#### Introduction

As we all know, 2019 was an interesting baseball season due to the abundance of home runs. For example, 58 players hit 30 or more home runs in 2019 which was remarkable. Here I am going to explore a more subtle pattern — which of these 30+ home run hitters were unusually streaky this season? (We’ll also find the hitters who were unusually consistent in 2019.)

#### What is Streaky?

Let me explain what I mean by streaky. Baseball hitters go through periods of good and bad performance. In this context, hitters will appear “hot” and hit a good number of home runs in some weeks. In other weeks, they will seem “cold” and go through periods with no home runs. The question is: how does their pattern of home run hitting deviate from what one would see in coin tossing? We’re making a distinction between a player’s observed streakiness and the player’s true streakiness. If a player’s observed streakiness resembles the same patterns as repeated flips of a coin, then the player is not truly streaky. Instead, if the chance of a home run in a PA depends on the player’s performance in recent PAs, a player is truly streaky.

#### Coin Tossing

In a typical computer game, a particular player is programmed to have a home run chance of, say 0.05, in a plate appearance. In this game, the computer simulates a coin flip where the chance of heads is 0.05 — if one observes heads on the simulated coin flip we’ve observed a home run. The outcomes of these coin flips are independent in the sense that the chance of a home run for one flip is not dependent on what happens in previous flips. If one simulates the performance of this computer baseball hitter for, say 600 PA, we’ll see some streaks of good home run performance and some home run slumps. Can we distinguish the pattern of streaks and slumps of this simulated player from the streaks and slumps of home run hitting of a real hitter?

#### Is Mike Trout Truly Streaky?

To illustrate our method of detecting “real” streakiness, consider Mike Trout. In the 2019 season, Trout had 45 home runs in 586 plate appearances with a HR rate of 45 / 586 = 0.077. Looking at his gaps between home runs, we notice that Trout had a “ofer” of 53 PA — that is, a streak of 53 PA without a home run. Is this unusual?

To answer this question, we consider a “Trout coin”, that is a coin where the probability of heads is 0.077. We simulate 586 flips of this Trout coin and record the ofers, that is, the gaps between successive home runs. We record the length of the longest ofer. If we repeat this simulation 1000 times, we get the following distribution of the longest ofer for 586 flips of the Trout coin.

Remember the 2019 Trout’s longest ofer was 53 PAs. Looking at this distribution we see that Trout’s streakiness (as measured by the longest ofer value of 53) is **not extreme** relative to a Trout coin. So Mike Trout’s pattern of hitting home run is not unusual — it is pretty consistent with the pattern of streakiness of a Trout coin.

#### Streakiness of All 2019 Sluggers

Were any of the 2019 sluggers unusually streaky? I repeated this procedure for each of the 58 sluggers who had 30 or more home runs in 2019. Using the same number of PA and chance of home run success, I simulated coin flipping for each player, recording the maximum length of the ofer. I constructed a distribution of maximum ofers from repeated simulations and looked to see if the player’s maximum ofer was extreme relative to this distribution. My measure of extreme is the p-value, the chance of observing a maximum ofer at least as large as the observed measure. A small p-value means the player is unusually streaky (relative to coin flipping) and a large p-value (close to 1) means the player is unusually consistent (relative to coin flipping).

Here I graph the p-values from smallest to largest. The streakiest home run hitter among this group was Austin Meadows with a p-value close to 0. Looking at his ofers, we see that he had a remarkable streak of 158 PA without a home run. On the other extreme, the most consistent HR hitter using this criterion was Bryce Harper. His maximum ofer was 49 which is unusually small relative to coin flipping — Harper’s p-value was close to 1. Mike Trout’s p-value was close to 0.5 which means that his pattern of home run hitting (as measured by the maximum ofer) resembles the pattern of streakiness of a Trout coin.

#### Some Takeaways

Here are some observations from this activity.

**Is the coin flipping model realistic?**When I simulated the data, I am assuming that the chance of a home run is constant for each plate appearance. This is clearly unrealistic. I believe a player’s chance of home run will change during the season and certain change against different pitchers.**Is the coin flipping model helpful?**Although this model is a oversimplification of reality, I think this model is useful for making sense of sizes of ofers in home run hitting. For example, in our exploration of the 58 sluggers, most of the largest ofers seem consistent with the largest ofers of random coins.**Streaky measure?**To measure streakiness, we used the maximum length of the ofers, that is, the gaps between successive home runs. Other measures of streakiness could be used. For example, in the R code with link below, I try using the standard deviation of the ofer lengths as a measure of streakiness.**Streaky hitters?**We did find several players who displayed patterns of home run hitting that were unusual relative to coin flipping. But I wouldn’t get excited about these players until I explored their streakiness tendencies in other seasons. Actually, I did this type of study some years ago and showed that Hank Aaron was generally very consistent (that is, more consistent than coin flipping) in the pattern of his home run hitting in his career.**Who cares?**Who cares about the streakiness tendencies of hitters? I think most managers are sensitive to streakiness. A manager often has to decide on who to pinch hit in a given situation and he wants to choose the player who is most likely to exceed. This decision may be based on the perceived “hot” or “cold” behavior of players on his bench. In other words, the manager’s opinion about true streakiness may guide his decision making during a game.**Got code?**All of the R code, including a link to the source Statcast data, can be found on my Github Gist site.

Jim,

Would love to know the process you use for putting together a blog entry. Do you author your posts inside RStudio or are you using something else?

Thanks.

Darren Triplett

Atlanta, GA

Darren, I don’t do anything fancy currently. It is pretty easy to write in the WordPress system and I paste in graphics and output. There are packages that will automatically create blog postings from Markdown files, but I haven’t found them that simple to use. But now that I’m having some success with the bookdown package, I should try out the blogdown package.

Hey Jim, I’m curious why you simulated coin flips instead of computing the negative binomial distribution and deriving p from it.

Maxwell, you’re right — I could have simulated negative binomial spacings. But I think my simple coin tossing was more descriptive for what I was doing.