Effective sample size

Correlated observations

Suppose a sample of several observations y i is drawn from a distribution with mean μ and standard deviation σ . Then the mean of this distribution is estimated by the mean of the sample:

In that case, the variance of μ ^ is given by

However, if the observations in the sample are correlated, then Var ⁡ ( μ ^ ) is somewhat higher. For instance, if all observations in the sample are completely correlated ( ρ ( i , j ) = 1 ), then Var ⁡ ( μ ^ ) = σ 2 regardless of n .

The effective sample size n eff is the unique value (not necessarily an integer) such that

n eff is a function of the correlation between observations in the sample. Suppose that all the correlations are the same and nonnegative, i.e. if i ≠ j , then ρ ( i , j ) = ρ ≥ 0 . In that case, if ρ = 0 , then n eff = n . Similarly, if ρ = 1 then n eff = 1 . More generally,

The case where the correlations are not uniform is somewhat more complicated. Note that if the correlation is negative, the effective sample size may be larger than the actual sample size. Similarly, it is possible to construct correlation matrices that have an n eff > n even when all correlations are positive. Intuitively, n eff may be thought of as the information content of the observed data.

Weighted samples

If the data has been weighted, then several observations composing a sample have been pulled from the distribution with effectively 100% correlation with some previous sample. In this case, the effect is known as Kish's Effective Sample Size