What this answers

This test answers "does my sample provide statistical evidence against the assumption that it was drawn from a normal distribution?" A high W close to 1 is consistent with normality; a low W, paired with a small p-value, is evidence against it.

How it is calculated

W compares a weighted linear combination of your sorted (order-statistic) values against the total sum of squared deviations from the mean. The weights come from the expected values of order statistics under a standard normal model, following Royston's published 1992 algorithm, the same computational method statistical software packages use. The p-value uses Royston's 1995 approximation, which transforms W to an approximately normal scale using formulas that differ slightly for small (n at or below 11) versus larger samples.

Worked example

For a small, evenly spaced sample like 1, 2, and 3 (n=3), the exact closed-form weight vector gives W equal to exactly 1, the maximum possible value, since with only 3 points the middle value drops out of the calculation entirely and the two extreme points are perfectly evenly spaced. Larger, more irregular samples produce W values below 1, with lower values indicating stronger departures from a normal shape.

Assumption audit

Calculated from your data: the W statistic itself and its approximate p-value, using the sample size to select the correct branch of Royston's formula.

Evidence to review: a visual check alongside this number, using the Histogram Maker; a single test statistic cannot show you the shape of a departure from normality the way a plot can.

You must verify: that your sample was drawn independently and that a normal model is actually the relevant question for whatever analysis comes next; many common statistical methods are reasonably robust to mild non-normality, especially with larger samples.

What this result does not mean

A non-significant Shapiro-Wilk result never proves your data is normal; it means the test did not detect enough evidence against normality at your sample size, which is a much weaker claim, especially with small samples where the test has limited power. A significant result with a very large sample can also flag trivially small, practically unimportant departures from normality, since power increases with n.

Limitations

This calculator implements Royston's algorithm using the Blom-based approximate expected order statistics rather than the full small-sample exact tabulated values some reference implementations use for very small n; results match published software closely across typical sample sizes but may differ in the last reported digit for very small n. Treat this as one piece of evidence to combine with a visual check, not a standalone proof either way.