The sample error (see eq. (17)) can now be written in terms of the autocorrelation function:

$$\begin{align} \mathrm{err}_X^2 &= \frac{1}{n}\mathrm{var}(x)+\frac{2}{n}\cdot\mathrm{var}(x)\sum_{d=1}^{n-1} \frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=& \left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\ &=\frac{\tau}{n}\cdot\mathrm{var}(x) \tag{18} \end{align}$$

and we see that $\mathrm{err}_X$ can be expressed in terms the uncorrelated sample variance times a correction factor $\tau$ which accounts for the correlation between measurements. We call this correction factor the autocorrelation time:

$$\begin{equation} \tau = 1+2\sum_{d=1}^{n-1}\kappa_d \tag{19} \end{equation}$$