Kurtosis

Statistical Definition of Kurtosis

The shape of a distribution can be characterized by its moments. If a variable is denoted by X and its distribution has the mean µ, its rth moment about the mean is $m_r={\displaystyle \sum [{(X-\mu )}^r]/n}$ in the case of a frequency distribution or $m_r={\displaystyle \int {(X-\mu )}^r}f(x)dx$ for a probability distribution. Either way, the first moment about the mean is zero by definition, and the second moment about the mean is the variance, σ2. Subsequent moments about the mean can be standardized by dividing them by ${\sigma }^r\text{(or}{m}_2^{r/2})$. Pearson labeled the standardized third moment about the mean, $m_3/m_2^{3/2}$, as β1, and this is often used as a measure of the skewness of the distribution. Similarly, Pearson labeled the standardized fourth moment about the mean, $m_4/m_2^2$, as β2.

Pearson noted that β2 = 3 for a normal distribution, and he defined the degree of kurtosis of the distribution as η = β2 − 3 (other writers have used the symbol γ2). This is sometimes known as “excess kurtosis” (i.e., beyond that of the normal distribution). Different writers have adopted either β2 or γ2 to refer to kurtosis itself, and for clarity, these symbols will be used for the rest of this entry. Pearson identified platykurtic distributions as those for which β2 was less than 3 and leptokurtic distributions as those for which β2 was greater than 3. It can be shown that $m_4\ge m_2^2$, and thus β2 ≥ 1 and γ2 ≥ −2, with equality when a random variable takes only one of the two different values with equal probabilities. It can also be shown that β2 > 1 + β1 and, in the case of unimodal symmetric distributions, that β2 ≥ 1.8. However, there is no upper limit to β2, which may be infinite for certain distributions.

Student’s Memoria Tecknica

In 1927, another British statistician, William S. Gosset (who published under the pseudonym of “Student”), suggested a memoria tecknica or [Page 940]mnemonic for remembering the difference between platykurtic and leptokurtic distributions. He presented the illustration shown in Figure 1 along with the following explanation: