How many wins can you attribute to skill, and how many to luck, in the course of a baseball season?
Baseball is a game of luck and skill, and, fortunately, we can measure both quite accurately. The amount of luck involved is probably higher than most people think, and it’s higher than the Lords of Baseball would like to admit.
First, let’s look at team performance over a season. Take 16 evenly balanced teams that have an equal likelihood of winning or losing any game. The actual results should show a normal distribution with a mean of .500, with some variation. A common measure of variation is the standard deviation, called sigma, which you can calculate by simply taking the sum of the squares of the difference between the team’s actual wins and the expected value, which would be 81 wins in a 162-game season. For a simple example, take four teams with wins of 89, 84, 77, and 73. The mean is 81 and the sum of the squares is 64 + 16 + 16 + 64, or 160. Divide 160 by the number of teams (4) and take the square root, and you get ≈ 6.
Calculating the expected sigma for team wins involves a formula derived for a binomial distribution. The formula is the square root of p x q x n, where p is the probability of success (.5), q is the probability of failure (also .5), and n is the number of samples (162). This gives a value of the square root of .25 x 162 (40.5), or 6.36.
Another characteristic of sigma in a normal distribution is that approximately two-thirds of the ...