Binary Bet Skewness and Excess Kurtosis

[Ganchrow]

With all the recent rehashing of the relationship of variance and betting odds, some might be curious how a bet's characteristics affect its higher order moments.

Well it turns out that both skewness and kurtosis can be expressed as functions of win probability alone, without regard to odds, edge, or even bet size (this because by convention skewness and kurtosis are standardized measures). Results are as for the binomial distribution with n = 1 and trivial to prove via the definitions of each and a little simple algebra.

Skewness:

γ₁ = (1 - 2p)/ √p*(1-p)

Excess Kurtosis:

γ₂ = 1 p + 1 (1-p) - 6

Notable about both is that they're symmetric about p = ½, with only a change in sign for γ₁ (negative/left skew for p > ½, positive/right skew for p < ½).

p = ½ ± 1/√12 yields a mesokurtic bet, with kurtoisis going negative as p goes to ½, where the bet is minimally kurtotic, γ₂ = -2.

[LT Profits]

He's baaaaaaaaaaaaaaaack!

Time to dust off the old thesaurus, or at the very least brush up on my Ganchenese.

[Dark Horse]

Note to self:

Skewness and Kurtosis

A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.

[Dark Horse]

Ganch, what do you mean when you say 'a bet's characteristics affect its higher order moments'?

And what are the practical uses of skewness and kurtosis in sports betting? What does the use of these formulas result in, in everyday betting?

[Wrecktangle]

Skewed distributions are what usually drive you to using medians as measures of central tendency vs means. Selections drawn from "peakier" Kurtosis are more valuable vs those from broader, "flatter" dists. Both can bite you when combining dists. as you can wind up with bi/tri/multi modal distributions depending upon how many freq. dists. are involved.