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title: "Normal approximation to the Chi Squared distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575265/Normal%20approximation%20to%20the%20Chi%20Squared%20distribution"
format: markdown
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The [Chi Squared](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575221/) distribution ChiSquared(*n*) can be approximated by a [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575244/) distribution for large n. The ChiSquared(*n*) distribution is the sum of *n* independent (Normal(0,1))2 distributions, so ChiSquared(*a*) + ChiSquared(*b*) = ChiSquared(*a*+*b*). A (Normal(0, 1))2 = ChiSquared(1) distribution is highly skewed (skewness = 2.83). [Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/) says that ChiSquared(*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 ChiSquared(*n*) distribution in a Normal distribution):

 

ChiSquared(*n*) » Normal

![image](media://582d6528-ce93-492c-986b-1d0e8617cc2e)

> Macro (mathinline)



 

The ChiSquared(*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.

 

![image](media://7576de61-09e6-4758-a480-b4e9330587f2)

 

Examples of a Normal approximation to a ChiSquared distribution

 

 

 

 

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