However, if the observations in the sample are correlated, then is somewhat higher. For instance, if all observations in the sample are completely correlated (), then regardless of .
The effective sample size is the unique value (not necessarily an integer) such that
is a function of the correlation between observations in the sample. Suppose that all the correlations are the same and nonnegative, i.e. if , then . In that case, if , then . Similarly, if then . 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 even when all correlations are positive. Intuitively, may be thought of as the information content of the observed data.
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