25
Sep
Distribution Of Sample Variance. Is a chi-square1 random variable. 2 Similarly the expected variance of the sample variance. The expected value of for a sample size is then given by. The objective of this paper is to derive a general formula for the mathematical expectation of sample variance.
The sampling distribution of the sample variance is a chi-squared distribution with degree of freedom equals to n 1 where n is the sample size given that the random variable of interest is normally distributed. The expected value of sample variance is often derived by deriving its sampling distribution which may be intractable in some situations. 1 where is the sample mean. Where n is the number of categories. Chi-Square Distribution of Sample and Population Variances Given a random sample of n observations from a normally distributed population whose population variance is and whose resulting sample variance is s2 it can be shown that has a distribution known as the chi-square distribution with n 1 degrees of freedom. The variance of this sampling distribution can be computed by finding the expected value of the square of the sample variance and subtracting the square of 292.
The sample mean m is simply the expected value of the empirical distribution.
Now lets get to what Im really interested in here - estimating σ 2. Suppose that we have a bivariate normal distribution where both measures have the same variance σ 11 σ 22 σ 2 and correlation ρ. 2 Similarly the expected variance of the sample variance. Thats because the sample mean is normally distributed with mean mu and variance fracsigma2n. The probability distribution for the sample variances is shown next. Note that without knowing that the population is normally distributed we are not able to say anything about the distribution of the sample variance not even approximately.