Chi-Squared Distribution and the Central Limit Theorem
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This Demonstration explores the chi-squared distribution for large degrees of freedom 
, which, when suitably standardized, approaches a standard normal distribution as 
by the central limit theorem. In this Demonstration, can be varied between 1 and 2000 and either the PDF or CDF of the chi-squared and standard normal distribution can be viewed. You can also see the difference between the two distributions, which becomes useful for large .
Contributed by: Peter Falloon (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
If 
are independent standard normal variables, then the random variable 
follows the chi-squared distribution with mean and standard deviation 
. Taking the standardized variable 
, the central limit theorem implies that the distribution of 
tends to the standard normal distribution as .
It can be seen that the chi-squared distribution is skewed, with a longer tail to the right. However, the PDF has maximum value at 
(for 
), which is equivalent to 
, so the skew disappears in the limit as .
Permanent Citation
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