Hypothesis Tests

Wilcoxon Signed-Rank Test

Nonparametric alternative to the one-sample t-test for the median.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Nonparametrics > 1-Sample Wilcoxon

Description

The Wilcoxon signed-rank test is a nonparametric test that evaluates whether the population median equals a hypothesized value. It is based on the ranks of absolute differences and is used when the normality assumption of the t-test is not met. For large samples, a normal approximation is used for the p-value.

Formulas

W⁺

W^{+} = i : d_{i} > 0 \sum R_{i}

Sum of ranks of positive differences

z = \frac{W - μ _{W}}{σ _{W}}, μ_{W} = \frac{n ( n + 1 )}{4}, σ_{W} = \frac{n ( n + 1 ) ( 2 n + 1 )}{24}

z approximation for large samples

H₀

H_{0} : \tilde{μ} = μ_{0}

Null hypothesis: The population median equals the target value

Assumptions

Data are continuous and symmetric about the median
Observations are independent

Limitations

Less powerful than the t-test for truly normal data
For very small samples (n < 10), the z approximation may be inaccurate

References

Wilcoxon, F. (1945), Individual comparisons by ranking methods, Biometrics Bulletin, 1(6), 80–83
Hollander, M. & Wolfe, D. A. (2013), Nonparametric Statistical Methods, 3rd Ed.