) when extreme values barely change its value, if they change it at all. Drag one data point far out to the right and a resistant statistic stays where it was, while a nonresistant one follows the point.

The Fall 2026 AP Statistics framework uses this exact language in Unit 1 topic 1.7. It calls the median a resistant measure of center and the interquartile range (IQR) a resistant measure of variability, because outliers do not greatly affect their values. It calls the mean a nonresistant measure of center, and the range and standard deviation nonresistant measures of variability.

Resistance describes only how sensitive a statistic is to a few unusual values. It is not a synonym for accurate, and it does not make a statistic better. A resistant summary buys its stability by throwing away information about how far the extreme values actually are, and sometimes that information is the point of the study.

Which statistics are resistant and which are not

Here is the split, with the symbol each one uses.

Statistic	Resistant?	Symbol	What it summarizes
Median	Yes	$M$	Center
First and third quartiles	Yes	$Q_1$ and $Q_3$	Position
Interquartile range	Yes	IQR, equal to $Q_3 - Q_1$	Variability
Mean	No	$\bar{x}$ , read "x-bar"	Center
Standard deviation	No	$s$ for a sample, $\sigma$ (sigma) for a population	Variability
Range	No	no standard symbol	Variability
Correlation	No	$r$	Strength of a linear association
Least-squares regression line	No	$\hat{y}$ , read "y-hat", the predicted response	Prediction

Two of them give the reason away in their definitions. The range is the maximum minus the minimum, built out of nothing but the two most extreme values, so it moves with every change to either end. The mean divides a total by $n$ , the number of data values, so every value votes, and it votes with its actual size.

Why the mean chases and the median holds

Every value owns an equal share of the mean, exactly $1/n$ of it. So if you change one value by an amount $d$ and leave the rest alone, the mean shifts by exactly $d/n$ . There is no change so large the mean can shrug it off; pick a big enough $d$ and you can drag the mean anywhere you like.

The median works on positions instead of sizes. It is whichever value sits in the middle of the sorted list, so it asks only how many values fall above and below, never how far away they are. Make the largest value ten times larger and the middle position has not changed, so the median has not either.

The quartiles behave the same way, because $Q_1$ and $Q_3$ are themselves medians, of the lower half and the upper half of the sorted data. That makes the IQR, $Q_3 - Q_1$ , resistant too. On this site quartiles follow the median-excluded rule that the TI-84 uses: when the number of values is odd, leave the median itself out of both halves before finding $Q_1$ and $Q_3$ .

The standard deviation is the opposite case, and it is worse off than the mean. It squares each value's distance from the mean before averaging, so a point sitting far out contributes its distance squared, and one outlier can dominate the entire calculation.

Which pair to report: median and IQR, or mean and SD

The practical rule is short. Report the median and the IQR when the data are skewed or contain outliers. Report the mean and the standard deviation when the distribution is roughly symmetric with no extreme values.

Keep the pairs together. A median reported next to a standard deviation is a mismatched summary, because you would be giving a center that ignores the outlier alongside a spread that the outlier controls. Center and spread should agree about how much weight the tails deserve.

Symmetric data are the mean's home ground for a second reason. Nearly all of the inference later in the course, confidence intervals and t-tests for means, is built on $\bar{x}$ and $s$ . Choosing the mean is not only a description choice; it is what keeps the rest of the toolkit available.

When the two summaries disagree sharply, the gap is itself information. A mean well above the median points to right skew or a high outlier, the pattern worked through in mean vs median. To confirm that a suspicious value really counts as an outlier, apply the 1.5 times IQR rule.

The five-number summary and boxplots are built from resistant pieces

The five-number summary is the minimum, $Q_1$ , the median, $Q_3$ , and the maximum. Three of those five are resistant, and they are precisely the three that form the box. The two that are not resistant, the minimum and the maximum, are the two pushed out to the ends.

That structure is why a boxplot survives an outlier so gracefully. The box spans the middle 50% of the data, from $Q_1$ to $Q_3$ , with a line at the median, and none of those three positions move when a far-out point moves farther out. Only a whisker changes. When the 1.5 times IQR rule flags the point, the whisker stops at the most extreme value that is not an outlier, and the outlier is plotted as its own mark.

So a boxplot shows you an outlier without letting it distort the picture of the center, which a histogram or a dotplot does not do automatically. The construction details are in how to make a boxplot, and you can watch resistance happen by dragging a point around in the descriptive statistics sandbox.

Resistance in regression: correlation and the least-squares line

The idea does not stop at one-variable summaries. The correlation coefficient $r$ , which summarizes the strength and direction of a linear association between two quantitative variables, is computed from z-scores, and a z-score is built from the mean and the standard deviation. Neither of those is resistant, so $r$ is not resistant either, and a single point far from the pattern can move it a long way.

The least-squares regression line inherits the same weakness for a sharper reason. The line is chosen to make the sum of the squared residuals as small as possible, and squaring means one point with a large residual counts for more than several points with small ones. A single unusual point can tilt the line noticeably, especially when its x-value sits far from the other x-values.

Be honest about the label, though. The Fall 2026 framework applies the words resistant and nonresistant only to the Unit 1 measures listed above, so treat the regression version as the same idea extended rather than as exam vocabulary. What the framework does ask, in topic 5.1, is that you describe unusual features of a scatterplot, including points that do not fit the general pattern. Dragging such a point in the regression influential point interactive shows the effect on the line immediately.

How the AP exam tests resistance

Resistance runs through Unit 1 topics 1.6 to 1.8, and Unit 1 carries 20% to 30% of the multiple-choice section. Topic 1.6 asks you to describe a distribution, including unusual features such as outliers, gaps, and clusters, in context. Topic 1.7 defines the summary statistics and contains the learning objective that asks you to justify the selection of a summary statistic for describing quantitative data (skill 4.B). Topic 1.8 covers the five-number summary and boxplots.

That justification objective is where resistance earns points. Naming the median is only half an answer. The other half is the reason: the distribution is skewed with an outlier, so the median resists that value while the mean chases it. Write the reason every time, not only the choice.

The same phrasing works when you compare two groups. If one group's distribution is skewed, compare medians and IQRs rather than means and standard deviations, and say why you switched. The topic reference is 1.7 Summary Statistics, and the official course framework is at AP Central.

Move one point far out: the median holds while the mean chases

Seven students report how many books they read over the summer: 3, 5, 6, 8, 9, 11, 14. Find the mean, median, quartiles, IQR, range, and sample standard deviation. Then one student's count is corrected from 14 to 63, and the data become 3, 5, 6, 8, 9, 11, 63. Recompute every summary and say which ones moved.

The data are already sorted: 3, 5, 6, 8, 9, 11, 14, and $n = 7$ , where $n$ is the number of data values.
Add them up: $3 + 5 = 8$ , $8 + 6 = 14$ , $14 + 8 = 22$ , $22 + 9 = 31$ , $31 + 11 = 42$ , $42 + 14 = 56$ .
Mean: $\bar{x} = 56 / 7 = 8$ books.
Median: with 7 values the middle position is the 4th, so the median is 8 books.
Quartiles, using the median-excluded (TI-84) rule: drop the median itself, leaving a lower half of 3, 5, 6 and an upper half of 9, 11, 14. The middle of the lower half is $Q_1 = 5$ , and the middle of the upper half is $Q_3 = 11$ .
Spread: IQR $= Q_3 - Q_1 = 11 - 5 = 6$ books, and range $= 14 - 3 = 11$ books.
Sample standard deviation: the deviations from the mean of 8 are -5, -3, -2, 0, 1, 3, 6. Their squares are 25, 9, 4, 0, 1, 9, 36, and $25 + 9 + 4 + 0 + 1 + 9 + 36 = 84$ .
Divide by $n - 1 = 6$ : $s^2 = 84 / 6 = 14$ , so $s = \sqrt{14} \approx 3.742$ books.
Now change the 14 to a 63. The sorted data are 3, 5, 6, 8, 9, 11, 63, still with $n = 7$ .
New sum: $56 - 14 + 63 = 105$ . New mean: $\bar{x} = 105 / 7 = 15$ books.
New median: the 4th value of 3, 5, 6, 8, 9, 11, 63 is still 8 books. The change happened above the middle position, so the middle position never moved.
New quartiles: the lower half is still 3, 5, 6, giving $Q_1 = 5$ , and the upper half is now 9, 11, 63, giving $Q_3 = 11$ . Both are unchanged, so IQR $= 11 - 5 = 6$ books, also unchanged.
New range: $63 - 3 = 60$ books, up from 11.
New standard deviation: the deviations from the mean of 15 are -12, -10, -9, -7, -6, -4, 48. Their squares are 144, 100, 81, 49, 36, 16, 2304, and $144 + 100 + 81 + 49 + 36 + 16 + 2304 = 2730$ .
So $s^2 = 2730 / 6 = 455$ and $s = \sqrt{455} \approx 21.33$ books.
Check the mean's mechanics: the one value rose by $63 - 14 = 49$ , and $49 / 7 = 7$ , which is exactly the jump from 8 to 15.
Check that 63 really is an outlier: $Q_3 + 1.5(\text{IQR}) = 11 + 1.5(6) = 11 + 9 = 20$ , and $63 > 20$ , so 63 is an outlier by the 1.5 times IQR rule.

Before: mean 8, median 8, $Q_1 = 5$ , $Q_3 = 11$ , IQR 6, range 11, $s \approx 3.742$ . After 14 becomes 63: mean 15, median 8, $Q_1 = 5$ , $Q_3 = 11$ , IQR 6, range 60, $s \approx 21.33$ . The median, both quartiles, and the IQR did not move at all. The mean rose by 7, the range rose by 49, and the standard deviation grew by a factor of about 5.7.

Pick the summary for a set with one huge value

Eight donations to a school fundraiser, in dollars, are 180, 195, 205, 210, 220, 235, 250, 905. Find the mean, the median, the quartiles, the IQR, and the range. Then decide which pair of summaries to report, and justify the choice.

The values are already sorted and $n = 8$ .
Add them up: $180 + 195 = 375$ , $375 + 205 = 580$ , $580 + 210 = 790$ , $790 + 220 = 1010$ , $1010 + 235 = 1245$ , $1245 + 250 = 1495$ , $1495 + 905 = 2400$ .
Mean: $\bar{x} = 2400 / 8 = 300$ dollars.
Median: with 8 values, average the 4th and 5th: $(210 + 220) / 2 = 430 / 2 = 215$ dollars.
Quartiles: with an even count there is no middle value to exclude, so the lower half is 180, 195, 205, 210 and the upper half is 220, 235, 250, 905. Then $Q_1 = (195 + 205)/2 = 400/2 = 200$ and $Q_3 = (235 + 250)/2 = 485/2 = 242.5$ dollars.
Spread: IQR $= 242.5 - 200 = 42.5$ dollars, and range $= 905 - 180 = 725$ dollars.
Outlier check: $1.5 \times 42.5 = 63.75$ , so the upper fence is $242.5 + 63.75 = 306.25$ . Since $905 > 306.25$ , the 905 dollar gift is an outlier.
Compare the two centers: the mean of 300 sits above every donation except the 905, while the median of 215 sits among the bulk of them. Compare the two spreads: the range of 725 is created almost entirely by that one gift, while the IQR of 42.5 describes the middle 50% of the donations.

Mean 300, median 215, $Q_1 = 200$ , $Q_3 = 242.5$ , IQR 42.5, range 725, all in dollars. Report the median of 215 and the IQR of 42.5, because 905 is an outlier by the 1.5 times IQR rule and the distribution is right-skewed, so the mean and the range are both distorted by that single donation.

Frequently asked questions

Is resistant the same thing as robust?

Yes. The AP Statistics course framework uses resistant and robust as synonyms, and nonresistant and non-robust likewise. Textbooks and software documentation lean toward robust, while AP questions and rubrics use resistant, so recognize both.

Is the median always the better summary?

No. For roughly symmetric data with no outliers, the mean uses more of the information in the data, and it is what the standard deviation, confidence intervals, and t-tests are built on. Resistance is a reason to prefer the median when the shape calls for it, not an argument that the median is generally superior.

Why is the standard deviation less resistant than the mean?

The mean adds each value once, so an outlier contributes in proportion to its distance. The standard deviation squares each distance from the mean before averaging, so a point twice as far out contributes four times as much. One extreme value can therefore dominate the standard deviation even when it only nudges the mean.

Can the IQR be affected by outliers at all?

Usually not much, and it depends where the value started. A value that already sits beyond a quartile can move as far out as you like without touching Q1 or Q3, because the middle positions of the two halves do not change. A value pulled out from inside a half is different: 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. So a single extreme value usually does not move Q1 or Q3, though it can nudge one. If more than about a quarter of the data sit far out on one side, the quartile on that side shifts in earnest, so resistance has limits.

Should I just delete outliers instead of using resistant statistics?

No. Investigate the value first, since it may be a recording error, a different population, or the most interesting observation in the data. Deleting it changes the data set you are describing, while reporting the median and IQR keeps every observation and still gives a summary the outlier cannot distort.