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In 1989 Tversky and Gilovich published a paper entitled [“The Cold ...
#### Bernoulli trials In probability and statistics a *Bernoulli t...
### Analyzing over 600k free throws between 2006 and 2016 I foun...
### Simpson’s Paradox The **Simpson’s paradox** is a statistical p...
#### Hypergeometric distribution Discrete probability distribution...
I again used the 2006-2016 NBA free throw dataset that I mentioned ...
$Z$ represents the number of standard deviations ($\sqrt{var(a)}$) ...
If we recompute these tables for the data referenced in the previou...
A very interesting recent article on this (with some great referenc...
If once again we look at our new data set (as mentioned in previous...
Simpson’s Paradox and the Hot Hand in Basketball
A number of psychologists and statisticians are interested
in how laypersons make judgments in the face of uncer-
tainties, assess the likelihood of coincidences, and draw
conclusions from observation. This is an important and
exciting area that has produced a number of interesting
articles. This article uses an extended example to demon-
strate that researchers need to use care when examining
what laypersons believe. In particular, it is argued that the
data available to laypersons may be very different from
the data available to professional researchers. In addition,
laypersons unfamiliar with a counterintuitive result, such
as Simpson’s paradox, may give the wrong interpretation
to the pattern in their data. This paper gives two recom-
mendations to researchers and teachers. First, take care
to consider what data are available to laypersons. Sec-
ond, it is important to make the public aware of Simpson’s
paradox and other counterintuitive results.
KEY WORDS: Hot hand phenomenon; McNemar’s test;
Multiple analyses; Simpson’s paradox.
Schoolchildren routinely learn to identify optical illu-
sions. It is arguably as important that the general public
learn to identify statistical illusions. Many outstand-
ing researchers have addressed this issue. As examples,
Diaconis and Mosteller (1989) investigate computing the
probabilities of coincidences; Kahneman, Slovic, and
Tversky (1983) consider judgments made in the presence
of uncertainty; and Tversky and Gilovich (1989) inves-
tigate the popular belief in the hot hand phenomenon in
basketball. This article examines some of the data pre-
sented by Tversky and Gilovich.
Suppose that a basketball player plans to attempt
shots, with each shot resulting in a hit or a miss. A statis-
tician might assume tentatively that the assumptions of
Bernoulli trials are appropriate for this experiment. Sup-
pose next that the experiment
performed and the player
obtains the following data:
these data provide convincing evidence against the
tentative assumption of Bernoulli trials? Are the three
occurrences of three successive hits convincing evidence
of the player having a “hot hand”? These are difficult
questions to answer because of the myriad of possible
alternatives to Bernoulli trials that exist. It is mathemat-
ically and conceptually convenient to restrict attention to
alternatives that allow the probability of success on any
trial to depend on the outcome
the previous trial or,
perhaps, the outcomes of some small number of previous
Wardrop is Associate Professor, Department of Statistics,
University of Wisconsin-Madison, Madison,
The author
thanks the referees and associate editor for helpful comments.
American Statistician, February 1995,
trials. (This restriction may be unrealistic, but that issue
will not be addressed in this article.) With the restrictive
class of alternatives described here, Tversky and Gilovich
devised a clever experiment to obtain convincing evidence
that knowledgeable basketball fans are much too ready to
detect occurrences of streak shooting-the hot hand-in
sequences that are, in fact, the outcomes of Bernoulli trials.
Having established that basketball fans detect the hot
hand in simulated random data, Tversky and Gilovich
next examined three sets of real data. The data sets are:
shots from the field during National Basketball Associa-
tion (NBA) games; pairs of free throws shot during NBA
games; and a controlled experiment using college varsity
men and women basketball players. Using the restrictive
alternatives described above, Tversky and Gilovich found
no evidence of the hot hand phenomenon in any of their
data sets. In addition, using a test statistic that is sensitive
to certain time trends in the probability of success, they
again found no evidence of the hot hand phenomenon.
This article examines the free throw data presented by
Tversky and Gilovich. Tversky and Gilovich began by
asking a sample of 100 “avid basketball fans” from Cornell
and Stanford: “When shooting free throws, does a player
have a better chance of making his second shot after mak-
ing his Erst shot than after missing his first shot?” A
“Yes” response was interpreted as indicating belief in the
existence of the hot hand phenomenon, and a “No” as
indicating disbelief. (Actually, a
response com-
bines persons who believe in independence with those who
believe in a negative association between shots; but the
researchers apparently were not interested in separating
these groups.) Sixty-eight of the fans responded “Yes” and
the other 32 “No.” Thus, a large majority of those quest-
ioned believed in the hot hand phenomenon for free throw
shooting. Tversky and Gilovich investigated the above
question empirically by examining data they obtained on
a small group
well-known and widely viewed basket-
ball players, namely, nine regulars on the 1980-1981 and
1981-1982 Boston Celtics basketball team.
After their analysis of the Celtics data, Tversky and
Gilovich concluded that “These data provide no evidence
that the outcome of the second shot depends on the out-
come of the first.” Section 2 of this article will examine the
Celtics data with the goal of reconciling what Tversky and
Gilovich found and what their basketball fans believed,
In particular, it will be shown that, in a certain sense, the
prevalent fan belief in the hot hand is not necessarily at
odds with Tversky and Gilovich’s conclusion.
The analysis presented in Section 3 of this paper indi-
cates that several Celtics players were better at their second
shots than at their first.
It is instructive to begin by considering just two
nine Boston Celtics players who are represented in the free
throw data, namely, Larry Bird and Rick Robey. During
1995 American Statistical Association