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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 .

**Previous:** Estimator Variance**Next:** Sample-Variance Variance

**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.