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There are often differences between results of Skewness & Kurtosis and normality tests, and I have always doubts if it is better to choose parametric or nonparametric tests (I use SPSS). Sometimes histograms show if distribution looks normal or not, and I noticed that most often S&K are better pointers but when I did analysis last time it was different and I really don't know what to do... I read that e.g. when groups are equinumerous, in choosing between t-Student test and nonparametric ones it is better to choose t-Student's even if distributions aren't normal. Is that true?

kjetil b halvorsen
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Zuzanna Kowalska
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    Velcome to the site! But you should explain what you mean by S&K. – kjetil b halvorsen May 15 '14 at 13:51
  • S&K mean as in the question: skewness and kurtosis. – Zuzanna Kowalska May 15 '14 at 17:43
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    The decision should not be based on the outcome of a formal hypothesis test, which answers entirely the wrong question. Your question of interest here is about how much impact the non-normality in your data will have on your inference, and hypothesis tests don't speak to that at all. Indeed, they're most likely to reject when you have a nice large sample size... in which case, for things like ANOVA, say, the non-normality may barely matter. If your sample sizes are small, you may have highly non-normal data (and a big impact on your inference) but little power to reject it. – Glen_b May 15 '14 at 19:21
  • Recommended reading: [Is normality testing 'essentially useless'?](http://stats.stackexchange.com/q/2492/32036) – Nick Stauner May 15 '14 at 21:05

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You should maybe give more details about your application for us to be able to give specific advice. Yes, normal-based tests (for means, not for variances) are usually quite robust. But even slight differences from a normal distribution may destroy their optimality. So, if in doubt, you should use the nonparametric tests!

A big advantage with normal-based theory is its larger flexibility. So, if you need this flexibility, you can combine the normal-theory tests with suitable transformations of the data (log, in case of skewed distributions, for instance).

Nick Cox
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kjetil b halvorsen
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You should use skewness and kurtosis because they are not so dependent on sample size.

However, I am not absolutely sure that they cover all deviations from normality. If you have doubts, you can do both parametric and non-parametric tests and see if they differ substantially.

Peter Flom
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    I am absolutely sure that skewness and kurtosis *do not* cover all deviations from normality. A thought experiment: symmetric distribution (0 skew), bimodal. Now imagine such a distribution with heavy tails (negative kurtosis); you can also imagine it in peaky form with thin tails (high positive kurtosis). There are zero kurtosis zero skewness bimodal distributions, and therefore kurtosis and skewness tests are not definitive for normality. As a bonus there are an infinite number of symmetric >2-modal distributions with zero skew and zero kurtosis also. – Alexis May 15 '14 at 14:22
  • Good point! I hadn't thought of the bimodal cases. – Peter Flom May 15 '14 at 14:23
  • So what's the answer?:( – Zuzanna Kowalska May 15 '14 at 17:44
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    Skewness and kurtosis are normally defined using third and fourth m oments, respectively. You will need huge samples to be able to estimate them with any precision. Maybe some alternative definitions of S/K, based on quantiles, might behave better. – kjetil b halvorsen May 15 '14 at 17:47
  • I am having a hard time imagining high positive kurtosis "in peaky form with thin tails." – BigBendRegion May 01 '19 at 11:30