<P> It is often of interest to estimate the variance or standard deviation of an estimated mean rather than the variance of a population . When the data are autocorrelated, this has a direct effect on the theoretical variance of the sample mean, which is </P> <Dl> <Dd> V a r (x _̄) = σ 2 n (1 + 2 ∑ k = 1 n − 1 (1 − k n) ρ k). (\ displaystyle (\ rm (Var)) \ left ((\ overline (x)) \ right) = (\ frac (\ sigma ^ (2)) (n)) \ left (1 + 2 \ sum _ (k = 1) ^ (n - 1) (\ left (1 - (\ frac (k) (n)) \ right) \ rho _ (k)) \ right).) </Dd> </Dl> <Dd> V a r (x _̄) = σ 2 n (1 + 2 ∑ k = 1 n − 1 (1 − k n) ρ k). (\ displaystyle (\ rm (Var)) \ left ((\ overline (x)) \ right) = (\ frac (\ sigma ^ (2)) (n)) \ left (1 + 2 \ sum _ (k = 1) ^ (n - 1) (\ left (1 - (\ frac (k) (n)) \ right) \ rho _ (k)) \ right).) </Dd> <P> The variance of the sample mean can then be estimated by substituting an estimate of σ . One such estimate can be obtained from the equation for E (s) given above . First define the following constants, assuming, again, a known ACF: </P>

What is a good unbiased estimator for the var(x)