Pearson/Fisher Kurtosis

[Karl L. Wuensch]

B2, the expected value of the distribution of Z scores which have been raised to the 4th power, which has a value of 3 for a normal distribution, is often referred to as "Pearson kurtosis."  Subtract 3 from this quantity and you get a parameter which is often referred to as "Fisher kurtosis."  For example, at http://www2.chass.ncsu.edu/garson/pa765/assumpt.htm, you can find the following:   Kurtosis is the peakedness of a distribution. A common rule-of-thumb test for normality is to run descriptive statistics to get skewness and kurtosis, then divide these by the standard errors. Kurtosis also should be within the +2 to -2 range when the data are normally distributed (a few authors use +3 to -3). Negative kurtosis indicates too many cases in the tails of the distribution. Positive kurtosis indicates too few cases in the tails. Note that the origin in computing kurtosis is 3 and a few statistical packages center on 3, but the foregoing discussion assumes that 3 has been subtracted to center on 0, as is done in SPSS and LISREL. The version with the normal distribution centered at 0 is Fisher kurtosis, while the version centered at 3 is Pearson kurtosis. SPSS uses Fisher kurtosis. Various transformations are used to correct kurtosis: cube roots and sine transforms may correct negative kurtosis.       Now I happen to think that this statement is dead wrong in associating negative kurtosis with "too many cases in the tails" and positive kurtosis with "too few cases in the tails," but I am not looking to start an extended discussion of what the "tails" of a distribution really are.       My query to this group is:  Why is B2 called Person kurtosis and (B2 - 3) called Fisher kurtosis?  Recently I read Pearson, K. (1905). Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson. A Rejoinder. Biometrika, 4, 169-212, and in that interesting article Pearson defined kurtosis as (B2 - 3).   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Karl L. Wuensch, Department of Psychology, East Carolina University, Greenville NC  27858-4353 Voice:  252-328-4102     Fax:  252-328-6283 [EMAIL PROTECTED] http://core.ecu.edu/psyc/wuenschk/klw.htm