Are MLB player skills normally distributed?

There's a paper in the most recent JQAS on evaluating outfielder and cather throwing arms. It's called "Evaluating Throwing Ability in Baseball," by Matthew Carruth and Shane Jensen.

I'm still going through it, but the first thing that struck me was that the authors assumed that throwing skills in the baseball population are normally distributed.

That seemed to be wrong. In the 1985 Abstract (Blue Jays comment, page 113), Bill James argued, convincingly, that major-league player skills should be shaped not like the normal bell curve, but like the right tail of that curve. The general population is shaped like the normal distribution, but only the very best players make the majors. Those are the ones at the extreme right tail, which doesn't have a bell-curve shape at all.

So shouldn't outfielder and catcher arms also be shaped like the right tail? If they are, the Carruth/Jensen study shouldn't work at all.

After thinking about it a bit, I wonder if individual skills might not be close to normal after all.

Suppose a player's overall talent was the sum of 1,000 independent skills, each of which is normally distributed in the population. Wouldn't each of those skills be almost normally distributed in MLB? Consider skill number 502. If you were average in skill 502, it would barely affect your chance of making the majors – you'd have 999 other skills to be good at. And so if you took a MLB-wide profile of skill 502, it would barely be different from the general population. The same, of course, would be true for each of the other skills. Each skill would look very close to normal, but the *sum* of all skills would be shaped like Bill James' right tail.

Of course, the example of 1000 different skills is farfetched. But, it turns out, the "right tail" disappears much sooner than 1000. It goes away also with a much more realistic model of baseball skills.

Suppose there are three basic skills: hitting, range, and throwing. All three are normally distributed in the general population with mean 100 and SD 16 (which I think puts them on the same scale as IQ). All are independent. A player's overall value is the sum of 70% of his hitting score, 20% of range, and 10% of arm. Any player with an overall score of 146 or more makes the majors.

Under those conditions, you'd still expect the MLB distribution of overall score to be shaped like the right-tail of the normal distribution. But what about the three individual skills? Will they be right-tail shaped, or bell shaped? Probably batting, being 70% of the total, will be tail shaped. But what about range? And arm?

To check, I ran a simulation. I created a general population of 2,000,000 baseball players (I would have used more, but the VB random number generator apparently started repeating). Of those two million, only 87 players made the majors.

I then plotted graphs for those 87 players. As you would expect – and as the simulation forced – the overall rating of those 87 players looked like the right tail. 29 players scored at 146. Only 18 scored at 147. 12 scored at 148. 9 were at 149, 11 at 150, and then they trickled off, so that only 8 were between 151 and 161. Definitely a right-tail picture.

That was the overall rating. So I then plotted just batting. And what happened? The distribution looked more like a normal curve! Even though batting comprised 70% of the "right tail" rating, that 70% was bell shaped!

(I'm not good enough with HTML to show the curve here, but I've posted it on my website. Take a look here.)

The other two skills – range and arm – looked even "more" normal. They're shown in the above link also.

I didn't run any formal statistical tests for normality. Indeed, it's probably easy to show that none of the three skill distributions should be normal. But they're *approximately* normal, pretty good bell shapes. A normal distribution has a "skewness" (measure of symmetry) of zero, and a "kurtosis" (measure of peakedness) of 3. Here are the stats for these four curves:

Normal : Skewness = +0.0, Kurtosis = 3.0
----------------------------------------------------------
Overall: Skewness = +2.5, Kurtosis = 11.5
Batting: Skewness = +0.4, Kurtosis = 3.6
-Range : Skewness = -0.1, Kurtosis = 2.8
-- Arm : Skewness = +0.1, Kurtosis = 2.4


The above numbers don't tell you anything the graphs don’t – they're just a numerical way of summing up the pictures. I have no idea if the last three are statistically significantly different from normal (0.0, 3.0), but they're pretty close in real-life terms.

I can't help but conclude that most individual player skills, so long as they're not overwhelmingly correlated with the player's overall value, could be pretty close to normally distributed.

But a few points. First, I assumed that all three skills were independent of each other in the general population. In real life, that's obviously not true: good athletes will have both good range and good arms. The more correlated the skills, the more likely they'll be in line with overall value, which is right-tail shaped. So that might change things.

Second, the assumption was that an arm of –2 SDs is worth twice as much (badness) as an arm of –1 SD. That's again not true: a minus-two player can be taken advantage of by baserunners, and can therefore cost his team three or four times as much as the minus-one guy. That might mean there are fewer –2s in real life than the model, which would make the distribution look more right-tailed.

(Finally, my sample may be too big. 87 out of two million is the equivalent of some 4,000 players in the U.S. male population of baseball-playing age. Throwing out half the players may make everything more right-tailed. Hang on, let me check ... nope, results stayed roughly the same. Never mind!)

By the way, a few more interesting notes from the sample of 87:

1. The players had an average batting rating of 163. The average range rating was only 119. The average arm rating was barely above average at 105. This is as you'd expect: if you can hit 4 SDs above average, you're a major leaguer, no matter how bad your arm. But if you can throw 4 SDs above average, so what? Unless you can hit, your arm just isn't that valuable to the team.

This works for other sports too – in golf, as I understand it, there are guys who can drive a ball 400 yards. But the other aspects of their game aren't very good, so they're doing driving contests for a living instead of playing on the PGA tour.

2. The best player in the sample, with an overall 161, had only an 88 arm. Again, this makes sense. It would probably take months of going through people at the mall before you found one who can hit like a major-leaguer outfielder. But you could probably find some guy who can throw like a major-league outfielder much more easily.

3. There was a high negative correlation (r = -0.75) between hitting and range. That makes sense too. Players who make it to the majors with their bat don't have to have good range to keep their jobs. And players who earn a job with their glove are unlikely to be among the best hitters.

The correlation between range and arm was –0.14, and between bat and arm was –0.18.

4. Every one of the 87 players had a better hitter rating than his overall rating. This is probably indicative of the study being oversimplified, since there are numerous players in MLB whose value comes mostly from defense. A better study is probably called for, but I think the conclusion – that individual skills may indeed be normally distributed – is still supported.