"Jason Giambi went 1-for-4, flailing miserably in his first three at-bats before lining a ninth-inning single to right to end a 4-for-41 slump..."

This was taken from the Associated Press recap of Friday night's baseball game featuring the New York Yankees and Oakland Athletics. This type of reporting bothers me considerably. It is primarily an issue of sample size. Four at-bats is not enough data points to make the determination that Jason Giambi's slump is over. This is especially true when going 1-for-4; it is not that different statistically from getting 4 hits over 41 at-bats. We just cannot tell the difference between a .300 hitter and a .100 hitter from a 1-for-4 night. Let me explain this.

First, let's model the probability of whether a player gets a hit during an at-bat using a binomial random variable. For those of you who don't like math, you can stop reading now...or you can think of this as replacing the player with a coin. Heads and he gets a hit. Tails and he heads back to the bench. In this scenario, we are not dealing with a fair coin meaning that the probability of getting a hit (heads) is not 50%. For most players, it is much lower than half—somewhere between 25% and 30%. We can think of a slump as replacing the normal coin for that player with one that is even more weighted towards making outs. Breaking the slump would mean returning to the normal coin. In Giambi's case, let's say that his normal coin returns a hit 30% of the time since his career average is .295. In his current slump, he is getting hits around 10% of the time. Now the question is how certain can we be that he has gone back to the normal coin (broken the slump) given the evidence of a 1-for-4 game.

We plug the numbers into our handy binomial distribution formula and get some answers. A player who gets hits 10% of the time will get a single hit in four at-bats about 30% of the time. A player who hits around .300 will go 1-for-4 around 40% of the time. Given the small sample size, it is basically impossible to tell the difference between a .100 hitter and a .300 hitter if they both have a 1-for-4 night. The same holds true for telling whether Giambi has broken his slump or not. Now, if he had gone 2-for-4 or 3-for-4, the numbers would have come out differently. In four at-bats, a .300 hitter is about five times more likely to get two hits, twenty times more likely to get three hits and eighty times more likely to get a hit every time. Then maybe we could do a hypothesis test to determine how much confidence we have that Giambi broke his slump. As it is, I am expecting an 0-for-4 today.