Sample Size for Testing One Population Median Using the Sign Test

Example clinical scenario

A hospital wants to test whether its median emergency-department waiting time differs from a 120-minute service benchmark using a one-sample sign test.

Formula and methodological note

The exact sign statistic is S = Σ I(Xᵢ > m₀), conditional on non-tied observations. Under the null, S follows Binomial(n, 0.5). The calculator enumerates each integer n and uses exact equal-tail rejection regions for two-sided tests.

The sign test is distribution-free under the null after conditioning on non-tied observations, but planning requires a specific alternative sign probability. Exact observations equal to the hypothesised median are excluded. The exact binomial test is conservative because attainable type I error is discrete.

For the Normal-shift option only, p₁ = Φ(Δ/SD) translates a planned median difference into a sign probability; the analysis remains an exact sign test. It is not a Wilcoxon signed-rank test and does not use ranks or distances.

Realised power may be lower when the observed exact-tie rate or non-response/incomplete-outcome rate exceeds the planning value. A sign probability close to 50% represents a small effect, so the required sample is sensitive to small changes in that assumption.

These tools support research planning and do not replace statistical consultation.

Method references

1. Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods. 3rd ed. Wiley; 2014.
2. Conover WJ. Practical Nonparametric Statistics. 3rd ed. Wiley; 1999.
3. Lehmann EL, D'Abrera HJM. Nonparametrics: Statistical Methods Based on Ranks. Springer; 2006.
4. Wan X, Wang W, Liu J, Tong T. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Medical Research Methodology. 2014;14:135. doi:10.1186/1471-2288-14-135.