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 $\Delta x$, and we'll get[@catipovic2013Improving the quantification of Brownian motion]

$$\text{Var}(\mathrm{\Delta }x)=⟨\mathrm{\Delta }{x}^2⟩-{⟨\mathrm{\Delta }x⟩}^2$$

For standard Brownian processes, we also have $\left<\Delta x\right> = 0$

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

$$⟨\mathrm{\Delta }{x}^2⟩=2Dt$$

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