Normal approximation to the Chi Squared distribution


The Chi Squared distribution ChiSq(n) can be approximated by a Normal distribution for large n. The ChiSq(n) distribution is the sum of n independent (Normal(0,1))2 distributions, so ChiSq(a) + ChiSq(b) = ChiSq(a+b). A (Normal(0, 1))2 = ChiSq(1) distribution is highly skewed (skewness = 2.83). Central Limit Theorem says that ChiSq(n) will look approximately Normal when n is rather large. A good rule of thumb is that n > 50 or so to get a pretty good fit. In such cases, we can make the following approximation by matching moments (i.e. using the mean and standard deviation of a ChiSq(n) distribution in a Normal distribution):

ChiSq(n) » Normal  

The ChiSq(n) distribution peaks at x = n-2, whereas the Normal approximation peaks at n, so acceptance of this approximation depends on being able to allow such a shift in the mode. Of course as n gets large, the difference becomes relatively small.

 

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