Normal Distribution - Kurtosis

[Biju George]

Hi All,
Quick question : Is there a relation between Kurtosis and Standard deviation for normally distributed curves.
For example there can be normally distributed curves with 10%,20%,30% standard deviations for which Kurtosis will be varying for each .Can somebody explain further .
Thanks in advance

 

[Arka Bose]

Hi,

Its actually a very good question and its been discussed for quite some time now in the stats world.

Since kurtosis is standardized, it is unit-less, that should mean it must not depend upon variance of the distribution.
https://www.riskprep.com/all-tutorials/36-exam-22/145-understanding-kurtosis

Above is a good link which actually says you just cannot judge whether a distribution is normal or non-normal just by looking at the diagram, i would like you to go through.
But, the page says 'In order to compare kurtosis between two curves, both must have the same variance', which I DO NOT agree because the point of calculating a standardized kurtosis is after all, comparing distribution of different shapes and sizes.

So, in short, kurtosis is not dependent upon variance as i see it.

However, I would certainly like David to look into your question.

 

[Biju George]

Thanks .will wait for further comments too.
I was trying to understand why Kurtosis is constant (=3) fro normal distribution. With varying 10%,20%,30% standard deviations the shape of the normal curve (though symmetrical ) in terms of flattening/peak will vary , hence Kurtosis will vary ?

 

[Arka Bose]

With your question now (and from reading somewhere else) I now understand why they they wrote 'In order to compare kurtosis between two curves, both must have the same variance' is correct.
So yes, you are right, the kurtosis will vary.

 

[David Harper CFA FRM]

@RiskGuy Consistent with what @arkabose says, the fourth central moment does change with variance or standard deviation because it is given by 3*sigma^4, see central moments @ http://mathworld.wolfram.com/NormalDistribution.html. If sigma (standard deviation) is 1.0, then 4th central moment = 3*1^4 = 3; if sigma doubles to 2.0, then 4th central moment = 3*2^4 = 48. But kurtosis is 4th central moment/sigma^4; i.e., standardized 4th central moment. To illustrate by way of a silliness, consider that the variance is the 2nd central moment which equal sigma^2; if we were to treat this analogous to kurtosis, we would "standardize" it by dividing it by sigma^2. So the standardized 2nd central moment = sigma^2/sigma^2 = 1.0, which is the standard normal. I hope this helps,