Friday, March 02, 2012

50th Anniversary of Wilt Chamberlain's 100-Point Game

Fifty years ago today, March 2, 1962, Wilt Chamberlain of the then-Philadelphia Warriors scored an NBA record 100 points in a single game. The game was played in Hershey, Pennsylvania, as the Warriors defeated the New York Knicks, 169-147. Chamberlain, who lived from 1936-1999, stood 7-foot-1 and was associated with many amazing statistical feats (e.g., a 33-game winning streak with the 1971-72 Los Angeles Lakers; playing over 1,000 games without ever fouling out). However, it is the 100-point game that is his signature accomplishment.

Despite the magnitude of Chamberlain's performance, information on his 100-point game is relatively scarce. No video or film of the game seems to exist and, because it was played 95 miles outside of Philly in a much smaller city, attendance was low (4,125) and there was less media coverage than usual. One really has to do some work to learn the details of the game, and luckily writer Gary M. Pomerantz did so for readers of his 2005 book Wilt, 1962: The Night of 100 Points and the Dawn of a New Era (there's an excellent website accompanying the book, which includes audio of the radio broadcast of the game).

Here at the Hot Hand website, our focus is statistical analysis, so that is what we shall pursue. As can be seen from the box score of the game, Chamberlain hit 36-of-63 from the floor for 72 of his points and went 28-of-32 from the free-throw line.It is the latter performance, a nearly 90% success rate (.875), that is really amazing. Wilt's free throw percentage during the 1961-62 season was .613, which actually was the best single-season free-throw accuracy rate of his career. Thus, on this night, he really overcame his free-throwing difficulties to make the 100-point game possible.

Noting that Chamberlain had made 21-of-22 free-throws through the first three quarters of play, Pomerantz asks in his book, "How does a sixty percent free-throw shooter throughout the season convert ninety-five percent on a night in Hershey?" (p. 116). Updating the percentages to reflect the full game and rephrasing the question a bit, we can ask how likely it is that a .613 shooter would hit 28 (or more) free throws out of 32, assuming independence of outcomes (i.e., what happens on one shot having no impact on the next shot, like coin tossing).

This binomial-probability calculator allows one to type in the underlying probability of an outcome (p = .613), the number of opportunities (n = 32), and what the website calls the number of "stipulated" successes (k, in this case 28 made free-throws). Because the probability of exactly 28 (or any other specific number) made free-throws would tend to be very small, statisticians typically inquire into the probability of a particular result or one more extreme. Hence, we have the reference above to "28 (or more)" successful free-throws out of 32.

In the following chart, I've graphed the probability of Chamberlain making anywhere from 9 to 32 free-throws out of 32, given his season-long percentage of .613 (below 9, the likelihood is extremely low). You may click on the graphic to enlarge it.

As can be seen, the probabilities of Chamberlain making 28, 29, 30, 31, or 32 free-throws were all tiny, so that the probability of 28 or more made shots was around 1 or 2 in 1,000, according to this annotated screen caputre from the statistical calculator website:

With a roughly 61% free-throw percentage, Chamberlain would have been most likely to make, you guessed it, 61% of his attempts, which translates into 19.5 out of 32 (note how the bars for 19 and 20 are the highest in the graph above). Thus, he greatly exceeded statistical expectations.

Pomerantz tried to answer his original question about how Chamberlain could have been so successful on free-throws (which we now know to be a roughly 1-in-1,000 phenomenon) by claiming that the baskets in Hershey featured "soft," rickety rims. When a ball hits a soft rim, the reasoning goes, it will bounce gently around and possibly fall through the hoop, rather than banging hard against it and flying off. Whether that's a valid explanation or not, I can't say. Still, Wilt must have been shooting the ball close enough to the basket on his free-throw attempts to (allegedly) benefit from the soft rims.

A famous quote from the writings of the late Harvard paleontologist Stephen Jay Gould is that, "Long streaks always are, and must be, a matter of extraordinary luck imposed upon great skill." Indeed.

1 comment:

Gur Yaari said...

Very interesting!

Although a p value of 0.001-2 sounds small, one has to correct for multiple testing ( This means that if one has 100 games for example, then the probability of having a p value of 0.001 in one of the games is not so small (could be approximated by p*100 = 0.1-0.2 in this case).

However, the other fact mentioned in this post is way more unlikely to occur by chance - namely, the free throws percentage Wilt Chamberlain had in the entire 1961-62 season: 0.613.

To calculate how unlikely this is to occur by chance, one can use the same binomial distribution, assuming the probability of success is 0.511 (his career free throws percentage) and ask what is the p value to obtain 835 successes out of 1363 attempts. The answer is 10^-14 (0.00000000000001) which stays extremely low even after correcting for multiple testing ... 1961-62 was indeed an incredible season for Wilt Chamberlain's free throws performance.