Read the output in four steps
- Identify the test and exactly what was estimated from the data.
- Confirm whether raw observations or model residuals were tested.
- Read the statistic and p-value against the stated null.
- Inspect a QQ plot and assess whether the visible departure matters for the planned analysis.
Never translate p > .05 into “the data are normal.” Failure to reject can reflect low power, especially in small samples.
The tests emphasize different discrepancies
| Test | What the statistic compares | Important caution |
|---|---|---|
| Shapiro–Wilk | Ordered observations against expected normal order statistics | Designed specifically for normality; sample-size implementation limits vary. |
| Kolmogorov–Smirnov | Largest distance between empirical and specified cumulative distributions | Standard reference values assume the comparison distribution is fully specified. |
| Lilliefors-type K-S | K-S adjustment when normal parameters are estimated from the sample | Do not read it as the unadjusted K-S test. |
| Anderson–Darling | Weighted cumulative-distribution discrepancy with extra tail weight | Critical values and p-values depend on the fitted distribution and implementation. |
R’s shapiro.test documentation reports W and p and documents its supported sample-size range.2 Official ks.test documentation warns about continuity, ties, and the distinction between one- and two-sample forms.3
Sample size changes the answer
With large n, a test can detect small, practically harmless deviations. With small n, serious skew or tail behavior may remain undetected. Ghasemi and Zahediasl therefore recommend combining formal tests with visual methods rather than allowing either to stand alone.4
That is why the QQ plot usually carries more diagnostic information than a binary p-value. Skewness and kurtosis describe aspects of shape, and heteroscedasticity diagnostics address a different assumption. Provenance labels keep software-specific decisions separate from universal methodological rules.
Test the assumption the model actually makes
Many regression procedures assume something about conditional errors, not the marginal distribution of every variable. Testing the raw outcome can answer the wrong question. Independence, mean structure, unequal variance, and influential cases can matter more than modest nonnormality.
If the plot shows a departure, do not automatically transform or abandon the analysis. Identify the feature, assess its influence on the target estimate, and use a method or sensitivity analysis suited to the design.
A defensible reporting sentence
“Shapiro–Wilk gave W = .97, p = .03 for the model residuals. The QQ plot showed that the departure was concentrated in the upper tail, so inference was checked with a method robust to that feature rather than treating the p-value as a complete model diagnosis.”