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One Score All or All Score One?
Points Scored Inequality and NBA Team Performance
Patryk Perkowski
Fall 2013
Professor Ryan Edwards!
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Introduction
There has recently been a proliferation of research on inequality and the impact on
resource distribution on societal performance. Dely et al. 2001 found that almost fifty percent of
the variation in state homicide rates could be accounted for by variation in relative income
inequality. Matlack and Vigdor 2006 concluded that income inequality has a statistically
significant impact on the shortage of affordable housing in the United States, while Wilkinson
2013 demonstrated links between increased income inequality and shorter life expectancies.!
Such studies have not been limited to just analyzing the impact of inequality on society. Various
studies have extended this analysis to understand how resource distribution affects firm
performance. The NBA is one such example, where a plethora of literature exists that
investigates the relationship between wage inequality and team performance. However, no
studies have looked at the impact of points scored inequality on team performance.
Method and Data
In this paper, I investigate the relationship between the distribution of points scored by a
team and their performance to determine whether teams perform better when one player scores
all of their points or when all of their players score an equal number of points. My goal is to
compare the impact of points scored inequality on the performance of the Miami Heat with the
impact of points scored inequality on the performance of all of the other teams in the NBA
during the 2012-2013 NBA season. The Miami Heat were chosen because they won their second
consecutive championship during this season and went on a 27-game winning streak. I
investigate the performance of the Miami Heat throughout the entire season but due to time
constraints, I only investigate the performance of the others teams during the first 82 games of
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the season. I obtained box score data for games played during the 2012-2013 NBA season from
www.basketball-reference.com.
In this paper, points scored inequality is measured by the Gini coefficient method. The
1
Gini coefficient g ! [0,1] for a quantity distribution dq is defined as
" [x ! L(x)]dx
0
1
or
" x dx
0
1
1! 2 " L(x)dx. , where L(x) plots the cumulative percentage of use of some quantity q against the
0
cumulative number of individuals that use that quantity from least to greatest, where x ![0,1].
(See Farris (2010) for a more rigorous look at Gini coefficients.) A Gini index of g = 0 represents
perfect equality where each player scores the same number of points while a Gini index of g = 1
represents complete inequality where one player scores all of the points in a game.
Let a be any NBA team with 12 dressed players and no more than 2 injured players. Let
pt denote the total points scored by team a and let pi denote the points scored by the ith dressed
player on team a, where a dressed player’s ordering is determined by the number of points
12
scored from least to greatest. Thus for any team a, we have
!p
i
= pt and 0 ! pi ! pi+1 ! pt for i
i=1
![1,11] (it may be the case that
12
n
i=1
i=1
! pi " ! pi = pt but the difference between constructed Gini
coefficients in negligible). Let L(x) be the function that maps the cumulative proportion of
ordered dressed players onto their corresponding cumulative proportion of total points. Thus L(x)
i
"
%
p
!
j
$ i
'
j=0
' , where 0 ! i ! 12. In order to calculate the Gini
is generated by the 13 coordinates $ ,
$ 12 pt '
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coefficient, I approximate L(x) as a fifth order polynomial in the form L(x) = m0x5 + m1x4 + m2x3
+ m3x2 + m4x + m5 subject to conditions m5 = 0 and m0 + m1 + m2 + m3 + m4 = 1 to satisfy L(0/12)
= 0 and L(12/12) = 1. Team performance is measured by three proxies: victory status, margin of
victory, and team shooting percentage.
Results
During the 2012 – 2013 NBA season, the Miami Heat scored on average almost 103
points per game with an average field goal percentage of 49.7% and an average Gini coefficient
of .5484. When playing against the Miami Heat, non-Heat basketball teams scored on average
about 95 points per game, with an average field goal percentage of 44.1% and an average Gini
coefficient of .5118. During the first 78 games of the season, non-Heat teams averaged a Gini
coefficient of .5053 when playing against other non-Heat teams. This suggests that when playing
against the Miami Heat, teams tolerate, on average, higher levels of points scored inequality to
boost team performance.
First I examined the relationship between Gini coefficient and win status. Appendix 1
contains the graph of Gini coefficient versus win status where a win is denoted as the number 1
and a loss as the number 0. When the Miami Heat lost, their Gini coefficient varied between with
.55 and .70 with an outlier of .465 when they lost to the New York Knicks on November 2, 2012.
When a non-Heat team lost to a non-Heat, however, their Gini coefficient varied between .3 and
.7, suggesting that points scored inequality has a greater impact on the performance of the Miami
Heat than the combination of the other teams in the NBA. However, when the Miami Heat won,
their Gini coefficients were in a range similar to that of the Gini coefficients of other teams when
they won, suggesting that points scored inequality has better predictive power when the Miami
Heat lose and not when they win. When the Miami Heat won, the average of their Gini
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coefficient was .5329, while when they lost it was .5935. A two-tailed unpaired t-test found the
difference between Gini coefficients when the Miami Heat won (M = .5329; SD = 0.0670) and
when the Miami Heat lost (M=.5935; SD = 0.0703) to be statistically significant at the 1% level,
t(71) = 3.16, p = 0.0023 < 0.01.
Next I analyzed the relationship between Gini coefficients and team success by taking
into account the margin of victory and not just using a binary variable. Appendix 2 shows the
graph of the Gini coefficient versus the margin of victory for the Miami Heat during the entire
2012 – 2013 NBA season and for non-Heat teams during the first eighty-two games of the
season. When a non-Heat team plays another non-Heat team, there seems to be no aggregate
relationship between the Gini coefficient and the margin of victory. If we estimate a linear
relationship between Gini coefficient and margin of victory, then the slope coefficient is -.0003
while R2 is less than .1%. For the Miami Heat, however, there seems to be a negative
relationship between margin of victory and Gini coefficient, as the team has lower Gini
coefficients with larger margins of victory. With an R2 value of .191, I find that the distribution
in player points scored accounts for 19% of the variation in margin of victory for the Miami
Heat.
Next I analyzed the relationship between Gini coefficients and team shooting percentage.
Appendix 3 contains this graph. The relationship between these two variables is not as well
defined as the relationship between Gini coefficients and margin of victory. The data seems to
cloud around the coordinate (0.5, 0.5), so I cannot conclude whether the relationship between
points scored inequality and team-shooting percentage is positive or negative, as it seems that
there is no relationship between these two variables.
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Finally I considered the Miami Heat’s 2013 almost unprecedented win streak. From
February 4, 2013 to March 25, 2013, the Miami Heat won 27 games in a row before falling to the
Chicago Bulls on March 27th. Appendix 4 contains the graph of the Gini Coefficient during the
Heat’s 27-game win streak. Throughout this run, the Heat averaged a Gini coefficient of .5542,
as compared to .5457 when they were not on this winning streak. A two-tailed unpaired t-test
found the difference between Gini coefficients for the Miami Heat during this winning streak (M
= .5542; SD = 0.05781) and for the Miami not during this win streak (M=.5456; SD = 0.0785) to
not be statistically significant, t(80) = 0.5057, p =0.6145 > 0.05. Similarly, a two-tailed unpaired
t-test found the difference between Gini coefficients when the Miami Heat won during the win
streak (M = .5542; SD = 0.05781) and when the Miami Heat won not during this win streak
(M=.5456; SD = 0.0785) to not be statistically significant, t(80) = 0.5057, p = 0.6125 > 0.05.
Conclusion
My results indicate that for the Miami Heat, points scored inequality has a statistically
significant negative impact on victory status, accounts for almost 20% of the variation in margin
of victory, and is not related to the Heat’s field goal percentage. For the rest of the league,
however, there seems to be no aggregate relationship between points scored inequality and
victory status, margin of victory, and field goal percentage. Future work could look into this
relationship for individual teams and control for other factors that may influence team
performance, such as home court advantage and the relative ranking of the other team.
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References
Farris, Frank. “The Gini Index and Measures of Inequality.” Santa Clara University. (December
2010). Retrieved 26 October 2013 from <http://math.scu.edu/~ffarris/MonthlyFinal.pdf>.
“NBA Games Played.” Basketball Reference. Retrieved November 27, 2013 from
<http://www.basketball-reference.com/boxscores/>.
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Appendices
Appendix 1
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