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Re: Determining if samples are normal


From: Paul Kienzle
Subject: Re: Determining if samples are normal
Date: Mon, 26 Sep 2005 03:55:52 -0400

The Anderson-Darling test is claimed to be pretty good:

        http://www.itl.nist.gov/div898/handbook/prc/section2/prc213.htm
        http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm

Here's the relevant section from the R manual:

        http://www.maths.lth.se/help/R/.R/library/nortest/html/ad.test.html

The Anderson-Darling test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is

A^2 = -n -frac{1}{n} sum_{i=1}^{n} [2i-1] [ln(p_{(i)}) + ln(1 - p_{(n-i+1)})],

where p_{(i)} = Phi([x_{(i)} - overline{x}]/s). Here, Phi is the cumulative distribution function of the standard normal distribution, and overline{x} and s are mean and standard deviation of the data values. The p-value is computed from the modified statistic

        Z=A^2 (1.0 + 0.75/n +2.25/n^{2})

according to Table 4.9 in Stephens (1986).

Here are the critical values I found elsewhere on the net.

        90%     0.631
        95%     0.752
        97.5%   0.873
        99%     1.035

For example, if A^2 > 0.752 you can say that your data set is not normally distributed with 95% confidence.

This is implemented in octave-forge as 1-p = anderson_darling_test(x,'normal'). That is, if anderson_darling_test returns a value of 0.05 then you can say your data set is not normally distributed with 95% confidence.

- Paul

On Sep 25, 2005, at 1:59 PM, Søren Hauberg wrote:

Hi,
Does anybody know how I can test wether or not some samples are normaly distributed? I tried graphical methods, such as looking at histograms and qqplots, but I don't trust my own judgement enough to use graphical methods.

/Søren




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