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We all know

Leptokurtic ~ Kurtosis > 3 and Platykurtic ~ Kurtosis < 3

I am bit confused about the shape of the curve. Somewhere I had read that since the area under the curve should be 1, for leptokurtic distributions (since the peak is higher than the Normal distribution), the tails approach the X - axis much earlier than in case of Normal distribution i.e. when the pdf is plotted, the tails in case of leptokurtic lie below Normal curve tail so I guess leptokurtic has got a thin tail.

On the other hand, in case of platykurtic, the peak is less than the normal curve, however it's tail will lie above the Normal curve tail and is fat tailed. This is my understanding and I might be grossly wrong.

However, when I had referred many sites or went through Google images, I came across many instances where the Peak was shown much higher than the Normal distribution and also the tail lying above Normal distribution curve.

So I am very confused now as I am not a hard core statistician but will certainly love to clarify my doubts and learn few things. Also, I am not sure if FAT tail is synonym for HEAVY tail. Can someone explain this in simple words so that a moron like me will be enlightened.

kjetil b halvorsen
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Katherine Gobin
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    In fact, higher kurtosis is associated with both increased peakedness and *heavier* extreme tails, but there's no necessary relationship in either case (you can find counterexamples to higher peak or heavier extreme tails, even though both are typical with higher kurtosis). A variety of shapes is possible in either case. – Glen_b Sep 17 '13 at 11:10
  • Kendall and Stuart (I think in Vol I) present a collection of counterexamples to the various combinations of peak and tails. See also the discussion [here](http://stats.stackexchange.com/questions/57203/kurtosis-of-a-standardized-students-t-distribution/57221#57221) – Glen_b Sep 17 '13 at 11:16
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    Also see the paper by Kevin P. Balanda and H.L. MacGillivray. "Kurtosis: A Critical Review". The American Statistician 42:2 [May 1988], pp 111–119. (And yes, generally speaking, fat-tailed and heavy-tailed are synonyms, except for contexts where one or both has been given an explicit definition.) – Glen_b Sep 17 '13 at 11:23
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    You might want to read the following paper: **DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2(3), 292.** – Alecos Papadopoulos Sep 17 '13 at 23:31
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    A caveat about the DeCarlo paper - the very first sentence of the abstract is wrong! Even for symmetric, unimodal distributions, high kurtosis does not mean "peakedness," and low kurtosis does not mean flatness. There are symmetric, unimodal flat-topped distributions with extremely high (even infinite) kurtosis; there are also infinitely peaked distributions with negative excess kurtosis. See specific examples given here: https://en.wikipedia.org/wiki/Talk:Kurtosis#Why_kurtosis_should_not_be_interpreted_as_.22peakedness.22 – BigBendRegion Oct 21 '17 at 12:20
  • Glen_b, your "you can find counterexamples to heavier extreme tails, though both are typical with higher kurtosis" is only correct if you define "heavier extreme tails" in a particular way out of infinite ways. There is no counterexample to the mathematical facts that (i) as kurtosis tends to infinity, E(Z^4*I(|Z| >b))/kurtosis ->1, for all b; and (ii) kurtosis is within +- .5 of E(Z^4*I(|Z| > 1)) +.5; both of these statements explain how kurtosis is related to tails of the distribution, and not the peak. If you define "heavier tails" as larger "E(Z^4*I(|Z| > 1))" then there you have kurtosis. – BigBendRegion Nov 15 '17 at 23:32
  • Glen_b, your comment "In fact, higher kurtosis is associated with both increased peakedness ..." seems patently false in the face of numerous counterexamples. Can you provide a theorem to justify it? – BigBendRegion Nov 20 '17 at 01:43

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