On April 11th, 2014 the Minnesota Timberwolves played the Houston Rockets in what was assumed to be a normal game between a lottery-bound team against a team fighting for home-court advantage at the top of a very competitive Western Conference. Kevin Love, who was by far the best player on the team, was out and the Timberwolves had nothing to play for except pride or, if they lost, a better position in the lottery. With a bleak season coming to an end for their opponent, Rockets were a heavy favorite in the matchup. But Corey Brewer clearly did not get the memo and he poured in an outlandish 51 points and led the Timberwolves to victory.

Before this game, his career high in a single game was 29 points and in the 2013-2014 season his single game high was 27 points. His season average was 12.3 points per game and his highest scoring season barely topped that at 13 points per game. It was such an unlikely scoring performance that the NBA ‘randomly’ drug-tested Brewer immediately after the game. In short, this man was not on the basketball court for his ability to score.

So how did he do it?

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In this game alone, Brewer reached his career highs in six major statistical categories (which are highlighted in the box score). The biggest factor in reaching his career high in points is the number of field goals attempted. Taking 30 shot attempts is usually a luxury reserved for only superstars and rarely for a roleplayer. One reason Brewer was able to take so many shots this game because of his uncanny ability to create transition offense. In the 2014-2015 season he had the highest frequency of transition possessions of all players that averaged at least one per game and was in the 82.6 percentile of players in points-per-possession in transition, per NBA.com. In this incredible game, he scored 25 points off of transition shots alone.

He also managed to create transition offense with his defense. After only three and half minutes, Brewer’s superb individual defense prompted Dave Benz, the Minnesota Timberwolves play-by-play announcer, to yell, “[He] is like a man possessed on defense here!” He ended the game with a career high of 6 steals, which entered into an elite club, and converted those steals into 8 easy points. These transition opportunities also created many attempts at the free-throw line. He had 3 And-Ones and had a career high in both free throws made and attempted.

Brewer took almost two-thirds of his shots inside the paint, the closest and most efficient area to shoot, and managed, as illustrated in the shot chart below, to convert 84.2% of those shots.

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You may be wondering why one of the made field goals is not accounted for in the shot chart. Brewer, once again defying the odds, banked in an improbable buzzer-beating half-court shot  that reduced the gap from seven to four at half-time.

Corey Brewer also benefitted from the absence of both his teammate Kevin Love and the Houston Rockets’ Dwight Howard. Love was their leading scorer and with him gone many more shots were available for the rest of the Timberwolves. Dwight Howard, who is known as one of the league’s top rim protectors, was  also injured. Brewer’s efficiency and volume at the basket would be much worse if the league’s seventh-best shot blocker that season was on the floor.

All of these low-probability events culminated in Corey Brewer’s amazing performance. But just how unlikely was it? It’s possible to find out the exact probability of his scoring outburst with the Central Limit Theorem.

The Central Limit Theorem states that the sampling distribution of a random data set will be normal if the sample size is large enough. Our sample size (n) is eighty-one, the number of games Brewer played in that season. The sample size is sufficiently large enough to assume a normal distribution. The Central Limit Theorem is especially useful because it can be used to calculate the probability that a value higher than X will appear. The components of the equation that calculates the probability includes the sample size, X, the standard deviation (σ_ and the mean (μ), which are calculated from the all points he scored in the 2013-2014 season. Below is a table of the values:

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These values are then plugged into the equation: screen-shot-2016-10-16-at-6-51-38-pm and the result is the Z-score, which is -14.927. One note: X is equal to 50 because the Central Limit Theorem calculates the probability above X. Using a standard normal table, the Z-score is converted into a probability, which in this case is .00006.

The probability that Corey Brewer scores 51 points is only .00006! He would score that many points only once in 16,667 games, or 203 full regular seasons. To put how unlikely that probability is in context, Robert Parish holds the NBA record for most games played in a career with 1,611 games. Corey Brewer would need to play more than ten times the number of games that Parish played.

The beauty of the Central Limit Theorem is that the probability of any career high can be found with easy to find components and only a few simple calculations. But of course there are certain flaws with this method. I only use the sample of one season to calculate the probabilities in order to remove the many variables in a player’s career including injuries, playing on different teams, and many others. There are many endogenous variables not accounted for but the Central Limit Theorem is still a good tool that illustrates the chances of an event happening.

Another quick example is Kobe Bryant’s 81-point game in his dominant 2005-2006 season when he averaged 35.4 points. So how unlikely was that game?

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The game resulted in a Z-score of -29.277 and a probability of .00003; Kobe Bryant’s career-high is somehow less likely than Corey Brewer’s. Other examples include Scott Skiles’ 30-assist game (P = .00003), Lebron James’ 61-point game (P = .00005), and Kevin Love’s 31-rebound game (P = .0024)

By Andrew Haruki Hill

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