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

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

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.

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.