Sample Mean and Variance

Sample Mean and Variance

Consider a set of independent and identically distributed random variables

X₁, X₂, ..., X_n, where E(X_i) = m and Var(X_i) = s², i = 1, 2, ..., n. These random variables can be considered as elements of a random sample from an infinite population having a probability distribution with mean m and variance s². For this random sample, the sample mean and variance are given by the following equations:

The sampling distribution of the mean is the probability distribution of the mean of a random sample. Its mean and variance can be easily calculated as follows:

The sampling distribution of the mean has the same mean as the original population, but its variance is smaller than that of the original population by a factor of 1/n. The square root of the variance of the sample mean is also known as the standard error of the mean.

The sampling distribution of the sample variance is the probability distribution of the sample variance of a random sample. Its mean is calculated as follows:

The above computations show why the factor in the equation for the sample variance is (1/(n-1)) instead of (1/n)--this is true so that the expected value of the sample variance will equal the variance of the population.

The variance of the sample variance has a mathematical form that depends on the probability distribution of the parent population. Thus, its value will differ from one probability distribution to another, even if the distributions have the same mean.