09: Kurtosis Inversely related to Peakedness/Flatness

Music City Center 

Kurtosis continues to be misinterpreted as "peakedness/flatness"; large kurtosis "indicating" peakedness, and small "indicating" flatness. The "indicates" terminology appears in the first sentence of the abstract of DeCarlo, 1997 Psychological Methods, "On the Meaning and Use of Kurtosis."

A downside of using AI for research is that it can provide faulty information when its sources are wrong. A case in point is "indicates peakedness/flatness," which was shown false in statistical literature showing how kurtosis measures tails. However, AI reports this phrase, as do current scientific research articles.

In this presentation I show graphs of families of symmetric, unimodal distributions for which *smaller* kurtosis determines *greater* peakedness, and where *larger* kurtosis determines *flatness.*

The only logic ever given for "high kurtosis indicates peakedness; low kurtosis indicates flatness" is the existence of families of distributions for which this is true. By the same logic, based on the given families, one can as easily conclude "high kurtosis indicates flatness; low indicates peakedness."

(Both interpretations are wrong; examples do not generalize.)

Kurtosis

Distribution

Probability

Heavy Tails

Peakedness 

History of Statistics Interest Group