Variance and Standard Deviation

Let $\sigma^2$ be the variance of a random variable. Suppose we have a simple random sample from the population, Xi, i=1,...,N. One estimate of $\sigma^2$ is the sample variance:

$s^2 = \frac{1}{N-1} \sum_{j=1}^{J}[\bar{x} - \mu]^2$

This is an unbiased estimator of the population variance. The standard deviation (σ) can be estimated by taking the square root of the variance.

After we have collected a random sample, x_i, i=1,...,N, we can calculate a sample value for s². If we collect a different random sample, we will calculate a different value for the sample variance.

Since s² is a function of random variables, it is also a random variable and has a distribution. If the population is normally distributed, then the sampling distribution of s² is given by:

$\frac{N-1}{\sigma^2}s^2 \sim \chi^2(N-1)$

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Variance and Standard Deviation

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