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The Anomaly of Lebron James: A Case Study Using Interval Estimation (May 26, 2018)

Updated: Jun 17, 2022

(Per Geoff Burke USA Today Sports)

Father time is the term used to denote when age and mileage has caught up to an NBA player, and their level of production begins to decline. This is a phenomena that has been seen in most players, including all time greats such asKobe, Shaq, and Tim Duncan. Lebron James, on the other hand, has proven to be an outlier to this phenomena. During James’ 15th NBA season, he averaged 27.5 points per game(ppg), 8.6 rebounds, and 9.1 assists on 54.2% shooting from the field, and with a player efficiency rating of 28.6. It is unprecedented to see a player perform so well this late into their career.

LeBron’s astounding level of play has led me to ponder the question of how I could quantify just how impressive his production has been. One way to answer that question is to create a 95% confidence interval measuring the scoring output in Year 15 (or the last year of a player’s career if they did not play that long) of the Top 50 scorers in NBA history. This will allow Lebron’s level of play to be juxtaposed with 50 of the greatest scorers in basketball history(output shown in table below). Furthermore, a confidence interval will essentially show that in thelong run, 95% of the intervals constructed will contain the true mean, μ, of the population represented by the 50 yi’s shown in the table below. The remaining 5% of intervals will lie either entirely to the left of μ or entirely to the right.

Let y1, y2, …, y50 denote the scoring outputs recorded by each of the Top 50 scorers in their 15th season (or final year). I will assume that the yi’s are normally distributed with an unknown mean, 𝝻. Y= 12.206, and I also found 𝞼= 5.586 ppg. Thus, fy (y;𝝻)= -∞< y < ∞. From this I will be able to construct a 95% confidence interval for the true mean of the population represented by the 50 yi’s in the Table below.The endpoints for a 95% confidence interval for μ are given by the general formula. If Lebron’s scoring output of 26.5 is outside of the interval, it can be argued just how unprecedented his performance this year has been.

Using the formula for a confidence interval, I got (10.658, 13.754). So, in the long run, I am 95% confident that the true mean of points scored per game of the 50 greatest scorers in basketball history (in 15th season) lies within the interval of 10.7 ppg to 13.8 ppg.

Lebron’s scoring output of 27.5ppg is more than 2 standard deviations away from the upper limit. Additionally, the large difference between 27.5 and y=12.206 further emphasizes just how much Lebron’s play has been an anomaly from the norm of player production in year 15. Since the μ = 27.5 is not contained in the 95% confidence interval (or even close to being contained), we could conclude that Lebron’s 26.5ppg is not likely to have come from a normal population where μ = 12.206 (and σ = 5.586).It would appear, in other words, that Lebron’s play this year is a definite outlier. From here I have concluded that his unprecedented greatness is something that should be appreciated and should continue to be celebrated.

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