# how to study a distribution that is not identically distributed?

My distribution histogram looks like it is not identically distributed, as in the negative counts have a different shape than the positive counts. Here is an image:

The chart has 800 data points, and the tallest count of 40 is for 0.

I don't know a formal approach, but looking at it, it appears to me it is not identically distributed. My guess is that the negative counts (to the left of 0) are closer to a normal distribution, while the positive counts (to the right of 0) are closer to a Laplace distribution, or have higher kurtosis than the left side.

Is this possible, are there cases where this happens, is there a topic in probability that studies it? Or, is this just my eyes playing tricks on me, or not having enough sample data or some other simple reason?

Thanks in advance for any help.

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Do you mean symmetrically instead of indentically? – cardinal Apr 17 '12 at 1:23
"Identically distributed" is a standard term in probability theory, and its meaning in standard usage is not at all what seems to be meant by it here. Two or more random variables are identically distributed if they have the same distribution. That has nothing to do with what's being asked about in this problem. – Michael Hardy Apr 17 '12 at 1:31
Thank you cardinal and Mr Hardy for your comments, I was using the wrong term. I think "symmetrically" is closer to my meaning. – d l Apr 17 '12 at 1:41
dl, "shape parameters" are used to describe changes, such as "skewness", in a family of distributions other than those generated by "location" or "scale" parameters. See wikipedia (en.wikipedia.org/wiki/Shape_parameter) for some examples of families with a shape parameter. Your distribution actually looks like a product of two normal distributions with different means, like $N(-1,1)\cdot N(1,1)$. – B R Apr 17 '12 at 1:55
Thank you B R, I think your comment points me in the right direction. I wish I could mark it as the accepted answer. – d l Apr 17 '12 at 2:03

"Shape parameters" are used to describe changes, such as "skewness", in a family of distributions other than those generated by "location" or "scale" parameters. There are many examples of distributions with shape parameters. Your distribution actually looks like a product of two normal distributions with different means, like $N(−1,1)⋅N(1,1)$.

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As @cardinal wrote, you probably don't mean "identically distributed": that's something you can't really tell from a histogram. If you want to test whether this is a normal distribution, you might use a D'Agostino-Pearson test, a Shapiro-Wilk test, or a Kolmogorov-Smirnov test. See e.g. http://www.graphpad.com/faq/viewfaq.cfm?faq=959 and references there.

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Thank you for the comments and guidance, I suppose I should rephrase the question as: Can a distribution have 2 different PDF formulas for the right and left side, or different shape parameters? – d l Apr 17 '12 at 1:36
Yes, of course it can. The only requirements on a PDF are that it is nonnegative everywhere and its integral over the real line is $1$. – Robert Israel Apr 17 '12 at 6:17