## xHR%: Questing for a Formula (Part 2)

*Part 2 of a series of posts regarding a new statistic, xHR%, and its obvious resultant, xHR, this article will examine formula 1. The primer, Part 1, was published March 4.*

As a reminder, I have conceptualized a new statistic, xHR%, from which xHR (expected home runs) can be derived. Furthermore, xHR% is a descriptive statistic, meaning that it calculates what *should *have happened in a given season rather than what *will* happen or what *actually* happened. In searching for the best formula possible, I came up with three different variations, all pictured below with explanations.

*HRD – Average Home Run Distance. The given player’s HRD is calculated with ESPN’s home run tracker.*

*AHRDH – Average Home Run Distance Home. Using only Y1 data, this is the average distance of all home runs hit at the player’s home stadium.*

*AHRDL – **Average Home Run Distance League. Using only Y1 data, this is the average distance of all home runs hit in both the National League and the American League.*

*Y3HR – The amount of home runs hit by the player in the oldest of the three years in the sample. Y2HR and Y1HR follow the same idea. In cases where there isn’t available major league data, then regressed minor league numbers will be used. If that data doesn’t exist either, then I will be very irritated and proceed to use translated scouting grades.*

*PA – Plate appearances*

*(Apologies for my rather long-winded reminder, but if you really forgot everything from Part 1, then you should really invest in some Vitamin E supplements and/or reread the first post.)*

The focus formula of this post is the first one, which also happens to be the one I think will work the least well because it relies too heavily on prior seasons to provide an accurate and precise estimate of what should have happened in a given season.

In the second piece of the formula, with only fifty percent of the results from the season being studied taken into account, it likely fails to take into account the fact that breakouts occur with regularity. As a result, it probably predicts stagnation rather than progress.

### Methodology

Luckily for myself and the readers, the process was an incredibly simple one. Pulling data from FanGraphs player pages, ESPN’s Home Run Tracker, and various Google searches, I compiled a data set from which to proceed. From FanGraphs, I collected all information for Part Two of the formula, including plate appearances and home runs. Unfortunately, because a few of the players from the sample were rookies or had fewer than three years of major league experience, I had to use regressed minor league numbers. In some cases, where that data wasn’t applicable, I dug through old scouting reports to find translatable game power numbers based off of scouting grades (and used a denominator of 600 plate appearances).

Then, from ESPN’s amazingly in-depth Home Run Tracker website, I obtained all relevant data for player home run distance, average home run distance for the player at home, and league average home run distance. Due to my limited time, I only used players that qualified for the batting title during the 2015 season, yielding an iffy sample of only 130 players. Additionally, before anyone complains, please realize that the purpose of my research at this point is only to obtain the most viable formula and refine it from there.

### Results

Using Microsoft Excel, I calculated the resultant xHR% and xHR. Some key data points:

*League Average HR% (actual): 3.03%*

*Average xHR%: 2.85%*

*Average Home Runs: 18.7*

*Expected Home Runs: 17.7*

*Please note that there is a significant amount of survivorship bias in this data. That is, because all of these players played enough to qualify for the batting title, they are likely significantly better than replacement level, which is why the percentages and home runs seem so high.*

Clearly, the numbers match up fairly well, with this version of the formula expecting that the league should have hit home runs at a .18% lower clip, and one fewer per player, which amounts to a significant difference. Over the course of a 600 plate appearance season, the difference between them is still only a little more than one home run, an acceptable distance.

*Correlation between xHR% and HR%: 0.960506092 *

*R² for above: 0.922571953 *

*HR% Standard Deviation: 1.5769373 *

*xHR% Standard Deviation: 1.3883746 *

*Correlation between xHR and HR: 0.966224253 *

*R**² for above: 0.933589307 *

*HR Standard Deviation: 10.43771886 *

*xHR Standard Deviation: 9.201355342 *

While xHR% using this formula apparently explains about 92% of the variance, correlation may not be the best method of determining whether or not the formula works adequately. This holds at least for between xHR% and HR%, because there’s only a minuscule difference between their numbers (but one that matters), meaning it’s not a particularly explanatory method and that it may not have the descriptive power I’m looking for. Nevertheless, it is important to note that the correlation is not a product of random sampling, as p<.005. Unsurprisingly, the standard deviation for xHR% is smaller than that of HR% (nearly insignificantly so), indicating that the data is clumped together close to the mean as a result of using this formula, a potentially good thing (in terms of regression).

A better indicator of the success of the formula is the correlation between xHR and HR, a relatively high value of ≈.97. Here, presumably because the separation between home runs and expected home runs is greater, the formula ostensibly explains approximately 94% of the variance in outcomes and resultant data. However, in this case, the standard deviation for actual home runs is about 10.4, while for xHR it’s about 9.2, suggesting that, after being multiplied out by plate appearances, xHR is spaced nearly as evenly as HR. Ergo, it likely serves as a decent predictor of actual home runs.

### Players of Interest

Mr. Bryce Harper – It’s likely there isn’t a better candidate for regression according to this formula than Bryce Harper, who the formula says have hit only 32 home runs as opposed to his actual total of 42. While he did lead his league in “Just Enough” home runs with 15, he’s also always been known for having prodigious power (or at least a potential for it). Furthermore, Mr. Harper dramatically changed his peripherals last season to ones more conducive to power. Suggesting this are the facts that he increased his pull percentage from 38.9% to 45.4%, his hard hit percentage from 32% to 40%, and his fly ball percentage from 34.6% to 39.3%. On their own, all of the previous statistics lend credence to the idea that Harper changed his profile to a more home-run-drive one, but when taken together they significantly suggest that. His season was no fluke, and the formula certainly failed him here because it weighted prior seasons far too heavily.

Mr. Brian Dozier – No surprises here. Mr. Dozier has certainly been trending upward for a long time, and in a model that heavily weights prior performance such as this one, upticks in performance are punished. Nevertheless, the data vaguely supports the idea that Dozier should have hit 24 home runs instead of 28. While he did significantly increase his pull percentage to an incredibly high 60% from 53%, he did play in a stadium where it’s of an average difficult to hit pull home runs as a right-handed hitter. Moreover, 10 of his 28 home runs were rated as “Just Enough” home runs, in addition to his average home-run distance being 12 feet below average (admittedly not a huge number, nor a perfect way of measuring power). If I were a betting man, I’d expect him to hit 4-6 fewer home runs this coming season.

Keep watch for Part 3 in the coming days, which will detail the results of the other formulas. Something to watch for in this series is the issue that the results of the formula correspond too closely to what actually happened, which would render it useless as a formula.

*Note that because I have never formally taken a statistics course, I am prone to errors in my conclusions. Please point out any such errors and make suggestions as you see fit.*

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A busy person, but one who spends his free time in front of a computer screen, fiddling with statistics. And yes, that describes everyone who regularly visits this website.

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