# The Anomaly of Lebron James: A Case Study Using Interval Estimation

Nov 3, 2017; Washington, DC, USA; Cleveland Cavaliers forward LeBron James (23) gestures after scoring against the Washington Wizards in the fourth quarter at Capital One Arena. The Cavaliers won 130-122. Mandatory Credit: Geoff Burke-USA TODAY Sports - 10388766

Author: Chibuzo Ikonte (Weinberg ’20)

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 as obe, haq, 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 lead 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 the long 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. he 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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