# Spectral Audio Signal ProcessingBeginning Statistical Signal ProcessingWhite NoiseEstimator VarianceSample-Mean Variance

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#### 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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.