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Nancy
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How to check if data is normally distributed

Asked by Nancy
on 7 Aug 2012
Latest activity Answered by 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

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3 Answers

Answer by 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

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Answer by 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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Thanks Sean. How can I use vartesn for a 2 sample variance test. The input is just X. My samples are of unequal sizes.
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.

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Answer by 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.

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Each sample has about 100 points. Would that be enough to test for normality? What can I use to check if their variances are independent.
The fewer points you have available, the less definitive the test is. If you run the previous set of sample code for a smaller set of data and reference points you should see what I mean. (e.g. The shape of the lines, is less well defined and more affected by random noise with a smaller sample set.)
Regarding a test for independence... you might try scatter plotting them with respect to each other.
For example:
data1=randn([100,1]);
data2=(data1.^2-3*data1+5)+0.01*randn([100,1]);
%data2 is a function of data1 + noise
ref=randn([100,1]);
subplot(1,2,1);scatter(data1(:),ref(:));
subplot(1,2,2);scatter(data1(:),data2(:));
As you can see, the independent reference variable is all across the plot, but the relationship between the two data samples is clearly evident.
Another way to look at this would be:
subplot(1,2,1);plot(conv(data1,data2))
subplot(1,2,2);plot(conv(data1,ref))
Note: I have not vetted/proved these methods in a rigorous way, so I would use it with the understanding that it MAY reveal some dependencies, but isn't guaranteed, especially if there is a real but weak relationship or a time delayed relationship.
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.
Thanks for your help Sean.

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