# t-distribution Kurtosis inconsistencies

Please feel free to correct me if i’m wrong, but to my knowledge a distribution that is platykurtic exhibit the following attributes: An excess kurtrosis < 0, being Less peaked and (thin tails) The t-distribution, however, is described by : It is less peaked than a normal distribution, with more probability in the tails (fatter tails). so my question is how could both exhibit a kurtosis of <0 ( being less peaked ), while one has thin tails and the other fatter tails ??

T- distribution has positive excess kurtosis (proportional to 1/df, but I’m too lazy to look it up exactly).

Why would it have a postive excess kurtosis ? When compared to the normal distribution, the t-distribution is flatter with more area under the tails (it has fatter tails). As the df for the t-distribution increases, however, its shape approaches that of the normal distribution. It is also described for being less peaked than a normal distribution. Therefore, according to the the definition of kurtosis (a less peaked distribution with thin tails = a platykurtic distribution -i.e. negative kurtosis), logically t-distributions should have thinner tails not fatter tails.

darkvont Wrote: ------------------------------------------------------- > Why would it have a postive excess kurtosis ? > Alright I looked it up. It’s 6/(df - 4) > When compared to the normal distribution, the > t-distribution is flatter with more area under the > tails (it has fatter tails). Yes > As the df for the > t-distribution increases, however, its shape > approaches that of the normal distribution. Yeah, sort of. > It is > also described for being less peaked than a normal > distribution. OK > Therefore, according to the the > definition of kurtosis (a less peaked distribution > with thin tails = a platykurtic distribution -i.e. > negative kurtosis), logically t-distributions > should have thinner tails not fatter tails. Check out that definition again. The T-distn is leptokurtic because it has fatter tails. The t-distn is defined as the distn of Z/Sqrt(chi-square/df) and then we work out stuff about kurtosis.

JDV, please look at the figures in first two links and then the fig in the third link (fig from cfa textbook). 1. http://mvpprograms.com/help/mvpstats/distributions/SkewnessKurtosis 2. http://www.public.asu.edu/~jwang2/portfolio/statistics/chapt6/chapt6.htm 3. http://picasaweb.google.com/8spirit86/Leptokurtic#5244340653501617314 When we refer to distribution having fat tails, we compare the tail from a given distribution with tail from normal distribution? Leptokurtic: Lepto- means ‘slim’ in Greek and refers to the central part of the distribution. Won’t a leptokurtic distribution have a thin tails and Platykurtic have a fat tails?

First off, these words suck and that’s why they are only used in things like CFA exams. Statisticians don’t use words like leptokurtic. Anyway, leptokurtic means fail tailed. It’s thin in the center so it has to be fat in the tails.

So if you want to play with the cool kids (statisticians) you have to call it excess kurtosis. platy & lepto are just, like *so* uncool.

Right. Totally, like, yesterday brother.