Describe › Normality

Anderson–Darling test

Anderson–Darling tests normality with more weight on the tails than Kolmogorov–Smirnov, making it better at catching heavy tails and outliers.

What is Anderson–Darling test?

Anderson–Darling is an EDF-based test like Kolmogorov–Smirnov, but the test statistic integrates the squared CDF gap weighted by 1 / (F(x)·(1 − F(x))). That weight blows up at the tails, so A² is much more sensitive to outliers and heavy tails than the supremum-based K–S statistic.

Among the modern normality tests, A² and Shapiro–Wilk are usually the top two in power studies. A² tends to win when the alternative is heavy tails (Cauchy, mixture distributions); Shapiro–Wilk tends to win against shape departures (skewness, kurtosis). The two together give you a complete picture.

When should I use Anderson–Darling test?

When you suspect heavy tails or a few extreme observations and want a normality test that pays attention to them.
Final normality check before bootstrap CIs or any procedure whose validity hinges on tail behaviour.

What data does it need?

One numeric column.

What does it report?

A² statistic and p-value. Small p rejects normality.

What does it assume?

Independent observations.
Continuous data.

Formula

A² = −n − (1/n) · Σᵢ (2i − 1)·[log Φ(zᵢ) + log(1 − Φ(zₙ₊₁₋ᵢ))]

How do I interpret the result?

A² > 0.752 (after small-sample adjustment) corresponds roughly to p < 0.05 for a Normal alternative — but rely on the reported p, not the raw statistic. If A² is significant but the Q-Q plot looks acceptable apart from one or two extreme values, ROUT or Grubbs on the suspected outliers may be a better next step than transforming the whole variable.

References

Anderson & Darling (1954). A test of goodness of fit. JASA 49(268).