Anderson-Darling Normality Test
The Anderson-Darling test for normality is one of three general normality tests designed to detect all departures from normality. While it is sometimes touted as the most powerful test, no one test is best against all alternatives and the other 2 tests are of comparable power. The p-values given by Distribution Analyzer for this test may differ slightly from those given in other software packages as they have been corrected to be accurate to 3 significant digits.
The test rejects the hypothesis of normality when the p-value is less than or equal to 0.05. Failing the normality test allows you to state with 95% confidence the data does not fit the normal distribution. Passing the normality test only allows you to state no significant departure from normality was found.
The Anderson-Darling test, while having excellent theoretical properties, has a serious flaw when applied to real world data. The Anderson-Darling test is severely affected by ties in the data due to poor precision. When a significant number of ties exist, the Anderson-Darling will frequently reject the data as non-normal, regardless of how well the data fits the normal distribution. Below is an example of data generated from the normal distribution but rounded to the nearest 0.5 to create ties. A tie is when identical values occurs more than once in the data set:
Both the Shapiro-Wilks test (p-value = 0.1311) and Skewness-Kurtosis All test (p-value = 0.9930) pass this set of data. The Shapiro-Wilks test is also affected by ties, but not nearly as bad as the Anderson-Darling test. The Skewness-Kurtosis All test is not affected by ties and thus the default test.