Monday, June 8, 2009

More Kurtosis

Until I figure out what's wrong with Nmix I'll just learn more about kurtosis.

Jen just asked me what the hell kurtosis was, so I should probably explain it a bit better. Technically it's defined as the "standardized fourth population moment about the mean," but that just sounds like mumbo jumbo to me.

Sample kurtosis is defined on this Wikipedia page.

Kurtosis is just another characterization of distributions, like the mean and variance. But while the mean is the center of the distribution and variance determines its wideness, kurtosis measures how pointy the peak is and thick the tails are. I'm not really sure how it does that, but it does. Kurtosis of 0 indicates a normal/Gaussian distribution, positive indicates a pointy peak and thick tails, and negative means the opposite.

Kurtosis and Bimodality
Since bimodal distributions typically have two peaks in close proximity, they essentially form a fat and flat peak to the distribution. Well flat peaks are typical of negative kurtosis, so negative kurtosis can be an indication of bimodality. The lowest possible value of kurtosis is -2, so any distribution with a kurtosis around there is definitely not normal. It might not be bimodal, but that is definitely a possibility, especially when dealing with data that is supposed to be normally distributed like velocities.

Limitations of Kurtosis
One problem is that different shaped curves can have the same kurtosis. Since it relies on both the pointedness of the peak and the thickness of the tails, changes in one or the other can cancel out the differences in the other factor. Thus a pointy distribution with thin tails can have the same kurtosis as a flatter curve with thick tails.
So if a data set has a lot of outliers there would be thicker tails and a higher kurtosis than the actual value.

But limitations like that are typical of all shape characteristics. For example, the mean by itself doesn't tell much about the actual shape of the curve. And the variance doesn't tell much about the skewness or kurtosis. But all together these characteristics give a pretty good estimate of the shape.

Based on:
On the Meaning and Use of Kurtosis (1997)
Lawrence T. DeCarlo
--Google it for a PDF.

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