Calculating diffusion coefficient from jump statistics

If we have a random process, such as when looking at Brownian Motion, we can look at the individual displacements $\mathrm{\Delta }x$, and we'll get[@catipovic2013Improving the quantification of Brownian motion]

For standard Brownian processes, we also have $⟨\mathrm{\Delta }x⟩=0$

And from here, we have that the variance is equal to the mean-squared displacement, which can be re-written as the known equation:

In this case $t$ is the time between frames in the measurement (which is given as constant).