Robust Location and Scale Estimates

Robust location and scale estimates, such as trimmed and winsorized means, Huber M-estimates, the Hodges–Lehmann estimator, MAD, IQR, Sn and Qn, summarise centre and spread while limiting outlier influence.

로버스트 위치·척도 추정은 절사·윈저화 평균, Huber M 추정, Hodges–Lehmann 추정, MAD, IQR, Sn, Qn 처럼 이상값의 영향을 줄여 중심과 퍼짐을 요약하는 방법이다.

My data have outliers; which measures of centre and spread should I report instead of the mean and SD?

자료에 이상값이 있을 때 평균과 표준편차 대신 무엇을 보고해야 하나요?

Robust estimators down-weight or cut extreme values so that a few outliers cannot move the estimate far, for example the 20% trimmed mean or the median absolute deviation. Comparing them with the mean and SD shows how much outliers or skewness drive the ordinary summaries.

When to use Robust Location and Scale Estimates

• The data contain outliers or heavy tails
• You want summaries that are stable under contamination
• You need a location estimate with a distribution-free confidence interval

When to use something else

• The data are clean and normal; the mean and SD are the most efficient summaries
• You need to compare groups; use Yuen's trimmed-means test or a rank test instead

Assumptions and requirements

• Observations are independent
• For the Hodges–Lehmann interval, the distribution is symmetric
• Normal-consistency constants assume an approximately normal core
• Hodges-Lehmann (n ≤ 3,000) and Sn/Qn (n ≤ 5,000) are computed only for moderate samples

What the analysis reports

• Mean, median, trimmed and winsorized means
• Huber M-estimate of location and scale
• Hodges–Lehmann estimate with confidence interval
• MAD, IQR, Sn and Qn scale estimates

How to report it

The 20% trimmed mean was [value] (median = [value], MAD = [value]); the Hodges–Lehmann estimate was [value], 95% CI [[value], [value]].

Related methods

• Bootstrap Confidence Intervals (BCa)
• Wilcoxon Signed Rank Test
• Yuen's Trimmed Means t Test

Run Robust Location and Scale Estimates on Quorant