Sn and Qn Robust Scale Estimators
Also known as: Sn estimator, Qn estimator, Rousseeuw-Croux scale estimators, robust scale estimation, Sn ve Qn Ölçek Tahmincileri
Sn and Qn are robust estimators of scale (spread) proposed by Rousseeuw and Croux (1993) as alternatives to the median absolute deviation (MAD). Both attain a 50% breakdown point while delivering higher statistical efficiency than MAD, so they measure dispersion accurately even when the data contain outliers.
Key highlights
- Attain a 50% breakdown point, so up to half the data can be contaminated before the estimate breaks down.
- Much higher statistical efficiency than MAD (Qn reaches about 82% asymptotic efficiency).
- Sn does not require a symmetric reference point, unlike MAD which is built around the median.
Intuition
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How it works
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When to use it
Use Sn or Qn when you need a robust measure of spread for continuous data and want better efficiency than MAD without sacrificing resistance to outliers. They suit symmetric or mildly asymmetric distributions and work best with at least about 20 observations. They are a strong default whenever a few contaminating points could distort the classical standard deviation.
Strengths & limitations
- Attain a 50% breakdown point, so up to half the data can be contaminated before the estimate breaks down.
- Much higher statistical efficiency than MAD (Qn reaches about 82% asymptotic efficiency).
- Sn does not require a symmetric reference point, unlike MAD which is built around the median.
- Become unstable in very small samples; with fewer than 20 observations MAD is preferable.
- With fewer than 10 observations scale estimation is unreliable and a permutation-based approach is safer.
- Estimate dispersion only — they do not by themselves provide a full model of the data.
Common pitfalls
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Applications
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Frequently asked
How are Sn and Qn different from MAD?
MAD measures deviations from the median, which assumes rough symmetry and has only about 37% efficiency at a Gaussian model. Sn and Qn instead use pairwise differences between observations, keep the same 50% breakdown point, and reach much higher efficiency — Qn around 82%.
What is a breakdown point?
The breakdown point is the fraction of contaminated observations an estimator can tolerate before producing arbitrary results. Sn and Qn both attain the maximum possible 50%, meaning nearly half the data can be outliers before the scale estimate is ruined.
When should I prefer Qn over Sn?
Qn generally has higher efficiency and does not even need an explicit centre, so it is often the better default. Sn remains a sound choice and is conceptually simple; both are far more efficient than MAD.
What sample size do I need?
At least about 20 observations for stable estimates. Below 20, MAD is preferable, and below 10 a permutation-based approach is safer because scale estimation becomes unreliable.
Sources
- 1.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). Sn and Qn Scale Estimators. ScholarGate. https://scholargate.app/statistics/sn-qn-estimators