Say I have a data set, how do I know if the distribution is normal or not? If not, how can I tell what type of distribution it is? Are there some tests for identifying distribution types?

  • $\begingroup$ Why do you want to know that? Answer: you probably don't want to know that. If anything you may want to look at the residuals, but even that is pretty much irrelevant in moderate to large datasets. $\endgroup$ – Maarten Buis Mar 22 '16 at 8:20
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    $\begingroup$ See also stats.stackexchange.com/questions/2492/… $\endgroup$ – Tim Mar 22 '16 at 9:29
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    $\begingroup$ If you're looking as some set of data and trying to decide if it was drawn from a normallity distributed population -- tou don't know it is, and you can't know it is. You can very often know it isn't (though it might be quite well approximated). $\endgroup$ – Glen_b -Reinstate Monica Mar 22 '16 at 11:27
  • Obtain the value of the mean and the standard deviation
  • Perform a z transform on the data using the standard deviation
  • Find the data fitting and whether 99% + of the data comes in 3 sigma
  • Perform a visual inspection using P-P plots and Q-Q plots
  • Perform the following tests of univariate normality include D'Agostino's K-squared test, the Jarque–Bera test, the Anderson–Darling test, the Cramér–von Mises criterion, the Lilliefors test for normality (itself an adaptation of the Kolmogorov–Smirnov test), the Shapiro–Wilk test and the Pearson's chi-squared test. See the Wikipedia entry for more details
  • $\begingroup$ It is not clear what is the use of the first two points -- they can be applied to any data with the same effect. Your third point does not prove normality anyhow. $\endgroup$ – Tim Mar 22 '16 at 9:30
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    $\begingroup$ Q-Q plots beat all other methods hands down in my view. Jarque-Bera implementations I have seen use asymptotic results for small samples, which here is a very bad idea. Chi-square is here long past its useful period and lingers for poor reasons. $\endgroup$ – Nick Cox Mar 22 '16 at 11:03
  • $\begingroup$ 1. I agree with Nick on Jarque Bera; I wouldn't use the asymptotic approximation much below about n=300 (see the prior paper on the subject by Bowman and Shenton, for example), and I have additional concerns with it besides that. 2. I'd advise against doing multiple tests; if you must test, pick one suitable one. 3. Note also that no goodness of fit test will tell you "if a distribution is normal", so this doesn't help with the title question. 4. In most situations I'd advise against actually testing goodness of fit at all where you're trying to identify a distributional model. $\endgroup$ – Glen_b -Reinstate Monica Mar 23 '16 at 0:00

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