The simplest case to study first is the sample mean:
![]() |
(C.29) |
![]() |
(C.30) |
![]() |
(C.31) |
Var![]() |
(C.32) |
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
.