The Shapiro-Wilk Test: A Formal Verdict on Normality

Q-Q plots and normal distribution both mention the Shapiro-Wilk test as the standard formal companion to a visual normality check, without showing how the test statistic is actually computed. The Shapiro-Wilk test produces a single number, W, that quantifies how closely a sample's shape matches what a normal distribution would predict — the numeric counterpart to the Q-Q plot's visual impression, and the test most commonly recommended specifically because of its strong statistical power relative to other normality tests.

The Formula

The denominator is simply the sum of squared deviations from the mean — the same quantity used to compute variance. The numerator is a weighted combination of the sorted data values, using coefficients (aᵢ) that come from published statistical tables specific to the sample size n — not something calculated from the data itself, similar to how a t-test or chi-square test looks up a critical value from a table rather than deriving it fresh each time. W ranges from 0 to 1, with values close to 1 indicating the data closely resembles a normal distribution.

Worked Example

Problem: Using the same 6 exam scores from the Q-Q plot worked example — 62, 68, 71, 75, 78, 95 — compute the Shapiro-Wilk W statistic by hand.

Sorted data: 62, 68, 71, 75, 78, 95. The mean is 74.83, and the sum of squared deviations from the mean is 642.83 — this is the denominator.

For n=6, published Shapiro-Wilk coefficients are a₆=0.6431, a₅=0.2806, a₄=0.0875 (with a₁=−a₆, a₂=−a₅, a₃=−a₄ by symmetry). The numerator combines the data in symmetric pairs — smallest with largest, second-smallest with second-largest, and so on:

b = 0.6431×(95−62) + 0.2806×(78−68) + 0.0875×(75−71) = 0.6431×33 + 0.2806×10 + 0.0875×4 ≈ 24.38

W = b² / 642.83 = 24.38² / 642.83 = 594.4 / 642.83 ≈ 0.9245

With n=6, this W value corresponds to a p-value of approximately 0.538 — well above the conventional 0.05 threshold, so the test fails to reject the null hypothesis of normality. Despite the visibly high value of 95 in the Q-Q plot pulling one point off the diagonal line, the formal test doesn't flag this small sample as significantly non-normal.

The same conclusion is visible without a p-value at all: the Shapiro-Wilk critical values table gives roughly 0.79 as the critical W for n=6 at α=0.05, and the computed W=0.9245 clears it comfortably.

Why the Test Doesn't Flag the Outlier

This result is genuinely worth sitting with, since it might seem to contradict the Q-Q plot's visual impression. With only 6 observations, the Shapiro-Wilk test has very limited statistical power — the ability to detect a real departure from normality when one exists. A single somewhat-high value in a sample this small simply isn't enough evidence, on its own, to statistically distinguish "genuinely non-normal" from "a normal distribution that happened to produce one higher value by chance." This is exactly why a Q-Q plot and a formal test are meant to be used together, not as competing verdicts: the Q-Q plot shows a pattern worth noting even when the formal test can't yet confirm it statistically, and the formal test provides the quantified caution that a visual impression alone can overstate, especially with a small sample.

Interpreting the Result

A small p-value (below the chosen α, typically 0.05) leads to rejecting the null hypothesis of normality — the data shows a statistically significant departure from a normal distribution. A large p-value, as in the worked example above, means there isn't sufficient evidence to conclude the data is non-normal — importantly, this is not the same as proving the data is normal, a distinction that runs through every hypothesis test on this site: failing to reject a null hypothesis is never the same as confirming it's true.

The Test's Real Limitation: Sample Size Sensitivity

The Shapiro-Wilk test's power scales directly with sample size, in both directions that matter. With a small sample (as in the worked example), even a real, meaningful departure from normality can go undetected. With a very large sample, the test can flag statistically significant — but practically trivial — departures from perfect normality, since real data almost never follows a distribution exactly, and a large enough sample gives the test the power to detect even tiny, inconsequential deviations. This is why the test's p-value should be interpreted alongside a visual check (the Q-Q plot, or a histogram) rather than treated as a standalone, automatic pass/fail gate.

Real-World Uses

Validating assumptions before a parametric test: as covered in exploratory data analysis, checking normality before running a t-test, ANOVA, or linear regression is standard practice — the Shapiro-Wilk test is the most commonly recommended formal check for this purpose across many fields.

Quality control and process validation: manufacturing processes are often assumed to produce output with normally distributed measurements around a target value — the Shapiro-Wilk test is a standard tool for confirming that assumption before applying control-chart methods or other normality-dependent quality techniques.

Financial and scientific data screening: researchers routinely run a formal normality test as a first-pass screening step on a new dataset, flagging series that may need a transformation (like a log transform) or a non-parametric method before further analysis proceeds.

Common Mistakes

Treating a non-significant result as proof of normality. As explained above, failing to reject the null hypothesis only means the data doesn't provide strong enough evidence against normality — it's a genuinely different claim from confirming normality holds.

Ignoring sample size when interpreting the result. A p-value from a test on 6 observations and a p-value from a test on 6,000 observations carry very different statistical power behind them — always consider sample size alongside the raw p-value, not just the p-value in isolation.