In the wake of the Lakers' mud-slogging Game 7 win in the NBA Finals over the Boston Celtics by the score of 83-79, some fans were incredulous that a team could shoot 32.5 percent (27 of 83) and still win. In fact, many of them felt that the Celtics lost the game, rather than the Lakers winning it. To me, that sounds a little silly, inasmuch as basketball is a head-to-head sport. If the Lakers were shooting that poorly, presumably the Celtics had something to do with that, and just as presumably, the Lakers were doing something else to win the game.

So what was that something? I'll give you a little hint. It begins with "offensive," and it rhymes with "rebounding."

In the unlikely event you haven't caught on, a major key to the Lakers' victory was their offensive rebounding; they won that battle 23-8 over the Celtics. To be sure, gathering 23 offensive rebounds is usually a dubious feat, for it requires the team to miss far in excess of 23 shots. So to a large extent, the dominance of the Lakers on the offensive boards was a reflection of their miserable 32.5 percent shooting clip.

However, the Celtics only gathered 32 defensive rebounds, meaning that of the 55 rebounds available after Lakers misses, the Lakers collected almost 42 percent of them. So not only did the Lakers get a lot of offensive rebounds, they got them at an stunning rate, and that doesn't depend on how many shots they missed. To give you an idea of just how stunning that is, the NBA league average is about 26 percent. The Lakers were more than half again as effective at getting offensive rebounds. By contrast, there were 38 rebounds available on the Celtics' offensive end, and they got only 8 of them, for an offensive rebounding rate of 21 percent, a bit lower than average.

That suggested the following little puzzle: All those offensive rebounds increased the Lakers' overall efficiency at the offensive end, by giving them extra shots at the basket on each possession. Can we express that increased efficiency in terms of shooting percentage—in effect, collapsing the two figures into one?

I believe we can. Suppose for the moment that we don't care about free throws, three-point shots, and all those aspects of scoring that in truth are rather important. We only care about the raw shooting percentage. The Lakers hit 0.325 of their shots. If their offensive rebounding rate was 0 percent, then the fraction of their shooting possessions (as opposed to possessions that end with a turnover, say) that they score on is 0.325.

However, in truth, they rebounded 0.42 of their misses. They miss 1 - 0.325 = 0.675 of the time, so out of all their shooting possessions, they end up with the ball again on 0.42 × 0.675 = 0.28 of the time. Then they'll score 0.28 × 0.325 = 0.09 of the time, and so on. If they miss, they can rebound again, which they'll do 0.28 × 0.675 × 0.42 of the time. And so on.

It's much more concise to put this symbolically, as follows:

Fraction of shooting possessions ending with a score = 0.325 + 0.675 × 0.42 × 0.325 + 0.675 × 0.42 × 0.675 × 0.42 × 0.325 + ...

Each time, there's an extra factor of 0.675 × 0.42, representing the Lakers missing and then picking up the rebound. Since this can happen an arbitrary number of times in a given possession, this equation can have an infinite number of terms (well, limited only by the length of the game). This is called a geometric series, and fortunately, there's a simple formula that allows you to calculate the sum without adding and multiplying an infinite number of terms. Therefore,

Fraction of shooting possessions ending with a score = 0.325 ÷ (1 - 0.675 × 0.42) = 0.45

That is to say, 45 percent of shooting possessions end in a score for the Lakers. Not to put too fine a point on it, that's still fairly awful. But not as awful as the original shooting percentage suggested.

But now, as I said, I'm going to try to combine the offensive rebounding and the shooting percentage into a single composite figure, by asking this question: Suppose the Lakers gathered only 26 percent of their misses as rebounds (the league average), instead of the 42 percent they actually gathered. How much better would their shooting have had to be in order to match that 45 percent per-possession efficiency? In symbolic terms, solve for x:

x ÷ (1 - (1- x) × 0.26) = 0.45

I'm not going to make you do that for homework; I'll just give you the answer: It turns out that x = 0.38. In other words, if the Lakers had crashed the offensive boards like an average team, they would have had to shoot 38 percent in order to score on 45 percent of their shooting possessions. Like I said: Bad, but not historically bad—not bad like 32.5 percent bad.

To put it another way, their tremendous offensive rebounding was worth 5.5 percentage points on their shooting. That's huge: 5.5 percentage points is usually worth about 10 points on the scoreboard by the end of the game.

We can turn this approach to the Celtics, too. They shot 41 percent (29 of 71), and picked up 21 percent of their offensive rebounds. That means that they ended 47 percent of their shooting possessions with scores:

0.41 ÷ (1 - 0.59 × 0.21) = 0.47

However, if they had just rebounded like an average team on their offensive end, they could have shot a bit worse and still matched that per-possession efficiency. Solve for y:

y ÷ (1 - (1- y) × 0.26) = 0.47

Again, I'll save you the algebra and give you a peek in the back of the book: y = 0.395. That is, if the Celtics were an average rebounding team, they would have achieved that efficiency by shooting just 39.5 percent. A bit better than the Lakers, but I think you'll agree that 39.5 to 38 is a lot closer than 41 to 32.5. Almost six times closer, even.

Now, the Lakers actually won, which means they must have done other things as well to get the win. For one, they turned the ball over somewhat less often, even with all the extra cracks at their offensive end: just 11 turnovers to Boston's 14. And the Lakers also visited the foul line more often (although some of those free throws were toward the end of the game, when the Celtics were fouling to stop the clock, and the Lakers shot poorly on their extra free throws just the same). Those two factors were enough to put the Lakers over the top. But the dominant factor in overcoming an awful shooting performance was their persistence in rebounding on the offensive end.

So what was that something? I'll give you a little hint. It begins with "offensive," and it rhymes with "rebounding."

In the unlikely event you haven't caught on, a major key to the Lakers' victory was their offensive rebounding; they won that battle 23-8 over the Celtics. To be sure, gathering 23 offensive rebounds is usually a dubious feat, for it requires the team to miss far in excess of 23 shots. So to a large extent, the dominance of the Lakers on the offensive boards was a reflection of their miserable 32.5 percent shooting clip.

However, the Celtics only gathered 32 defensive rebounds, meaning that of the 55 rebounds available after Lakers misses, the Lakers collected almost 42 percent of them. So not only did the Lakers get a lot of offensive rebounds, they got them at an stunning rate, and that doesn't depend on how many shots they missed. To give you an idea of just how stunning that is, the NBA league average is about 26 percent. The Lakers were more than half again as effective at getting offensive rebounds. By contrast, there were 38 rebounds available on the Celtics' offensive end, and they got only 8 of them, for an offensive rebounding rate of 21 percent, a bit lower than average.

That suggested the following little puzzle: All those offensive rebounds increased the Lakers' overall efficiency at the offensive end, by giving them extra shots at the basket on each possession. Can we express that increased efficiency in terms of shooting percentage—in effect, collapsing the two figures into one?

I believe we can. Suppose for the moment that we don't care about free throws, three-point shots, and all those aspects of scoring that in truth are rather important. We only care about the raw shooting percentage. The Lakers hit 0.325 of their shots. If their offensive rebounding rate was 0 percent, then the fraction of their shooting possessions (as opposed to possessions that end with a turnover, say) that they score on is 0.325.

However, in truth, they rebounded 0.42 of their misses. They miss 1 - 0.325 = 0.675 of the time, so out of all their shooting possessions, they end up with the ball again on 0.42 × 0.675 = 0.28 of the time. Then they'll score 0.28 × 0.325 = 0.09 of the time, and so on. If they miss, they can rebound again, which they'll do 0.28 × 0.675 × 0.42 of the time. And so on.

It's much more concise to put this symbolically, as follows:

Fraction of shooting possessions ending with a score = 0.325 + 0.675 × 0.42 × 0.325 + 0.675 × 0.42 × 0.675 × 0.42 × 0.325 + ...

Each time, there's an extra factor of 0.675 × 0.42, representing the Lakers missing and then picking up the rebound. Since this can happen an arbitrary number of times in a given possession, this equation can have an infinite number of terms (well, limited only by the length of the game). This is called a geometric series, and fortunately, there's a simple formula that allows you to calculate the sum without adding and multiplying an infinite number of terms. Therefore,

Fraction of shooting possessions ending with a score = 0.325 ÷ (1 - 0.675 × 0.42) = 0.45

That is to say, 45 percent of shooting possessions end in a score for the Lakers. Not to put too fine a point on it, that's still fairly awful. But not as awful as the original shooting percentage suggested.

But now, as I said, I'm going to try to combine the offensive rebounding and the shooting percentage into a single composite figure, by asking this question: Suppose the Lakers gathered only 26 percent of their misses as rebounds (the league average), instead of the 42 percent they actually gathered. How much better would their shooting have had to be in order to match that 45 percent per-possession efficiency? In symbolic terms, solve for x:

x ÷ (1 - (1- x) × 0.26) = 0.45

I'm not going to make you do that for homework; I'll just give you the answer: It turns out that x = 0.38. In other words, if the Lakers had crashed the offensive boards like an average team, they would have had to shoot 38 percent in order to score on 45 percent of their shooting possessions. Like I said: Bad, but not historically bad—not bad like 32.5 percent bad.

To put it another way, their tremendous offensive rebounding was worth 5.5 percentage points on their shooting. That's huge: 5.5 percentage points is usually worth about 10 points on the scoreboard by the end of the game.

We can turn this approach to the Celtics, too. They shot 41 percent (29 of 71), and picked up 21 percent of their offensive rebounds. That means that they ended 47 percent of their shooting possessions with scores:

0.41 ÷ (1 - 0.59 × 0.21) = 0.47

However, if they had just rebounded like an average team on their offensive end, they could have shot a bit worse and still matched that per-possession efficiency. Solve for y:

y ÷ (1 - (1- y) × 0.26) = 0.47

Again, I'll save you the algebra and give you a peek in the back of the book: y = 0.395. That is, if the Celtics were an average rebounding team, they would have achieved that efficiency by shooting just 39.5 percent. A bit better than the Lakers, but I think you'll agree that 39.5 to 38 is a lot closer than 41 to 32.5. Almost six times closer, even.

Now, the Lakers actually won, which means they must have done other things as well to get the win. For one, they turned the ball over somewhat less often, even with all the extra cracks at their offensive end: just 11 turnovers to Boston's 14. And the Lakers also visited the foul line more often (although some of those free throws were toward the end of the game, when the Celtics were fouling to stop the clock, and the Lakers shot poorly on their extra free throws just the same). Those two factors were enough to put the Lakers over the top. But the dominant factor in overcoming an awful shooting performance was their persistence in rebounding on the offensive end.

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