StatReason

Anderson-Darling Normality Test

Test for evidence against a normal-distribution model with extra weight on the tails, where many real departures from normality (heavy tails, extreme outliers) actually occur.

At least 8 values are recommended for a reliable result.

What this answers

This test answers the same underlying question as Shapiro-Wilk, "is this sample consistent with a normal distribution?", but weights departures in the tails of the distribution more heavily than departures near the center, making it especially sensitive to occasional extreme values.

How it is calculated

The statistic compares your sorted, standardized values against the normal cumulative distribution function at every point, with more weight given to points near the extremes. Because the mean and standard deviation used to standardize your data are themselves estimated from the same sample, this calculator applies Stephens' (1974) small-sample correction factor to the raw statistic before computing a p-value, using the standard D'Agostino and Stephens (1986) piecewise approximation formula.

Worked example

The published critical value for a 5% significance level is a corrected statistic of about .752; plugging that value into the D'Agostino-Stephens formula gives a p-value of almost exactly .05, confirming the formula's calibration against the standard reference table. A roughly evenly spaced sample, like the default example above, produces a small statistic and a large p-value, consistent with no detected evidence against normality.

Assumption audit

Calculated from your data: the corrected A* statistic and its approximate p-value, using your sample's own estimated mean and standard deviation for standardization.
Evidence to review: whether any single extreme value is driving the result; because this test weights tail departures heavily, one outlier can dominate the statistic even when the bulk of the data looks reasonably normal.
You must verify: that your sample was drawn independently, and that at least 8 observations are available, since the correction factor and p-value approximation are calibrated for that range and larger.

What this result does not mean

A non-significant result does not prove normality, only that this particular test, which is especially attentive to tail behavior, did not detect enough evidence against it. A significant result flags a departure, most often in the tails given how this test weights deviations, but does not by itself say the departure is too large to matter for whatever analysis comes next.

Limitations

This calculator requires at least 8 values; below that, both the correction factor and p-value approximation lose their calibration. For a second opinion using a different weighting of evidence across the whole distribution rather than emphasizing the tails, see the Shapiro-Wilk Normality Test.