# Normal Distribution and Sports

One core idea in sports analytics is to be skeptical of hot streaks as being evident of anything more than just something a normal distribution would account for.

What I've never understood, is would a normal distribution, as applied to sports, imply that luck randomly fluctuates, that a player's skill or ability level at any given time randomly fluctuates around some true, underlying mean, or some combination thereof?

• Yes: luck, by definition, randomly fluctuates. A player's skill level, on the other hand, does peak at a certain age and thereafter decline, but on any given performance randomly fluctuates around some mean value. Feb 5 '15 at 23:39

This is starting to head into territory that might be better handled at CrossValidated, but it's interesting here nonetheless.

One concept that's important in statistics is that even if a random variable is not normally distributed (could be uniform, binomial, gamma, who knows), a sample of its means will be normally distributed. Thus, while you cannot assume that a random sample of 50 at-bats from a player will appear normal, you can assume that if you take 50 samples of 50 samples that they will be approximately normally distributed, assuming the basis for the data doesn't change over time.

This, of course, isn't an assumption you can make over a long period of time - 2500 at bats I'm sure a player will change over - but over a shorter period of time you probably can, if the player doesn't get injured and isn't in the very beginning or very end of his career.

This is used by assuming a player's performance that year/etc. is a random sample from the inferential (but imaginary) population of all at-bats he might take, which would define his 'true' ability level. If a player is batting .280, then the odds of his true ability level being .300 or .260 under a naïve assumption would be identically likely (and, based on the sample size, could be determined.) Further, you could use other samples of his batting ability (say, he batted .250 last year, and .270 the year before that, and .230 the year before that) to determine what the likely overall skill level is - again assuming, naïvely, that his skill level is constant. In this example, the .280 sounds like an outlier on the high end - a hot streak, so to speak - since his performance has fluctuated up and down around an average of .260 or so.

However, that's a very naïve assumption, and is certainly well behind where we stand in sports analytics. The normal distribution assumption is still present in certain elements, but various people have used a much more powerful tool to determine what factors have predictive value as to future performance in other areas than simply a mean of batting averages.

Among other things, with just a simple confidence interval, you can't assume batting average has any effect on other elements of performance (including scoring runs, etc.); instead, sports statisticians have used various forms of regression analysis to determine how the different statistics available correlate with important performance elements (particularly, scoring runs, preventing runs from scoring, and winning games, for baseball as an example). That gives you more ability to identify whether a player is likely on a hot streak, or whether his performance has truly improved - separating the Mike Trouts of the world from the Chris Davises.

You're partially correct, but I think you're not applying the concept of a normal distribution in quite the right way for the context. In the context of hitting streaks, it would be more appropriate to look at the normal distribution as related to hitting streaks. In other words, assuming that hitting streaks follow a normal distribution with some mean indicating the average length of a hitting streak (say, 10 games). If you assume that hitting streaks are normally distributed, then to see a hitting streak of, say, 30 games won't happen very often, but it will happen. And even hitting streaks of 56 games, like Joe Dimaggio's record, will happen, just very, very rarely.

Assuming that hitting streaks follow a normal distribution does not, however, imply that a player's skill necessarily fluctuates. If you view batting average as a measurement of skill, for example, you can draw a parallel between batting average and probability. If a player has a batting average of .300, the parallel is that they have a 30% chance of getting a hit. If an event (at bat) has a 30% chance of success, and the event happens 3-4 times per game, you can calculate the expected "hitting streak" (i.e. minimum 1 success per game). That would be the mean of your distribution. You can also calculate the standard deviation of the streak and that gives you a normal distribution.

So we can still view a player as having a fixed skill level (which may or may not be correct) and it doesn't contradict the idea that long hitting streaks are expected to happen given the probability distribution of hitting streaks. I also think that doesn't necessarily take away from the value of a long hitting streak. A player with an "underlying skill" (i.e. Batting Average) of .300 is still much more likely to have a 20+ game hitting streak than a player with an "underlying skill" of .200.

• So if the "true" .300 hitter (as determined by whatever means) hits .320 for a month, are you sure the difference from the mean is truly randomness (luck), or is his skill level temporarily higher, for any number of possible reasons, physical, psychological, etc?
– Chas
Feb 12 '15 at 19:29
• With baseball, it's very hard to say whether it's luck or improved skill that causes fluctuations in statistics. You can do some analysis of how the player handles different situations (batting counts, certain pitches, etc.) to see if there are changes, but that's very tough. The point that most sabermetricians are making is that because of the number of batters in the league and the large number of games being played, if you assume hitting streaks are more or less normally distributed, you'd expect the occasional 25+ or higher streak. Feb 12 '15 at 21:57
• Of course, I'm well aware. I've just never felt like there was a clear explanation for the source of the randomness. Some writers and analysts treat it as if it is luck, but, that doesn't pass the smell test for me. But maybe that's just bias.
– Chas
Feb 13 '15 at 0:15
• Well, my interpretation of that particular viewpoint is that it's not "luck", it's just that even a .300 hitter will have times when they hit .200 or even .100. You don't expect a .300 hitter to get 3 hits out of every 10 exactly. I guess a probability view is flipping a coin. A coin is 50/50 heads/tails. That doesn't mean you expect the coin to alternate heads and tails. In probability theory, if you flip a coin long enough, you actually expect that any length streak of heads WILL happen eventually, no matter how long the streak is. I guess this is somewhat similar. Feb 13 '15 at 2:05

One core idea in sports analytics is to be skeptical of hot streaks as being evident of anything more than just something a normal distribution would account for.

That isn't true. Successes of three point shots for example, don't follow a normal distribution. They are discrete trials, each with two possible events as outcomes: success, or failure. This is best modelled using a binomial distribution. This approaches the normal distribution if you have enough samples, but for something like a hot streak of 3 pointers, you don't usually have enough samples for the binomial to start looking really like a normal distribution.

• ok, but a series of samples of a binomial distribution (say, sets of 10 3s in a row) will approximate a normal distribution. My point anyway is about randomness and fluctuation in sports. Is the randomness only of luck, or of skill/ability as well? The topic is often treated as only randomness in luck, but I am skeptical that's right.
– Chas
Feb 12 '15 at 19:25