#### Sample-Mean Variance

The simplest case to study first is the sample mean:

Here we have defined the sample mean at time as the average of the successive samples up to time --a running average''. The true mean is assumed to be the average over any infinite number of samples such as

or

Now assume , and let denote the variance of the process , i.e.,

Var

Then the variance of our sample-mean estimator can be calculated as follows:

where we used the fact that the time-averaging operator is linear, and denotes the unbiased autocorrelation of . If is white noise, then , and we obtain

We have derived that the variance of the -sample running average of a white-noise sequence is given by , where denotes the variance of . We found that the variance is inversely proportional to the number of samples used to form the estimate. This is how averaging reduces variance in general: When averaging independent (or merely uncorrelated) random variables, the variance of the average is proportional to the variance of each individual random variable divided by .

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Sample-Variance Variance
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