# Skewness, Kurtosis applied

1. Jun 5, 2014

### joshmccraney

Hey PF!

I had a quick question for you about skewness and kurtosis. Suppose that I am measuring velocity in one dimension over time. Can you help me understand how positive or negative skewness relates to velocity? How about Kurtosis?

What if at two different points in space, say point 1 and point 2, and we collected a bunch of wind speeds (again, in one direction) and took the kurtosis of one point's 1 and 2 and found that kurtosis of point 1 was higher than point 2? What could explain this?

Let me know if I've been unclear in this.

Thanks a ton!

Josh

2. Jun 6, 2014

### FactChecker

Skewness indicates if the tail on one side on the mean stretches out farther than on the other side. Suppose you were measuring the total velocities of dust particles in the air. Most dust is settled near the ground and the vast majority is moving slowly. So the mean is near zero. But there are fast moving dust particles going all the way up to the jet stream. So the PDF of velocities would have a very long tail on the fast side and a very short one on the slow side. The skew would be positive.
Kurtosis is always positive. It is closely related to variance. Usually a large kurtosis just indicates a large variance. In that case, the difference in variance should be understood first. But two distributions with the same variance can still have different kurtosis values. In that case, a larger kurtosis indicates that the PDF has fat tails while a small kurtosis indicates thin tails. So the first thing to do when comparing kurtosis of two PDFs is to explain any difference in variance. Then normalize them separately so they have the same variance. Then try to understand any difference in the kurtosis values of the normalized data.

Last edited: Jun 6, 2014
3. Jun 7, 2014

### joshmccraney

Ahh, thanks!

4. Jun 10, 2014

### Hornbein

You want to compare the skewness of two samples, each from a different population. To me, the only way this would be meaningful would be to normalize the data from each population to mean zero and variance 1. Then positive skewness would tell you that there are more unusually high positive speeds than negative.

I think that the skewness and kurtosis numbers are not all that useful or meaningful. Usually we just look at a graph.