Hypothesis Tests

Shapiro-Wilk Test

Tests whether a sample comes from a normally distributed population.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > Normality Test (Shapiro-Wilk)

Description

The Shapiro-Wilk test is one of the most powerful normality tests, especially for small samples (n = 3–50). It compares the ordered sample values with the expected values from a normal distribution. The test statistic W ranges from 0 to 1 — values close to 1 indicate normality. If p < α (e.g. 0.05), the null hypothesis of normality is rejected.

Formulas

W = \frac{( \sum _{i = 1}^{n} a _{i} x _{(i)} ) ^{2}}{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}

Test statistic W: ratio of the weighted sum of ordered values to the total sum of squares

a = \frac{m ^{T} V ^{- 1}}{∥ m ^{T} V ^{- 1} ∥}

Weight vector a, computed from the expected normal order statistics m and the covariance matrix V

H₀

H_{0} : X_{1}, \dots, X_{n} \sim N (μ, σ^{2})

Null hypothesis: The sample comes from a normal distribution

Assumptions

Data are independent and identically distributed (i.i.d.)
Sample size: 3 ≤ n ≤ 5000 (optimal for n ≤ 50)
Continuous data

Limitations

For large samples (n > 5000), even small deviations from normality can become significant
Not suitable for discrete data or data with many ties
The p-value approximation can be inaccurate for very small n (n < 7)

References

Shapiro, S. S. & Wilk, M. B. (1965), An analysis of variance test for normality (complete samples), Biometrika, 52(3–4), 591–611
Royston, P. (1992), Approximating the Shapiro-Wilk W-test for non-normality, Statistics and Computing, 2, 117–119
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.7