Hi all,

I'm reading Statistical Sports Models in Excel Vol. 1 by Andrew Mack, and had a question about how he is estimating some theoretical prediction ceilings for a sport.

All sports have random variance and he argues that a sport is only knowable/predictable up to the percentage of that sport that's attributable to skill rather than randomness. Makes sense to me.

So he starts with an equation where Observed Variance = Variance Due to Skill + Variance due to randomness, and backs into Skill Variance after calculating the other two variables. Again, this makes sense.

Using the NBA winning percentages as an example, he has Observed Variance = variance in league winning percentages over a given period.

To calculate random variance, he uses the formula of (Prob of An Average Team to Win * Prob of An Average Team to Lose)/# Games in a season, or (.5 * .5)/82 with the understanding that an average team is a .500 team and would have a 50/50 chance against another average team, all else considered.

I'm an idiot but apparently he is using the binomial approximation to normal, per this linked blog post, to calculate the theoretical % of variance that is due to randomness. And I do not understand *why* this formula represents the variance due to randomness.

Is there a quick and dirty explanation for why this is the case? Thank you for any help you can provide.

This is quick, dirty, and very possible wrong, but here's my attempt:

Mack credits Tango and Birnbaum for the idea, but I don't remember seeing a formal proof in their writings as to why the formula works. My guess as to why they think it works is related to their idea that the results of a single game are completely random, but the results of several games are not. (I'm paraphrasing, you'd have to explore their works further for a justification of that theory.)

Keeping that theory in mind, remember that we calculate the variance of the Binomial distribution as Np(1-p) where N is the number of trials and p is the probability of a successful trial occurring. Next, notice that the authors calculate the amount of variance due to randomness as p(1-p)/N. I guess that is their way of isolating the amount of variance due to the "pure randomness" of a single trial. So once they have that number, they just subtract it from the observed variance to isolate the amount of variance due to skill.