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Revista de Psicología del Deporte. 2015, Vol 24, Suppl 1, pp. 31-35
ISSN: 1132-239X
ISSNe: 1988-5636
Universitat de les Illes Balears
Universitat Autònoma de Barcelona
Sergio L. Jiménez Sáiz. Facultad de Ciencias del Deporte, Universidad Europea de Madrid. Madrid. Email: sergiolorenzo.jimenez@uem.es.
*Departamento de Educaciòn Física de la Universidad de Las Palmas de Gran Canaria
**Facultad de Ciencias de la Actividad Física. Universidad Católica de Valencia “San Vicente Martir”
***Facultad de Ciencias de la Actividad Física de la Universidad Europea de Madrid
Fecha de recepción: 25 de Septiembre de 2014. Fecha de aceptación: 3 de Noviembre de 2015.
Last minute in NBA games
Juan Manuel García-Manso*, Juan Manuel Martín-González**, Yves de Saá Guerra*,
Teresa Valverde*** and Sergio L. Jiménez****
LAST MINUTE IN NBA GAMES
KEYWORDS: Basketball, NBA, Fractal, Performance.
ABSTRACT: Apparently the behavior during a basketball game, as in other team sports, shows tremendous variability manifested in
both individual and collective ways. However, when a significant number of games are studied, we can observe the unpredictability
that characterizes the game. The degree of complexity of the game is not stable. Patterns change during all the game time, but the last
minute is completely different reality. Our aim was to test and evaluate the existence of these patterns and their apparent complexity,
by analyzing the NBA games scoring and substitution dynamics. Therefore, we examined the difference between the last minute and
the rest of the game from the collected scores (1, 2 and 3 points), substitutions and timeouts. The underlying chaotic behavior of non-
linear interactions is inherent in Complex Systems. The data showed the existence of symmetries and repeated patterns of play during
basketball games of the NBA but the last minute, which can be considered a completely different game.
Basketball is a collaboration-opposition sport, where teams
struggle to prevail over his rival. Throughout all the game time,
behavior patterns characteristic of each period are shown, up until
reach the last minute. At this stage, in the last minute of the game,
the dynamic of the game is different from the rest, emerging new
patterns of game where certain characteristic elements of the
game such us fouls, substitutions, timeouts, etc., acquire more
weight.
Despite its complexity, we observe patterns that explain the
nature of the game, this allow us establishing strategies that lead
to optimize performance in a game or competition. Non-linear
systems can display a variety of behaviors, including: self-
organization, emergency, order-chaos, networks, fractal, etc.
(Chatterjee and Yilmaz, 1999; de Saá Guerra et al., 2012; Gould,
1989; McGarry, Anderson, Wallace, Hughes and Franks, 2002;
Ribeiro, Mendes, Malacarne, Pîcoli and Santoro, 2010; Yilmaz
and Chatterjee, 2000).
In this work, we will study the evolution of game dynamic
from the collected scores (1, 2 and 3 points), substitutions and
timeouts. These systems are not random but characterized by high
sensitivity to initial conditions (e.g. rules, scores, game time, etc.)
therefore slight changes in the beginning may result in significant
different responses, what characterizes them as chaotic systems.
Initial conditions are never perfectly known but inherently
unpredictable.
This effect of chaos, is particularly evident in the last minute,
where if the game is competitive, the result is very unpredictable,
which becomes this sport in something particular and where the
last minute can be extended over time, creating a state of high
expectation on fans.
Method
Sample.
We analyzed 5 NBA regular seasons (6.150 games). We used the
official statistics published by NBA which describe all the events
that occur during the game: two and three point shots, free throws,
rebounds, turnovers, violations and substitutions. We also
analyzed the last minute of each quarter, without taking into
account the team. From the analysis of the histogram of the
differences in points at the end of each game, de Saá et al. (2013)
find a Power Law distribution which basically identifies two
situations related here: competed games, which finalized by less
than 11 points difference and the non-competed games.
Procedures.
The number of points scored per minute in principle should
follow a random process characterized by the Poisson
distribution. This distribution is characterized because the ratio
between the variance and the mean, known as the Index of
Dispersion (ID), is one. For other results there are two cases:
- The first one happens when the Index of Dispersion is less
than 1. This is considered as under-dispersed, which means that
the probability data are more clustered around the mean making
Juan Manuel García-Manso, Juan Manuel Martín-González, Sergio L. Jiménez, Yves de Saa and Teresa Valverde
32 Revista de Psicología del Deporte. 2015, Vol 24, Suppl 1, pp. 31-35
more predictable. This condition is related with more regular
patterns than the randomness associated with Poisson. Points are
scattered more regularly.
- The second one, when de Index of Dispersion is more than1,
called over-dispersed, indicates larger data dispersion. This case
usually treated by testing a negative binomial distribution. This
case points out the possible existence of clusters in data
throughout time.
Results
The number and kind of score changes regularly in each phase
of the game and at the beginning, half and end of each quarter
(Figure 1). Unlike what happens in each game phase, the last
minute presents a score significant increase, especially in 1-point
baskets. We also observed a totally different score distribution in
each phase.
The Figure-2 shows the evolution of the number of
substitutions of each player every ten seconds of the game,
emphasizing game key moments (upper plot).
In the next plot (Figure 3a) we show the comparison
substitutions evolution according to the level of uncertainty of
the final result (competed games vs. non-competed games). As a
consequence, we show graphically the relationship between this
parameter and the timeouts used by teams (Figure 3b). Note that,
idem with the points scored, there is a self-similarity between the
first and second half and between the first and third quarter.
Significant changes are only observed at the end of the last
quarter, where both parameters are clearly different from the
second.
Substitutions in the last minute vary according to the score
achieved during the last minute of the game. Games with a high
uncertainty present a higher number of events, in comparison to
those which are already decided, therefore, the number of events
is reduced.
Discussion
Apparently, basketball games show a high variability in
individual and collective actions. However, when we analyze a
significant sample of NBA basketball games, within their
unpredictability, we observe repetitive behavior patterns such
as free scale (de Saá et al., 2011; de Saá et al., 2013; Gabel and
Redner, 2012; Vaz de Melo et al., 2012). These behaviors are
detected repeatedly but specifically for each variable during the
NBA games of the regular season.
However, the degree of complexity is not always the same.
Patterns change between the first and second half of the game,
first and third quarter compared to the second and fourth, or the
first, half and end of each quarter. Furthermore, unlike what
happens during the rest of the game, when the game is
competitive, in the last minute the game dynamics change
significantly, thus has an influence in the substitutions of the
players, game systems, personal fouls and, of course, number
of baskets scored per unit time, ID (variance/mean) and the
result of the game.
Also, there is a clear difference in how points are scored
during the last minute of the game in comparison with the rest
of quarters. At the end, most points are scored at intervals of
≤10 seconds, while in the rest of quarters most of the points are
scored at intervals of 12 and 24 seconds. In table 1 note that
between zero and three points, the distribution of one, two or
three point shots are similar to any other stage of the game,
except that the percentages of triple is greater. Between 4 and 9
the number of Free Throws is higher, and over 10 is dominant,
being clearly a case of extreme complexity, dominated by fouls
and the time remaining as a limiting factor.
The use of timeouts is a special characteristic of basketball
that allows managing the game time depending on the needs of
each team (Saavedra et al., 2012a; Skinner, 2011). Some authors
consider this element as a team strategy to alter the game
Figure 1. Index of Dispersion of the point scored by minute. We can observe that the trend of the values is to rise over time. Only at
the end of each quarter there are a significantly increase, closer to 1, but only at the minute 47 reach the value 1 (pure Poisson). The
minute 48 is completely out the range of the rest of the game, reaching values higher than 1. The minute 48 requires special attention.
As we can observe, the minute 48 exceed the value 1 significantly (over-dispersed). This suggests that the last minute in a basketball
game is a completely different process than the rest of the game, meaning the game has changed its dynamic. The subplot “a” represents
the sum of points every minute throughout the game time until minute 45. The red line represents 2 points, the blue line 1 points and
the black line 3 points. The subplot “b” shows the difference the last minute, where it is emphasized that the number of failures is
much higher.
NBA last minute
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Revista de Psicología del Deporte. 2015, Vol 24, Suppl 1, pp. 31-35
Scored Points Point Shots Made (in percentage)
1 2 3
Game Total Points (%) 34.0% 55.0% 11.0%
Minute 48 Total 58.0% 31.0% 11.0%
0-3 Points / Mi 35.6% 49.3% 15.1%
04-sep Points / Mi 63.6% 27.3% 9.1%
More than 10 Points / Mi 73.1% 18.8% 8.1%
Table 1. Shows the point shots made (%) of 1, 2 or 3 points in the total game and in the last minute of the game. This table also shows the
distribution of the different kind of baskets when achieving 0-3, 4 to 9 or more than 10 baskets per minute.
Figure 2 evolution of the number of players substitutions every ten seconds of the game, highlighting the game key moments (upper plots).
Figure 3 : Subplot A: compares the evolution of substitutions according to the level of uncertainty on the final scores (competitive games
vs non-competitive games). Subplot B shows the relationship between changes of the players and the timeouts used by the teams throughout
the game time.
dynamics and influence the result (Mace et al., 1992; Roane et
al., 2004). However, Saavedra et al. (2012b), in the statistic
study performed with 3000 NBA games, suggests that this
opinion may be erroneous; indicating that the timeout reinforces
the game of the team that is dominating the game at that
moment, despite this team is not the one that requested the
timeout. In the NBA, to our understanding, timeouts availability
is complex.
From the results observed, timeouts significantly mark the
strategy that will follow each team at the end of the game, thus
possible changes in the game system performed. In NBA league,
each team is entitled to six charged timeouts during regulation
play. Each team is limited to no more than three timeouts in the
fourth period. If a team has two or three full timeouts remaining
when the fourth period reaches the 2:00 mark, one of the
timeouts will be changed to a 20-second timeout and it will
retain one full timeout. In overtime periods, each team shall be
allowed two 60-second timeouts. If a team has two full timeouts
remaining when the overtime period reaches the 2:00 mark, one
of the timeouts will be changed to a 20-second timeout. There
must be two 100-second timeouts in the first and third periods
and three 100-second timeouts in the second and fourth periods.
If neither team has taken a timeout prior to 5:59 of the first or
third period, it shall be mandatory for the Official Scorer to take
it at the first dead ball and charge it to the home team. If no
subsequent timeouts are taken prior to 2:59, it shall be
mandatory for the Official Scorer to take it and charge it to the
team not previously charged. If neither team has taken a timeout
prior to 8:59 of the second or fourth period, a mandatory timeout
will be called by the Official Scorer and charged to neither team.
If there are no subsequent timeouts taken prior to 5:59, it shall
be mandatory for the Official Scorer to take it at the first dead
ball and charge it to the home team. If no subsequent timeouts
are taken prior to 2:59, it shall be mandatory for the Official
Scorer to take it and charge it to the team not previously charged
(Official rules of the National Basket Association 2013-14).
Note how rules are a conditioning variable to employ in the
voluntary interruption of the game, affecting directly the
moments in which players are substituted during the game.
Certainly, this circumstance will significantly impact the
offensive and defensive potential of each team, their way of
solving the game situations and the evolution of the results. If
we add these two variables to the points scored in the last minute
of each quarter and, especially, at the end of the game, we
observe a change in the game complexity which looks forward
to solving quickly and favorably every ball possession.
Conclusion
Basketball, understood as a sport where the key to figure
out how to abort every offensive opponent action. Basketball
shows in the final moments of every competed games, situations
that often present some characteristics such as: shorter
possessions (especially by the disadvantage team), played with
fewer number of passes and participating players, higher
number of fouls, higher game stops and number of changes.
Juan Manuel García-Manso, Juan Manuel Martín-González, Sergio L. Jiménez, Yves de Saa and Teresa Valverde
34 Revista de Psicología del Deporte. 2015, Vol 24, Suppl 1, pp. 31-35
ÚLTIMO MINUTO EN LOS PARTIDOS DE LA NBA
PALABRAS CLAVES: Baloncesto, NBA, Fractal, Rendimiento.
RESUMEN : Aparentemente, el comportamiento del juego durante un partido de baloncesto, igual que en el resto de deportes de equipo, muestra una
enorme variabilidad que se manifiesta tanto en las acciones individuales y en las colectivas. No obstante, cuando se estudia una cantidad significativa
de partidos se detecta que, dentro de la impredecibilidad que caracteriza al juego, existen, al menos en el baloncesto NBA, patrones de comportamiento
repetitivos que, frecuentemente, son libres de escala. El grado de complejidad del juego no es estable. Diferentes investigaciones muestran como los
patrones cambian entre el primer y segundo tiempo del partido, así como, primer y tercer cuarto respecto al segundo y cuarto, o la parte inicial, mitad o
final de cada cuarto. Nuestro objetivo fue comprobar y evaluar la existencia de estos patrones, y su aparente complejidad, analizando el final de los
partidos de la NBA y comparando esta fase del encuentro con el final de cada uno de los cuartos. Los datos muestran la existencia de simetrías (fractalidad)
y patrones repetidos de juego durante los partidos de baloncesto de la NBA.
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