Drawing samples from a population when its elements do not enter into preferential pairings
results in zero covariance among these variables
as shown in the above diagram. The fact that the samples are not correlated makes the variance additive. This can be illustrated as follows. As a first step to compute the variance of the sampled means, we had to sum the sampled values, as
The variance of a sum equals the variance of the elements summed plus two times their covariance,
However, since the summed elements are not correlated,
the variance of the sum equals the sum of the variances of the variables X1 and X2. If the variances of the sampled elements are about the same,
Thus the above equation can be further simplified as
At this point, we traced down what happens to the variance of the sampled means when the elements of the samples are added together. What happened to this variance when the sampled elements were divided by n to obtain the means?
As shown previously, variance of a variable divided by a constant equals the variance of the variable divided by the square of that constant, thus
and
For the example of the sampling distribution of the means, the variance of the sampled means (.33) indeed equals the variance of the population sampled from (.67) divided by the size of the sample, n, equal to 2. The square root of the above expression
is also known as the standard error of the mean.