Kurtosis Inversely Related to Peakedness

Ku osis In e sely Rela ed o Peakedness Pe e H. Wes all Texas Tech Uni e si y (Eme i us) Abs ac In 2014 I published he a icle, “Ku osis as Peakedness, 1905 – 2014. R.I.P.,” in The Ame ican S a is ician, soundly debunking once and o all he cen u y-old no ion ha ku osis measu es peakedness and la ness. Howe e , sou ces on he web and in pee - e iewed publica ions con inue o p esen ku osis, inco ec ly, as a measu e o peakedness and la ness. This a icle p esen s new esul s since 2014 ha make he case e en mo e s ongly ha ku osis mos empha ically does no measu e peakedness/ la ness, as amilies o dis ibu ions exis whe e inc easing peakedness causes dec eased ku osis. Ins ead o peakedness, I show how ku osis is p ecisely in e p e ed as a measu e o ail weigh , ega dless o symme y o modali y. Finally, I ecommend isualizing ku osis using no mal q- q plo s, as densi y/his og am plo s do no show ail weigh clea ly. 1. Ku osis Does No Measu e Peakedness o Fla ness In 1905, Pea son s a ed he inco ec in e p e a ions, saying [depa u e om no mali y in ol es a] deg ee o la - oppedness which is g ea e o less han ha o he no mal cu e. Gi en wo equency dis ibu ions which ha e he same a iabili y as measu ed by he s anda d de ia ion, hey may be ela i ely mo e o less la - opped han he no mal cu e. I mo e la - opped I e m hem pla yku ic, i less la - opped lep oku ic, and i equally la - opped mesoku ic. This in e p e a ion appea s essen ially in se e al books and a icles, including om many p es igious s a is icians, up o his e y day wi h mo e on he web, despi e Wes all's (2014) debunking o i . Fo example, a (2025) a icle in Psychology and Ma ke ing, "A Tu o ial on Wha o Do Wi h Skewness, Ku osis, and Ou lie s," by Iacobucci, Roman, Moon, and Rouzies, claims ha ku osis measu es peakedness/ la ness, independen o ou lie s, and ci es DeCa lo o his in e p e a ion. Speci ically, hey s a e, inco ec ly, I [excess ku osis] e lec s he ex en o which he dis ibu ion appea s peaked (lep oku ic, posi i e) o la (pla yku ic, nega i e) (DeCa lo 1997; see examples in he Appendix). DeCa lo (1997) makes he e o in he e y i s sen ence o he abs ac o his pape in he jou nal Psy hological Me hods, s a ing inco ec ly, Fo symme ic unimodal dis ibu ions, posi i e [excess] ku osis indica es hea y ails and peakedness ela i e o he no mal dis ibu ion, whe eas nega i e [excess] ku osis indica es ligh ails and la ness. This s a emen is clea ly alse. The cu en Wikipedia page (English e sion), gi es wo dis ibu ions, bo h symme ic and unimodal. The i s has ku osis <3 (“pla yku ic”) bu is in ini ely peaked, and he second has nea in ini e ku osis, bu appea s pe ec ly la - opped. The i s dis ibu ion is ob ained by mixing a Be a(0.5,1.0) wi h i s e lec ion abou 0, hen s anda dizing. The second dis ibu ion is ob ained by mixing a U(-1,1) and T(4.0000001), wi h p obabili ies 0.999 and 0.001, hen s anda dizing. Bu i ge s wo se o he “peakedness/ la ness” in e p e a ion. You can ha e en i e in ini e amilies o symme ic unimodal dis ibu ions, F q , indexed by a single pa ame e 0 < q < 1, all ha ing mean ze o and a iance 1.0, whe ein ku osis anges om 2.24 (“pla yku ic”) o in ini e, whe e he ku osis dec eases as peakedness inc eases, and whe e he dis ibu ions become mo e la - opped as ku osis inc eases. He e a e g aphs o ou dis ibu ions wi hin such a amily. These dis ibu ions a e de ined as ollows: The only logic e e gi en o he “peakedness” and “ la ness” in e p e a ions is ha he e exis amilies o dis ibu ions o which la ge ku osis implies peaked, and lowe ku osis implies la . Howe e , using ha same logic, one could as easily s a e as ollows, based on he abo e amily o dis ibu ions: Fo symme ic unimodal dis ibu ions, posi i e [excess] ku osis indica es hea y ails and la ness ela i e o he no mal dis ibu ion, whe eas nega i e [excess] ku osis indica es ligh ails and peakedness. Like DeCa lo's inco ec in e p e a ion, his in e p e a ion is also w ong, because examples do no p o e gene ali ies. A bea is an example o a mammal, bu no all mammals a e bea s. 2. Why Ku osis Measu es Tails I now gi e h ee ma hema ical p esen a ions es ablishing ha ku osis measu es ail weigh /le e age. 2.1 The ex emes comple ely de e mine la ge ku osis. In 2014, Wes all p o ed ha la ge ku osis is de e mined by he ail, speci ically mass ha is > b s anda d de ia ions om he mean, whe e b is a bi a ily la ge. Speci ically, he p o ed he ollowing: In pa icula , he a angemen o he densi y wi hin b s anda d de ia ions o he mean is i ele an o la ge ku osis. 2.2 The a angemen o he densi y wi hin 1 s anda d de ia ion has li le impo Since ku osis is E(Z4), i can be exp essed as he in eg al o z4 (z), o he a ea unde he g aph o he unc ion z4 (z). Taken om he c oss alida ed s ackexchange si e, he ollowing igu e shows g aphs o z4 (z) o a ious dis ibu ions. z I is clea ha he cha ac e o he dis ibu ion (z) wi hin one s anda d de ia ion (z be ween -1 and 1), has li le e ec on ku osis, since i is “dampened” by z4. Wes all (2014) p o ed ha he a ea unde hese cu es be ween -1 and 1 is always <1, and <0.5 o ypical classes o dis ibu ions. 2.3 Ku osis is p ecisely in e p e ed as “ ail le e age” o all ypes o dis ibu ions Gi en wo dis ibu ions (o da a se s), one ha ing la ge ku osis han he o he , wha can you say? The e is always a p ecise compa ison in ol ing ail weigh , and i goes as ollows. Suppose he andom a iables a e X and Y, and le hei s anda dized e sions be V and W, espec i ely. These may be disc e e, so include he case o sample da a. Suppose he ku osis o Y is highe , so ha E(W4 ) > E(V 4 ). Now, d aw he g aph o he dis ibu ion o V 4 . Since he mean is he poin o balance, he g aph balances a he ku osis o X, k(X). Now, d aw a g aph o he he dis ibu ion o W 4 , i balances a k(Y). So i you place a ulc um a k(X) unde he dis ibu ion o W 4 , he g aph will “ all o he igh .” Now, wha makes i “ all o he igh ”? Is i g ea e peakedness o concen a ion o he mean in he dis ibu ion o Y? No, hose aspec s would make i “ all o he le .” I is ins ead he ex eme alues in he dis ibu ion o Y ha a e esponsible. See he ollowing g aphs. The inal g aph illus a es he “ alling o he igh ” when he ulc um is placed a he ku osis o he da a se wi h smalle ku osis. 3. Highe Ku osis Does No Mean “Mo e in he Tails” o “Mo e in he Cen e ” Ku osis measu es ails, as shown by he “poin o balance” g aphs in he p e ious sec ion. Howe e , sou ces on he web ha e desc ibed highe ku osis as meaning “mo e p obabili y in he ails.” This in e p e a ion is also alse, as ail weigh shown in he g aphs is a kind o le e age, a combina ion o bo h mass and ex ension. Less mass, coupled wi h highe ex ension, can jus as easily cause highe ku osis, as shown by he ollowing example. In his amily, he ail p obabili y, q , ends o ze o while he ku osis simul aneously inc eases o in ini y. Ano he misin e p e a ion o ku osis, ela ed o he peakedness misin e p e a ion, is ha highe ku osis implies mo e p obabili y in he ange wi hin a s anda d de ia ion. This is also alse, as he ollowing example shows. 4. Visualizing Ku osis The densi y o his og am g aphs ha you ind online o illus a e ku osis di e ences a e ypically e y bad. People hink ha hea iness o ails should be isible in hese g aphs, and hey g ossly exagge a e he ail hickness so much ha he dis ibu ion ac ually looks ligh - ailed, like a uni o m dis ibu ion. O hey show g aphs ha simply di e in a iance. O hey show g aphs whe e he densi y is uni o mly la ge o highe ku osis, which is impossible because hen he a eas canno be 1.0. O hey show di e ences in peakedness, pe pe ua ing he my h ha ku osis measu es peakedness. Tails, e en when "hea y," a e s ill e y close o ze o, and hus a e no clea ly shown in his og am/densi y plo s. The poin o hea y ails is ha a e ela i ely a om ze o compa ed o he no mal dis ibu ion ails. Fo example, 10−3 is ela i ely huge compa ed o 10−100, bu bo h a e e y close o ze o in an absolu e sense, and canno be dis inguished in a densi y g aph. The solu ion is simple: S op using densi y/his og am plo s o illus a e ku osis. Ins ead, use no mal q-q plo s. Tail beha io is clea ly isible in q-q plo s, and he e is a di ec ma hema ical connec ion be ween he appea ance o hese plo s and he ku osis s a is ic, desc ibed as ollows: 5. Conclusion I has been mo e han a decade since I bu ied he “ku osis as peakedness” in e p e a ion. This a icle simply adds mo e nails o he co in. Re e ences DeCa lo, L. T. (1997). On he meaning and use o ku osis. Psychological Me hods, 2(3), 292–307. Iacobucci, Roman, Moon, and Rouzies, (2025). A Tu o ial on Wha o Do Wi h Skewness, Ku osis, and Ou lie s, Psychology & Ma ke ing. Pea son K. (1905). Das Fehle gese z und seine Ve allgemeine ungen du ch Fechne und Pea son. A Rejoinde . Biome ika. 1905;4:169–212. Wes all, P. H. (2014). Ku osis as Peakedness, 1905 - 2014. R.I.P. The Ame ican S a is ician, 68(3), 191–195.