Summary statistics, whether mean, standard deviation or any other single number, are useful, but sometimes they can mislead - vastly different data sets can have similar or even identical summary statistics. In the video below Robert Kosara discusses this phenomenon in more detail.

Thus here we give you an opposite advice to the one we gave previously. You should plot the data to see whether summary statistic you have obtained make sense.

Well, you should actually always do both things - visualize the data and do some tests or summarization. If intuition from the visuals and the numbers differs, then you should look for things that may have gone wrong.

Last time we have explored the distribution of increments obtained from the interpolated Brownian motion. We have seen that the correct interpolation formula generates proper increments. Yet we have seen that with just our eyes. While the eyes are a good tool, they sometimes can lie. Thus it is wise to use a formal method to verify our intuition.

Here in this post we will discuss one of the simplest methods to verify whether data follows certain distribution - Kolmogorov-Smirnov test.

Last time we have discussed how to statistically correct interpolate Brownian motion. You have probably noticed that the correct interpolation formula has one peculiarity:

\begin{equation} W(0.5) = \frac{W(0) + W(1)}{2} + \sqrt{0.25} \cdot \xi . \label{eq:correct} \end{equation}

Why \( \sqrt{0.25} \) and not \( \sqrt{0.5} \)?

Can you increase wages without increasing spending? Yes, you can. More details and important implications outside just the field of economics in the video by Zoe Griffiths.

Suppose you have obtained a sample path of Brownian motion using discrete step size \( \Delta t \). Yet what happens at a finer scale? While in the experimental setup it would be impossible to say how the Brownian particle actually moved in between \( t \) and \( t + \Delta t \), one could numerically generate probable paths particle took in that time period. We do this by relying on the fractal nature of the Brownian motion.