# How to check if data is normally distributed

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Nancy on 7 Aug 2012
Answered: Sarutahiko on 11 Dec 2013
Hi all,
I want to run a f-test on two samples to see if their variances are independent. Wikipedia says that the f test is sensitive to non normality of sample (<http://en.wikipedia.org/wiki/F-test)>. How can I check if my samples are normally distributed or not.
I read some forums which said I can use kstest and lillietest. When can I use either? I get an answer h=0. Does that mean my data is normally distributed?
Thanks. Nancy

Sean on 7 Aug 2012
Hello Nancy,
You cannot tell from only 2 samples whether they are normally distributed or not. If you have a larger sample set and you are only testing them in pairs, then you could use the larger sample set to test for a particular distribution.
For example: (simple q-q plot)
data= randn(100); %generate random normally distributed 100x100 matrix
ref1= randn(100); %generate random normally distributed 100x100 matrix
ref2= rand(100); %generate random uniformly distributed 100x100 matrix
x=sort(data(:));
y1=sort(ref1(:));
y2=sort(ref2(:));
subplot(1,2,1); plot(x,y1);
subplot(1,2,2); plot(x,y2);
The first plot should be a straight line (indicating that the data distribution matches the reference distribution. The second plot isn't a straight line, indicating that the distributions do not match.
Nancy on 7 Aug 2012
The data samples you have given have equal sizes. What would I do if there are unequal sizes. I need to compare the variances across a lot of samples. I am wondering if there was a test like the t test for doing so. If I submit a report, I would just to write in the p values.

Sarutahiko on 11 Dec 2013
Assuming you agree with the Anderson-Darling test for Normality, I'd just use Matlab's prebuilt function for that. It is http://www.mathworks.com/help/stats/adtest.html

Tom Lane on 7 Aug 2012
The functions you mention return H=0 when a test cannot reject the hypothesis of a normal distribution. They can't prove that the distribution is normal, but they don't find much evidence against that hypothesis.
The VARTESTN function has an option that is robust to non-normal distributions.
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Tom Lane on 9 Aug 2012
Suppose you would normally do
x1 = randn(20,1); x2 = 1.5*randn(25,1);
[h,p] = vartest2(x1,x2)
Then you can do something like this instead:
grp = [ones(size(x1)); 2*ones(size(x2))];
vartestn([x1;x2], grp)
I believe the two-sample vartestn test is not identical to the vartest2 test, but the p-values are likely to be similar. Then you can add options to do a robust test using vartestn.