Last week I posted a note on Facebook about Red Sox shortstop Jose Iglesias. According to Baseball Reference, the light-hitting Iglesias has a career WAR

^{1} over three times larger than slugging third baseman Will Middlebrooks, despite having less than half as many at bats. This season has been particularly rough on Middlebrooks who sits at a WAR of minus 0.7 with a batting average hovering around the Mendoza line. In contrast, Iglesias has been (relatively speaking) tearing the cover off the ball by hitting well over .400 with a WAR of +1.8, all while playing stellar defense. To put this in context, Iglesias's stats, if extended over a 162-game season, would give him a WAR of 10.4, which is about 50% better than Miguel Cabrera's WAR during his MVP Triple Crown season last year. This inspired my buddy Adam to ask me about sample sizes, so I figured I should address this on the blog.

**In baseball, when should a sample size be considered significant?**
First off, it should be stated that there are no exact cutoffs in probability and statistics. As I explain in

*Ballparking*, even a career .200 hitter like Mario Mendoza has a (small) chance of hitting .400 over the course of an entire season. There's no magic number above which we can definitively say, "These results are statistically significant." Fluctuations happen in any sample size no matter how large. That said, if we have a random sampling of statistically independent events, we

*can *make definitive statements like the following:

^{2}
*There's a 95% chance that Jose Iglesias's average over his next 92 at bats will be between X and Y.*

Here, X and Y define what's called a

confidence interval. We have limited data, but given the information we do have, we're 95% percent certain that Iglesias's batting average over the next 92 at bats will be between two numbers X and Y. What are those two numbers? Wikipedia's entry for

sample size determination gives a good description of how to calculate them. The width

*W* of the confidence interval is given by

where

*n* is the sample size, i.e. the number of at bats. Since Iglesias has had 92 at bats so far, we have

*W = *0.045. Iglesias is currently batting .435. If we believe Iglesias's stats represent a random unbiased sample, then we would expect there to be a 95% chance Iglesias's next 92 at bats will give a batting average between .412 and .457.

Did I make a math mistake? Is Jose Iglesias the next Ted Williams? The problem lies in the fact that Iglesias's 92 at bats were not selected at random. I noticed Iglesias's batting average

*because *it was incredibly

* *large. If I took any 750 baseball players and gave them each 92 at bats, there's a good chance some of them would, by shear dumb luck, hit over .435. Even at a 95% confidence ratio, you'll still have 5% of players with batting averages that lie outside the confidence interval. Given 750 Major League players, this means roughly 38 players will be outside the confidence interval. Iglesias's .435 average is almost certainly an

outlier due to random fluctuations in the large population of baseball players. The only reason I singled him out is because he randomly (and luckily) happened to have one of the largest and therefore most attention-drawing fluctuations.

Despite the fact that Iglesias's average is almost certainly a random fluctuation, the shortstop still shows a lot of promise. If we look at his 2012 season during which he hit an abysmal .118 in 25 games, we notice he

*still *nets a positive 0.3 WAR. Extended over a 162-game season, he would get a not terrible WAR of 1.9. Why? His defensive capabilities more than adequately compensate for poor hitting. Over his career, he's averaging a 5.4 WAR per 162 games, which is more than double the 2.6 WAR averaged by current starting shortstop Stephen Drew and over seven times greater than 0.7 WAR averaged by current starting third baseman Will Middlebrooks. Even given the small sample size, it's tough to argue that Iglesias doesn't deserve a spot in the starting lineup.

If you like math and sports or know someone who does, make sure to check out my book

*Ballparking: Practical Math for Impractical Sports Questions*.

*Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions. Follow him on Twitter at @aarontsantos.*
[1] "WAR" is one of those newfangled stats that sabermetricians like to throw around. It stands for "wins above replacement" and is supposed to represent the number of extra wins a player is expected to contribute compared to a standard replacement player.

[2] Strictly speaking, it's a bit more complicated if we're talking about actual baseball players rather than mathematical probability distributions. For example, a player's theoretical batting average is not constant over time. It can increase or decrease depending on the player's age or health.