Chi-Square Distribution

Suppose a sample of size $n$ is drawn from a normal population. The chi-square statistic can be calculated using:

$$\chi^2 = \frac{\nu s^2}{\sigma^2}$$

Here:

• $\nu$ - degrees of freedom
• $s^2$ - sample variance
• $\sigma^2$ - population variance

When sampling is done for an infinite number of times, and by calculating the chi-square statistic for each sample, the sampling distribution for the chi-square statistic can be obtained. It is then called the chi-square distribution.

Uses

• To model how sum of sample variances behave

Properties

• Skewed for $\nu < 3$
• Always positive

Degrees of Freedom

Usually, the degrees of freedom is $\nu = n - 1$. $$

As $\nu$ increases, the chi-square distribution approaches a normal distribution.

Mean

The mean is $\mu = \nu$. $$

Variance

The variance is $\sigma^2 = 2\nu$. $$