The sample variance of the $mn$ experiments can now be written in terms of the autocorrelation function $$\begin{equation} \sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} \frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 \tag{12} \end{equation}$$ and we see that $\sigma_m$ can be expressed in terms of the uncorrelated sample variance times a correction factor $\tau$ which accounts for the correlation between measurements. We call this correction factor the autocorrelation time $$\begin{equation} \tau = 1+2\sum_{d=1}^{n-1}\kappa_d \tag{13} \end{equation}$$ For a correlation free experiment, $\tau$ equals 1.