The Lilliefors test (Conover 1999, pages 443-447) tests the following null and alternative hypotheses for data sets up to \(n\) = 1000:
\(H_0\): The data have been drawn from a normal distribution
\(H_a\): The data are drawn from a non-normal distribution
The test is conducted as follows.
Compute the sample mean $$\bar{x} = \frac{1}{n}\displaystyle\sum_{i=1}^{n}x_i$$
and the sample standard deviation $$s = \sqrt{\frac{1}{n-1}\displaystyle\sum_{i=1}^{n}(x_i-\bar{x})^2}$$
Compute the normalized sample values \(Z_i\), as $$Z_i = \frac{x_i-\bar{x}}{s}$$ \(i\) = 1, 2,..., \(n\) (Equation 1.0)
Compute the Lilliefors test statistic \(T\) as follows $$T = \text{sup}|F^*(x)-S(x)|$$
where
\(T\) is the supremum, over all x, of the absolute value of the difference \(F^*(x)-S(x)\), and
\(F^*(x)\) is the cumulative distribution function of a normal distribution with mean zero and standard deviation one, and
\(S(x)\) is the empirical distribution function of the values of \(Z_i\) computed using Equation 1.0 above.
Reject \(H_0\)and accept \(H_a\) at the \(\alpha\) significance level if \(T\) exceeds the critical value for the test, which can be obtained from Table A14 (Conover 1999, page 548).
Test provided courtesy of ProUCL.
Conover, W.J. 1999. Practical Nonparametric Statistics, 3rd edition, Wiley, NY.
Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring, Wiley, NY.
ProUCL. 2004. ProUCL Version 3.0 User Guide. Available for download from http://www.epa.gov/nerlesd1/tsc/tsc.htm