Median Absolute Deviation (MAD) Estimation
Also known as: median absolute deviation, MAD scale estimator, robust scale estimation, Medyan Mutlak Sapma (MAD) Tahmini
Median Absolute Deviation estimation is a robust measure of statistical dispersion that replaces the standard deviation when outliers are present. Rooted in the influence-curve framework formalised by Hampel (1974), it summarises the spread of a continuous variable using medians instead of means, so a single extreme value cannot distort the result.
Key highlights
- Highly robust: a single outlier cannot distort it, with a breakdown point of 50% (half the data can be contaminated before the estimate fails).
- Makes no symmetry or normality assumption about the distribution.
- Simple to compute and, after scaling by 1.4826, directly comparable to the standard deviation.
Intuition
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How it works
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When to use it
Use MAD estimation when you need a measure of spread for a continuous variable and you suspect outliers or heavy tails that would inflate the standard deviation. It needs at least about 10 observations to be stable and does not assume a symmetric distribution. It is the right tool when robustness matters more than maximal efficiency under perfect normality; for very small samples (fewer than 5-10 observations) a permutation or bootstrap approach is preferable.
Strengths & limitations
- Highly robust: a single outlier cannot distort it, with a breakdown point of 50% (half the data can be contaminated before the estimate fails).
- Makes no symmetry or normality assumption about the distribution.
- Simple to compute and, after scaling by 1.4826, directly comparable to the standard deviation.
- Becomes unstable in very small samples (fewer than about 10 observations).
- With fewer than 5 observations the median absolute deviation is essentially meaningless.
- Less statistically efficient than the standard deviation when the data really are clean and normal, and as a single deviation it discards directional information.
Common pitfalls
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Applications
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Frequently asked
Why multiply by 1.4826?
The constant 1.4826 (the reciprocal of the 0.75 quantile of the standard normal) rescales the raw MAD so that, for normally distributed data, it estimates the same quantity as the standard deviation. This lets you read MAD on the familiar σ scale.
What does a 50% breakdown point mean?
It means up to half of the observations can be arbitrarily corrupted before the estimate can be driven to a nonsensical value. This is the highest breakdown point achievable, which is why MAD is so resistant to outliers.
How is MAD different from the standard deviation?
The standard deviation averages squared distances from the mean, so a single extreme value can inflate it dramatically. MAD uses medians of absolute distances, so outliers have almost no effect, at the cost of some efficiency when the data are genuinely normal.
When should I use Sn or Qn instead of MAD?
Rousseeuw and Croux proposed the Sn and Qn estimators as more efficient robust scale measures that also reach a 50% breakdown point. They are worth preferring when you want higher efficiency and have at least about 20 observations.
Sources
- 1.Hampel, F. R. (1974). The Influence Curve and Its Role in Robust Estimation. Journal of the American Statistical Association, 69(346), 383-393.
- 2.Rousseeuw, P. J. & Croux, C. (1993). Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association, 88(424), 1273-1283.
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Cite this page
ScholarGate. (2026, June 1). MAD Estimation. ScholarGate. https://scholargate.app/statistics/mad-estimation