What is the 4-th central moment?

1. Jan 18, 2012

Chriszz

Dears,

The kurtosis consists of the 4th central moment and the square of variance.
It measures the peakedness of the distribution.

However, it is hard to understand why 4th central moment is used to measure 'peakedness' of the distribution.

What is the 4-th central moment? It looks similar to the variance (2nd central moment).
However, 2nd central moment represents the dispersion, and 4th central moment does the shape of the distribuion.

What is the difference between 2nd and 4th central moments?

Thanks.

2. Jan 18, 2012

MRBlizzard

I've never had a statistics course, but this is how I see it.

The second moment is the Standard Deviation. For a normal/Gaussian distribution this is a measure of the width of the peak. You can also think of this as a measure of the difference between a pure, single value data set, and a non single value data set.

The third moment goes as the cube. It is an odd term. It measures the difference between a symmetrical curve and a skewed, left or right, non symmetrical curve.

The fourth moment is even and measures the difference between a Gaussian function and the plot of the data as it really is, or seems to be, given the finite sampling. I believe that I read one time, that the fourth moment is the error on the Standard Deviation.

I'd welcome corrections here.