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In practice, we may always be asked to check the skewness and kurtosis of a data set. I have two questions.

Given a probability distribution, how can we determine/evaluate the skewness and kurtosis of this data set?

If we have the data sample itself, which kind of statistics can help us evaluate its skewness and kurtosis?

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    $\begingroup$ "Given a probability distribution, how can we determine/evaluate the skewness and kurtosis of this data set?" -- this question conflates two different things (data sets and probability distributions). Which do you mean? $\endgroup$
    – Glen_b
    Dec 8, 2013 at 2:24

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The skewness and kurtosis of particular distributions are known functions of the distribution. e.g. the Normal distribution has a skewness of 0 and a kurtosis of 3 (often given as an "excess kurtosis" of 0).

The formulas for skewness and kurtosis are widely available on the web, e.g. Skewness and kurtosis

For any given sample, skewness and kurtosis can be calculated by many programs including Rusing the moments package; SAS using PROC UNIVARIATE or even Excel (and doubtless also programs that I am not familiar with, such as MATLAB, SPSS and so on)

Graphically, tools such as the quantile quantile plot, the density plot and box plots can all be useful in visualizing distributions (including their skewness and kurtosis).

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  • $\begingroup$ A normal has kurtosis 3 (excess kurtosis 0). Negative sign there is just a typo. $\endgroup$
    – Nick Cox
    Dec 7, 2013 at 17:38
  • $\begingroup$ I think we need to be clear that both skewness and kurtosis are general ideas which permit several precise measures. For example, skewness is perhaps most commonly now measured via moments, but we could also use (mean - median) / SD, definitions using paired quantiles, L-moments etc. $\endgroup$
    – Nick Cox
    Dec 7, 2013 at 17:45
  • $\begingroup$ @NickCox re Particular measures vs. general concepts. I think both terms are used both as the measures I gave and as general concepts. This is confusing. $\endgroup$
    – Peter Flom
    Dec 7, 2013 at 19:24

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