The 68–95–99.7 rule is a convenient way of quickly getting an overview of the spread of some normally distributed data.
I am wondering if there is an equivalent rule in the multivariate case? and how would I go about proving/disproving this?
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Toke Faurby
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I guess you would like to use chisquared distribution with the corresponding quantiles, like qchisq(c(0.68, 0.95, 0.997), df). For 2 normals you will have: 2.278869; 5.991465; 11.618286. You can prove it using definition of chi squared distribution. Of course you will need to standartize your data before and use the covariance matrix if your RVs are not independent. http://i.stack.imgur.com/ClHmC.png – German Demidov Sep 05 '16 at 09:51
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If your question was more clearly formulated it would be more clearly a duplicate. See [here](http://stats.stackexchange.com/questions/149334/multivariate-standard-deviation/149346#149346) and [here](http://stats.stackexchange.com/questions/62092/bottom-to-top-explanation-of-the-mahalanobis-distance/62147#62147) and [here](http://stats.stackexchange.com/questions/69350/about-trivariate-normal-distribution/69358#69358) for some discussion of relevant concepts and calculations. If you can formulate a clear question that is not covered by these, please post a new question. – Glen_b Sep 05 '16 at 10:00
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Also see [here](http://stats.stackexchange.com/questions/67422/volume-of-the-95-confidence-ellipsoid/67429#67429) – Glen_b Sep 05 '16 at 10:08