Chi-Squared Distribution and the Central Limit Theorem

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 .

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 .