(note that the data must be put in order) comes from a distribution with cumulative distribution function (CDF) is:
where
:
The test statistic can then be compared against the critical values of the theoretical distribution (dependent on which is used) to determine the P-value.
The Anderson-Darling test for normality is a distance or empirical distribution function (EDF) test. It is based upon the concept that when given a hypothesized underlying distribution, the data can be transformed to a uniform distribution. The transformed sample data can be then tested for uniformity with a distance test (Shapiro 1980).
In comparisons of power, Stephens (1974) found to be one of the best EDF statistics for detecting most departures from normality. [cite journal
first = M. A. | last = Stephens | authorlink = | coauthors =
year = 1974 | month =
title = EDF Statistics for Goodness of Fit and Some Comparisons
journal = Journal of the American Statistical Association
volume = 69 | issue = | pages = 730–737 | id = | url =
doi = 10.2307/2286009 ] The only statistic close was the Cramér-von Mises test statistic.
Procedure
(If testing for normal distribution of the variable "X")
1) The data , for , of the variable that should be tested is sorted from low to high.
2) The mean and standard deviation are calculated from the sample of .
3) The values are standardized as
::
4) With the standard normal CDF , is calculated using ::
or without repeating indices as
::
5) , an approximate adjustment for sample size, is calculated using
::
6) If exceeds 0.752 then the hypothesis of normality is rejected for a 5% level test.
Note:
1. If "s" = 0 or any (0 or 1) then cannot be calculated and is undefined.
2. Above, it was assumed that the variable was being tested for normal distribution. Any other theoretical distribution can be assumed by using its CDF. Each theoretical distribution has its own critical values, and some examples are: lognormal, exponential, Weibull, extreme value type I and logistic distribution.
3. Null hypothesis follows the true distribution (in this case, N(0, 1)).
ee also
*Kolmogorov-Smirnov test
*Shapiro-Wilk test
*Smirnov-Cramér-von-Mises test
*Jarque-Bera test
External links
* [http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm US NIST Handbook of Statistics]
* [http://www.analyse-it.com/blog/2008/8/testing-the-assumption-of-normality.aspx Testing the assumption of normality] .
References