📊

Shapiro-Wilk Test Calculator

Q: Does a high p-value prove the data are normal?

No. With small samples the test has little power; it only fails to find evidence of non-normality. Always look at a histogram or Q-Q plot too.

Q: Why is my large sample “not normal”?

With hundreds or thousands of values the test detects tiny, harmless departures. For large samples rely more on plots and on how robust your planned method is.

Q: Shapiro–Wilk or Kolmogorov–Smirnov?

Shapiro–Wilk is considerably more powerful for testing normality. The KS test with estimated mean and SD is miscalibrated unless the Lilliefors correction is used.

Shapiro–Wilk normality test for 3 to 5,000 values — W statistic and p-value (Royston algorithm, same as R and SciPy), with mean, SD, skewness and a verdict.

Data

Significance level

Loading calculator…

How it works

W = (Σ aᵢ·x₍ₙ₊₁₋ᵢ₎ − aᵢ·x₍ᵢ₎)² ÷ Σ(xᵢ − x̄)², with the coefficients aᵢ from normal order statistics (Royston 1992 approximation); p from a normalising transformation of ln(1 − W) (Royston 1995, AS R94)

The Shapiro–Wilk test checks whether a sample could come from a normal distribution and is the most powerful of the common normality tests for small and medium samples. W compares the ordered data with the values expected from a normal distribution: W close to 1 means a good fit. A p-value below α rejects normality. The coefficients and p-value follow Royston’s algorithm AS R94, the one used by R’s shapiro.test and SciPy’s stats.shapiro.

Worked example

Shapiro and Wilk’s eleven body weights (148 … 236 lb) give W = 0.7888 and p = 0.0067 — the heavy value of 236 makes the data depart significantly from normality.

Sources: Shapiro & Wilk (1965), An analysis of variance test for normality, Biometrika · Royston (1995), Algorithm AS R94, Applied Statistics · SciPy — scipy.stats.shapiro

Frequently asked questions

How do I read the p-value?

p ≥ α: no evidence against normality, so normal-theory methods are reasonable. p < α: the data are probably not normal — consider a transformation or a non-parametric test.

Does a high p-value prove the data are normal?

No. With small samples the test has little power; it only fails to find evidence of non-normality. Always look at a histogram or Q-Q plot too.

Why is my large sample “not normal”?

With hundreds or thousands of values the test detects tiny, harmless departures. For large samples rely more on plots and on how robust your planned method is.

Shapiro–Wilk or Kolmogorov–Smirnov?

Shapiro–Wilk is considerably more powerful for testing normality. The KS test with estimated mean and SD is miscalibrated unless the Lilliefors correction is used.

Related calculators

Kolmogorov-Smirnov Test Calculator

Two-sample Kolmogorov–Smirnov test — the D statistic (largest gap between the two empirical CDFs), where it occurs, the exact or asymptotic p-value and the critical D.

Coefficient of Skewness Calculator

Measure how lopsided your data is — the moment coefficient of skewness plus Pearson’s median-based skewness from a list of values.

Kurtosis Calculator

Find the kurtosis and excess kurtosis of your data — how heavy the tails are compared with a normal distribution — from a list of values.

Normal Distribution Calculator

Find the probability that a normally distributed value falls below or above a point — with its z-score and probability density.

Educational tool: check the test’s assumptions (sample size, independence, distribution) before relying on a result for a real decision.