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How can I use my sample's skewness and kurtosis to determine whether I have a multivariate normal distribution?
There is a large amount of literature on this topic, although not much is yet implemented in SAS or SPSS.
In _Multivariate Analysis, Part 1, Distributions, ordination, and inference_, (1994) W.J. Krzanowski and F.H.C. Marriott review (section 3.16, p.58) tests of the null hypothesis that the data come from a multivariate normal distribution.
Mardia 1970 found that if the null hypothesis is true, then a simple function of the sample skewness has an asymptotic chi-squared distribution, and the sample kurtosis has an asymptotic normal distribution. Sample sizes greater than 50 are needed for approximations to be acceptably accurate.
SAS's PROC CALIS will output several measures of univariate and multivariate skewness and kurtosis.
Also, the PRELIS2 program developed by Joreskog and Sorbom will test both univariate and multivariate normality simultaneously, including separate tests of skewness and kurtosis at both the univariate and multivariate level. See page 24 of the PRELIS2 manual (the section titled, "New features in PRELIS2").
Mardia's test of multivariate normality can also be found in EQS, Structural Equation Modeling Software.
AMOS also includes a test of multivariate normality. For detailed instructions on performing this test in AMOS, see AMOS FAQ 7 on handling non-normal data at http://ssc.utexas.edu/consulting/answers/amos/amos7.html.
Macros for both SPSS and SAS can be downloaded. Lawrence DeCarlo, Ph.D., provides an SPSS macro for Mardia's test of multivariate skewness and kurtosis at http://www.columbia.edu/~ld208/. A SAS macro is available in the SAS online manual at http://support.sas.com/. To find the macro, go to the Knowledge Base section and click on Samples and SAS Notes. Click on Search Samples, search for multnorm, and choose Macro to test multivariate normality. This site gives a downloadable version of the macro as well as instructions on how to use the macro.
If you have further questions, send E-mail to stats@ssc.utexas.edu.