What this answers

This calculator answers "taking everything I can measure about this sample's shape together, how much evidence is there against a normal distribution?" combining skewness, kurtosis, and one or two formal tests into a single, deterministic evidence statement.

How it is calculated

Skewness and excess kurtosis describe the sample's shape directly. The Shapiro-Wilk test always runs; the Anderson-Darling test also runs when the sample has at least 8 values, since its correction factor is calibrated for that range and larger. This calculator then combines the results into one of three fixed evidence statements: a formal test finding evidence against normality, no formal test finding evidence but skewness or kurtosis still notably large, or no evidence and modest shape statistics. None of the three ever claims normality is proven, only how much evidence was or was not found against it.

Worked example

For a right-skewed sample like 2, 4, 5, 7, 8, 9, 12, 15, 18, and 25, skewness comes out clearly positive, and depending on exactly how strong the formal tests' evidence is at this sample size, the summary either flags a formal departure or at least notes the skewness is large enough to be a caution, never a clean pass.

Assumption audit

Calculated from your data: skewness, excess kurtosis, the Shapiro-Wilk statistic and p-value, the Anderson-Darling statistic and p-value when n is at least 8, and the combined evidence summary built deterministically from all of them.

Evidence to review: whether the different pieces of evidence agree with each other; when a formal test and the raw skewness or kurtosis point in different directions, that disagreement itself is informative about how much to trust either signal at your specific sample size.

You must verify: that your sample was drawn independently, and that whatever analysis comes next genuinely needs a normal-distribution assumption in the first place; many common methods are reasonably robust to mild departures, especially with larger samples.

What this result does not mean

No summary this calculator produces ever means normality is proven; the best possible result only means no evidence against it was found given the tests and sample size available, a meaningfully weaker claim, especially with small samples where every included test has limited power to detect a real departure.

Limitations

This calculator requires at least 4 values to combine skewness, kurtosis, and Shapiro-Wilk meaningfully, and the Anderson-Darling test is omitted entirely below 8 values. Treat this as a starting summary, not a replacement for looking at a histogram or Q-Q plot directly.