2.6.2. Kurtosis

The term Kurtosis is perhaps the most vague term, and still under debate. Before we get into this debate it is important to know that a normal distribution has three parts: the peak, shoulders and tails. These are illustrated in Figure 33.


Figure 33. Three parts of a normal distribution

The term kurtosis was first introduced by Pearson (1905) as another measure of shape, which indicates a level of tailedness and peakedness. A positive kurtosis (higher than 3) indicates a peakedness and/or heavy tailedness (called platykurtic), while a negative kurtosis indicates a flatness and/or a light tailedness (called leptokurtic) (DeCarlo, 1997). This description is however limited to unimodal and symmetrical distributions and is sometimes criticized (Balanda & MacGillivray, 1988).

Especially the peakedness and with it the classification as a measure of shape is heavily criticized, as becomes evident from Westfall’s (2014) article titled ‘Kurtosis as Peakedness, 1905 - 2014. R.I.P.’. He argues that the kurtosis could only be somewhat interpreted as tail extremity.

A definition that tries to keep it vague and therefor still captures both is given by Balanda & MacGillivray: “it is best to define kurtosis vaguely as the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails” (1988, p. 111).

To read more on Kurtosis a nice start might be a the answer given to a forum post by Glen_b (2014).

References
Balanda, K. P., & MacGillivray, H. L. (1988). Kurtosis: A Critical Review. The American Statistician, 42(2), 111–119. http://doi.org/10.2307/2684482 
DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2(3), 292–307. http://doi.org/10.1037/1082-989X.2.3.292 
Glen_b. (2014, December 2). Why kurtosis of a normal distribution is 3 instead of 0. Cross Validated. Retrieved from http://stats.stackexchange.com/questions/126346/why-kurtosis-of-a-normal-distribution-is-3-instead-of-0 
Pearson, K. (1905). “Das Fehlergesetz und Seine Verallgemeinerungen Durch Fechner und Pearson.” A Rejoinder. Biometrika, 4(1/2), 169–212. http://doi.org/10.2307/2331536 
Westfall, P. H. (2014). Kurtosis as Peakedness, 1905–2014. R.I.P. The American Statistician, 68(3), 191–195. http://doi.org/10.1080/00031305.2014.917055