A Cheat Sheet to Relate Hits and Batting Average
I am someone who kinda sorta knew how this works, but never worked it out precisely. What motivated me into action was analyzing the statistics of Al Kaline in 1954 and 1955. His Batting Average jumped 64 points. There was some change in his SO rate, but not a great deal: from Ks in 8.9% of his AB in 1954, to Ks in 9.7% of his AB in 1955. Where one really saw the difference, beyond basic BA, was in his home runs, which increased from 4 to 27. Therefore, I expected that, despite the 64-point improvement in his batting averages, going by BAbip (Batting Average on Balls in Play), there wouldn’t be much difference. However, the BAbip verdict was in fact that Kaline still did 44 points better in 1955 than 1954. This is what they call a reckoning, isn’t it?1
With hits being the minority occurrence except in special circumstances (such as with Shohei Ohtani at the end of 2024), I knew that hits helped BA more than outs hurt. Somehow I had it in my head that hits generally were worth a couple of points, and outs generally cost a point (although this exact ratio would only be true for a .333 hitter).
But I took Kaline’s 1954 season, when he hit just shy of .276 over 504 at-bats, added a hit, and saw that the impact of his hits at that juncture was 1.44 points. Give him 100 more at-bats, 604 on the season, keep him a .276 hitter, and the impact of his hits goes down to 1.20 points. At 1000 at-bats, each of his hits would be worth just 0.72 points.
I was thus collecting data points, preceding in the trial-and-error fashion that I do, given that I no have faculty for algebra. I next tested the idea I’d stated before that the batting value of hits decreases with a higher batting average. Compared to the 0.72 points that the mere mortal Kaline would gain with a hit at the 1000 AB mark, the .500 hitter would gain exactly 0.5 points with a hit. With 500 AB, he is on that line that particularly orients me, where a hit is worth a point.
Noting these cases, it seems we can make a rule, that for the theoretical .500 hitter, the value of his hits times his at-bats will always equal 500. Trying other at-bat totals, it works out.
Instead of working with Kaline’s .276, it was easier (and also more generally applicable for the common case today) to work with a .250 hitter to search for a wider rule. At those same 500 AB, a hit for the .250 hitter is not worth 1 point, as it is for the .500 hitter, but worth 1.5 points.2 At 1000 AB, a hit for the .250 hitter is worth 0.75 points. So, again, the ratio between impact and at-bats assumes a constant, even with this different batting average. And, for the .250 hitter, this constant is 750.
So, we have a 500 constant for a .500 hitter, and a 750 constant for a .250 hitter. What is the pattern? It turns out that the constant that determines BA impact at any AB level is a function of the complementary probability of the batting average. So, for a .100 hitter, the constant is 900. For a .400 hitter, it is 600.
Since the constant is in terms of batting average point gain for 1 at-bat, it also represents the point where 1 hit still generates 1 point of batting average gain. Once you have that number, you can adjust the projected gain based on the specific number of at-bats you are interested in. So, a .276 hitter like Kaline in 1954 crosses over into less than a point of BA gain with 724 AB. With 300 AB, then, when that AB total has been divided by 2.413, each hit he makes should be worth 2.413 points.
Let’s see if it works out. .276*300 = 82.8. With 1 more hit, Kaline would be 83.8/301, or .278405. The value does round off to the same 2.41 point gain. And, if you see footnote 2, and we start from 81.8/299 instead, the hit to the target nets him 2.421 points. The two gains, then, average exactly 2.413.
Whether a sample size is small or large, and a batting average can easily move up or cannot, is a double-edged sword. The effect of outs also diminishes with time, and since outs are bad, a hitter likes this. The math doesn’t recognize whether we are calculating a batting average or an out average, so it is the same when calculating the effect of an out. The .250 hitter has a .750 out average, which means that his outs have already dropped to less than 1 point in value when he clears 250 at-bats (.250 translates to 62.25 h in 249 AB; add an AB, and 62.25/250 is exactly .249).3 This is good, as his hits at this point are worth 3 times more, 3 points.
Seeing and understanding this ratio of the effect of hits and outs in light of batting average is relatively easy. But it is also edifying to think of any at-bat total, say, 100. At 100 AB, there are 10 BA points on the line for that at-bat. (Do 1/100, *1000). The player’s BA coming in determines what the relative value and cost to his batting average of a hit and out will be. We divvy up the points on the line using his batting average. So, if he’s a .350 hitter, he will gain 6.5 points with a hit, and he will lose 6.5 points with an out. Again, we work from the complements.
Realizing that I am repeating myself, but doing so with the hope that you come out of this with something concrete and practical, the AB total where the cost of an out drops under a point equals the player’s batting average x 1000. To demonstrate another one: A .213 hitter in 212 AB has 45.156 hits. 45.156 hits in 213 AB makes for a .212 hitter.
I never apologize for doing math that baseball provokes and is tough to channel back into baseball. The math still relates to baseball. But even if I did, this doesn’t strike me as one of those cases without obvious application. For one thing, I think it will be valuable to me to have a much finer idea of just what batting average differences mean in terms of hit differences when comparing players. It is always best to be literal, to be relating the statistics back to reality rather than keeping them in a separate statistical box. I know I at least am guilty of sometimes forgetting what batting average means. And, now knowing the math as I do, I’ll be able to project BAbips much better without painstakingly working them out, or looking them up. In the case of Kaline, for instance, we could get hit to the impact of his home runs on his batting average by knowing the approximate value of a hit for him at that (ball in play) AB range, then multiplying that by the increase his home run gain represents in terms of hits.
A trivial reckoning, I suppose. Which is an oxymoron. I dare to go there and take the word “reckoning” in vain.
It does matter whether we sample the impact to the particular AB total, or away from it. The impact at the lower AB total will be slightly more than the impact at the higher one. So, while the 124/499 hitter is hitting 1.503 points lower than the 125/500 hitter, the 126/501 hitter is hitting just 1.497 points higher than his mark.
Unlike the value of a hit, which never equals the projection for the AB total exactly but comes to it on average taking the previous and subsequent hits, the value of an out does match the projection exactly if the comparison is made to the average in the AB before the AB target. In figuring the effect of a hit, you add 1 to the numerator and to the denominator, while in figuring the effect of an out, you only add 1 to the denominator, so I guess that is the difference.