Sample Median Confidence Interval (Sign-Test / Order Statistic)
Distribution-free confidence interval for the median from order statistics — no normality assumed.
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The engineering
This is the distribution-free confidence interval for the population median: you sort the data and pick two order statistics symmetric about the center. Because each observation is above or below the true median with probability ½ (the sign test), the coverage is a pure Binomial(n, ½) calculation — no normality, no variance estimate, no outlier leverage.
Coverage comes in discrete jumps because order statistics are discrete, so the 'actual coverage' rarely lands exactly on 95%; it is the largest achievable level that does not exceed your target. For small n the interval can be wide or even collapse — you need roughly n ≥ 6 before a 95% interval exists at all, and the interval tightens slowly (∝ 1/√n) like all nonparametric bounds.
Use it for QA lots, fatigue life, and any right-skewed reliability data where the mean lies to the analyst but the median tells the truth.
Where this math comes from
John W. Tukey and John E. Walsh worked the nonparametric median interval into practical form in the late 1940s, when statisticians were building tools that survived without the tidy Gaussian assumptions that laboratory data so rarely honor. Walsh's 1949 papers laid out how order statistics bracket the median with an exact, distribution-free confidence level tied directly to the binomial sign test.
The idea is older in spirit — the sign test dates to Arbuthnot's 1710 study of birth ratios — but Walsh made the confidence-interval version usable at the bench: sort, count in from each end, and read the binomial tail for the coverage you actually got.
- 1710John ArbuthnotFirst recorded sign test, comparing yearly counts of male vs. female births.
- 1945Frank WilcoxonRank-based nonparametric tests renew interest in distribution-free inference.
- 1949John E. WalshPublishes exact distribution-free confidence intervals for the median from order statistics.
- 1963Hodges & LehmannIntroduce the related estimator refining nonparametric location intervals.
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