I tweeted the other day that as strong a start as Josh Hamilton's had, his OPS (1.334) in 31 games is actually lower than Barry Bonds' was in three full seasons (2001, 2002 and 2004). While this is true, it occurred to me that Hamilton's season pace (through Saturday) - 87 homers in a 150-game season with a .402 average - had to be more valuable than Bonds' years, despite the lower OPS.

So what gives? OPS stands for On Base plus Slugging. The formula for on base percentage is hits plus walks (BB) plus hit-by-pitch (HBP) divided by AB+BB+HBP+Sac Flies. In other words it's how many times one reaches base per plate appearance. By contrast, the formula for slugging percentage is total bases (singles + 2|STAR|doubles + 3|STAR|triples + 4|STAR| home runs) divided by at-bats. So the denominator in OBP is plate appearances, while the denominator in slugging is at-bats. When you have a player like Bonds who walked an average of 202 times during each of those years, plate appearances and at-bats diverge greatly.

Let's use a simple (and extreme) example so show what can happen. Say Player A has 11 plate appearances, hits one home run and draws 10 walks. Player B hits one home run and has 10 singles. Which player has been more valuable? Clearly Player B, as a single is slightly more valuable than a walk. But Player A's OPS is 1.000 on-base plus 4.000 slugging (4 bases in 1 at-bat) = 5.000. Player A's OPS is 1,000 OBP plus 14 bases/11 at-bats (1.273 slugging) = 2.273. So Player A's OPS is more than twice as high despite being the less productive hitter.

The problem is how OPS values walks. If someone walked every time up, he'd have a 1.000 OBP, but what would his slugging be? Zero? The formula for slugging is total bases divided by at-bats. So it's zero divided by zero which does not equal zero. It's actually undefined mathematically.

So Bonds' walks contribute to his OBP, but they don't affect his slugging at all. So he's getting 1.000 OBP 202 times but not losing anything from his slugging. That's worth a lot more than getting 1.000 OBP and zero for your slugging. Essentially it allows him to maintain his usual slugging while getting a 1.000 OBP factored in every time. Since his usual slugging over those seasons was .825, and all those walks beefed up his OBP, he put up truly monstrous OPS years. Hamilton, by contrast has drawn only 13 walks, a 63-walk pace in 150 games. So he has 139 less times where he's getting a 1.000 on-base that doesn't hurt his slugging.

That said, Bonds deserves credit for his walks, (though an average of 72 per year were intentional and therefore likely to be less valuable than an average walk because they presumably came in situations where a walk was substantially less valuable than a hit, e.g., runner on second, first base open, rather than unintentional walks which are more likely to advance runners and be closer in value to singles. Moreover, while Bonds' stellar OBP was compiled over 631 plate appearances, his slugging was compiled over a much smaller sample - 413 at-bats. Contrast that with Hamilton who in 150 games would be on a pace for 668 plate appearances and 590 at-bats. So Hamilton's .877 slugging would be even more valuable in real terms because it would be on 177 more at-bats at an elite level.

Of course, it's kind of odd to ding Bonds' seasons for their small slugging samples when we're taking a much smaller Hamilton sample and extrapolating it over 150 games. And I'm sure Bonds had five-week stretches that dwarf Hamilton's. But the point is that while I tweeted that Bonds had three whole seasons more impressive than Hamilton's huge start, it's actually not true.