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Measures of Asymmetry
4. Kurtosis

Instruction

Kurtosis is a measure of the peakedness in a dataset. For an n-element set, kurtosis is given by the formula:

Kurtosis=1n((x1x)4+(x2x)4++(xnx)4)(1n((x1x)2+(x2x)2++(xnx)2)23 Kurtosis=\dfrac{\frac{1}{n}((x_1-\overline{x})^4+(x_2-\overline{x})^4+\ldots+(x_n-\overline{x})^4)}{(\frac{1}{n}((x_1-\overline{x})^2+(x_2-\overline{x})^2+\ldots+(x_n-\overline{x})^2)^2}-3

where x\overline{x} is the arithmetic mean.

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Histograms with kurtosis 0\approx{0} are called mesokurtic. This is similar to normal distribution.

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Histograms with kurtosis smaller than 0 are called platykurtic. The tail of such a histogram is rather short and thin, while the peak is comparatively low and broad. The word platykurtic comes from Greek: platy means broad, flat, while kurtos means bulging. Platykurtic histograms are therefore relatively flat.

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Histograms with kurtosis greater than 0 are called leptokurtic. The tails of such a histogram is long and fat, but the peak is usually tall and sharp. The word leptokurtic comes from Greek: lepto means narrow. Leptokurtic histograms are prominently peaked.

In case you're wondering, meso means middle in Greek, thus mesokurtic histograms are in the middle between platykurtic and mesokurtic histograms.

Exercises

Analyze the histograms below, and decide whether a given histogram is mesokurtic, platykurtic, or leptokurtic.

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Kurtosis: 0.64874458016

Answer:

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Kurtosis: -1.07063711911

Answer:

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Kurtosis: 0.84693877551

Answer: