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# Anderson-Darling test

The Anderson-Darling test is an empirical distribution function omnibus test for the composite hypothesis of normality.

Compared to Lilliefors test, Anderson-Darling test gives more weight to the tails of the distribution. It is generally considered to be one of the most powerful tests of normality, even on quite small samples.

The procedure to follow for the test is as follows

1. Arrange the data in rank order.
2. Estimate the mean () and standard deviation (sY) of the data (variable Y).
3. Standardize each of the values of Y by subtracting the mean and dividing by the standard deviation to give z-values.
(N.B. it is only necessary to do this if you only have access to tables of the standard normal distribution.)
4. Calculate the test statistic A, where A2 is given by:

#### Algebraically speaking -

 A2 =   −n − 1 Σ (2i-1) [lnF(Yi) + ln (1 −F(Yn+1-i)] n
where
• A is the (uncorrected) Anderson-Darling statistic,
• n is the number of observations,
• Σ is the sum from i=1 to n,
• i is the rank of each value, assuming no ties.
• F is the specified normal cumulative distribution function. If values have been standardized, it is the standard normal. For example, for a standardized value of Y of 1.4, F(1.5) = 0.0808

5. Adjust for estimating parameters from sample data:
 A2c = A2 (1  + 0.75 + 2.25 ) n n2

6. If A2c is greater than 0.752, then the null hypothesis that the data conform to a normal distribution is rejected at the 5% level.

Data can be tested against other theoretical distributions by using the appropriate cumulative distribution function. Critical values and software applications are available for the normal, lognormal, exponential, Weibull and logistic distributions.