How the Participation Gap Biases Group Evaluation

It is misleading to use the top performing individuals to compare groups of unequal sizes. Say you wanted to know whether men or women are better at chess or which country has the best athletes; using the top performers as representatives for each group (gender or country) would bias the evaluation simply because of group size. All else being equal, if for example only 17% of rated chess players are female, it is much less likely to find the best performer in this group.

Here is the thought experiment to show the obvious intuition that the larger group has a higher probability of providing the top performer:

Try it in R:

sizeA <- 10
sizeB <- 2

M <- matrix(c(rep("A", sizeA), rep("B", sizeB), rep(NA, sizeA+sizeB)),
            nrow = sizeA+sizeB, ncol = 2)
M <- as.data.frame(M)
names(M) <- c("group", "score")


n <- 100000
maxA <- rep(NA, n)
maxB <- rep(NA, n)


for (i in 1:n){
  M$score <- sample(1:100, nrow(M), replace = T)
  maxA[i] <- max(M$score[M$group=="A"])
  maxB[i] <- max(M$score[M$group=="B"])
}
  
round(mean(maxA), 1);round(mean(maxB), 1)