Is the mean resistant to arbitrarily large outliers?

No; a single replacement can move it without bound.

Is the median completely unaffected by changing data?

No; its value can change when relevant ordered positions change.

Does a resistant method remove the need to investigate outliers?

No. It limits their effect on an estimate, not their importance.

AI Glossary / Statistics and probability / Intermediate

Resistance

Definition

Resistance is the degree to which a statistical estimate is insensitive to a small number of unusual or substantially changed observations. A resistant summary does not move excessively when a small portion of the data becomes extreme. The median is a familiar resistant measure of location, whereas the arithmetic mean can be moved arbitrarily far by changing a single observation sufficiently.

What it means

Resistance concerns how an estimator responds to contamination or unusual data values. It is a property of the calculation in relation to specified changes, not a claim that a dataset contains no errors.
The broader word robustness can refer to stability under violations of statistical assumptions, including distributional changes and efficiency across models. Usage overlaps, but a careful explanation states which kind of stability is intended.

How it works

Identify the quantity being estimated and the changes that are relevant. For a location summary, one might replace a small proportion of observations with very large values and examine how the estimate changes.
Compare the mechanisms of candidate summaries. The arithmetic mean uses every magnitude directly, so an extreme value contributes proportionally to its size. The median depends on the middle ordered positions; moving an already extreme value farther outward often leaves those positions unchanged.
Resistance does not mean complete invariance. Changing enough observations, or changing observations near the middle, can move a median. Different resistant procedures also have different tuning choices and behavior under contamination.
Breakdown point is one formal way to study resistance: roughly, it asks how much contamination can drive an estimate arbitrarily far under a specified setup. Influence functions describe local sensitivity to small contamination in a population model. These are related but distinct analytical tools.
For spread, the range and standard deviation can be strongly affected by extremes. Measures based on central quantiles or median absolute deviations can be more resistant. Scaling constants and target distributions must be stated when interpreting them as estimates of a particular population scale.
Investigate unusual observations rather than automatically deleting them. An extreme value may be a recording error, a valid rare event, a separate subgroup, or evidence of a changing process. A resistant summary can stabilize one estimate while the unusual cases still deserve operational attention.
Consider efficiency and the estimand. A median estimates a distribution's median, not generally its mean. Choosing a resistant summary can change the question, particularly in skewed distributions where the mean and median differ substantially.
Report sensitivity analyses where useful: compare a conventional summary, a resistant summary, and the reasons for their difference. Document any trimming, weighting, or tuning so the analysis can be reproduced.

Three practical examples

AI monitoring: a dashboard uses median latency to summarize a typical request while separately reporting extreme delays.
Measurement analysis: an engineer compares resistant location estimates when a few readings may be contaminated.
Customer analytics: analysts inspect whether a handful of unusually large transactions dominate an average.

Think of it this way

Think of the mean as a balance point pulled by both the position and weight of every observation. The median is the middle position in an ordered line. Moving one person already at the far end much farther away can shift the balance point without changing who stands in the middle.

Worked example

Consider values 8, 9, 10, 11, and 12. Their mean is 50/5 = 10 and their median is 10. The absolute deviations from the median are 2, 1, 0, 1, and 2, so the unscaled median absolute deviation is 1.
Replace only the largest value, 12, with 112. The new values are 8, 9, 10, 11, and 112. Their mean becomes 150/5 = 30, while the median remains 10. The unscaled median absolute deviation remains 1 because the ordered absolute deviations are 0, 1, 1, 2, and 102.
The one-observation increase of 100 raises the mean by 100/5 = 20. Making that observation still larger would keep moving the mean, while leaving this particular median unchanged.
This does not establish that 112 is erroneous or irrelevant. If the target is total workload or average cost, its contribution may be essential. The example demonstrates sensitivity, not a rule that the median is always the right summary.

Illustrative case study

Fictional case: Beech Service Analytics notices that average request time jumps after a few unusually long sessions. The team reports both the median and the upper tail, traces the long sessions to a real processing bottleneck, and fixes that issue. The resistant median helps describe typical use without hiding the valid extreme experiences from the operations team.

Common misconceptions and mistakes

Do not call a statistic resistant merely because it worked well on one dataset. Resistance is about its response to specified perturbations.
Do not equate resistance with immunity to bias, poor sampling, dependence, or broad data corruption. A resistant estimate can still be misleading for the target population.
Avoid automatically replacing a mean with a median when the business question explicitly concerns the mean or total.
A stable headline can hide important tail behavior. Report the tails when rare extremes carry substantial consequences.

Business use

Resistance is useful when a small number of unusual records could destabilize descriptive summaries or model fitting. Organizations benefit from separating data-quality investigation, typical-case reporting, and tail-risk reporting. The estimator should match the decision rather than simply produce the most reassuring number.

Check your understanding

Why does the example's mean rise by 20?

Answer: The total increases by 100 across five observations.

Why does the median stay at 10?

Answer: The third ordered value remains 10 after the largest observation is changed.

Related terms

Robust statistics
Median
Outlier
trimmed mean
median absolute deviation
breakdown point
influence function
efficiency

Learn next

Learn next how resistance differs from efficiency under a model, how breakdown point and influence quantify different sensitivities, and how robust regression handles unusual response values and leverage points.

Robust regression

Sources

Sources: Howell, Statistical Methods for Psychology glossary, https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html ; NIST Dataplot, Biweight Midvariance, https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/biwmidv.htm ; NIST/SEMATECH, Detection of Outliers, https://itl.nist.gov/div898/handbook/eda/section3/eda35h.htm . Primary references explain resistance and outlier accommodation. The perturbation example and fictional case are original.

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