The central limit theorem states that the PDF \( \tilde{p}(z) \) of the average of \( m \) random values corresponding to a PDF \( p(x) \) is a normal distribution whose mean is the mean value of the PDF \( p(x) \) and whose variance is the variance of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).

The central limit theorem leads then to the well-known expression for the standard deviation, given by $$ \begin{equation*} \sigma_m= \frac{\sigma}{\sqrt{m}}. \end{equation*} $$

In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.