Shapiro–Wilk normality test

Tests whether a sample is consistent with a normal population by measuring how well the ordered data line up with expected normal order statistics.

W = \frac{\left(\sum_{i=1}^{n} a_i\,x_{(i)}\right)^2}{\sum_{i=1}^{n}(x_i - \xbar)^2}

W is at most 1; values well below 1 indicate non-normality. p-values from Royston's (1995) approximation, 3 ≤ n ≤ 5000.

$x_{(i)}$: The i-th smallest observation.
$a_i$: Coefficients derived from the expected values of normal order statistics.

When to use

As supporting evidence, together with a Q–Q plot, when a method assumes normality (small-sample t procedures, residuals of a regression or ANOVA). H₀: the population is normal.

Assumptions

Independent observations; continuous data (many tied values lower W).

How to interpret

Small p: evidence against normality — look at the Q–Q plot to see how (skew, heavy tails, outliers) and whether it matters for your method.
Large p: the sample is consistent with normality, which is not proof of normality; small samples rarely reject anything.
In large samples, trivial departures give tiny p-values while the CLT makes t procedures robust anyway.

Common mistakes

Testing the raw outcome instead of the residuals (ANOVA/regression) or the differences (paired t).
Treating p > 0.05 as "the data are normal".
Switching method mechanically on the test result instead of judging the size and impact of the departure.

Example

Ten values 2.1, 3.4, 1.9, 5.6, 2.8, 3.1, 4.0, 2.5, 3.3, 2.9: $W = 0.899$ , $p = 0.21$ . No strong evidence against normality, but with n = 10 the test has little power: inspect a Q–Q plot, where the single large value 5.6 hints at right skew.