# Variance and kurtosis

1. Jun 9, 2014

### joshmccraney

hello again pf!

as a really simple question, can someone talk to me about the difference between variance and kurtosis? i know as kurtosis decreases from 3 (normal distribution) our pdf is shorter and fatter, with less weight in the tails. i also know variance tells us how dispersed data is.

but, how to these compare?

please, can someone explain what high kurtosis and low variance means?

thanks!

josh

2. Jun 9, 2014

### FactChecker

High kurtosis => terms of (x-μ)4 tend to be large
Low variance => terms of (x-μ)2 tend to be small

Since the 4'th power is more important than the 2'nd power when abs(x-μ) is large, this implies that out towards the tails on both sides, the tails are fatter (higher probability) than the typical PDF with that variance. And the fact that the variance is small implies that the tails start a little thinner than average.

So the conclusion is: Thinner tails than average (normal?) near the mean, and fatter tails than average farther away from the mean -- but not necessarily fatter than they were near the mean, just not getting thin as fast as the normal distribution would.

3. Jun 10, 2014

### joshmccraney

thanks a ton!