in my mind, namely, the collapsed table for all players

I

have seen. Just like the Celtics data, my collapsed ta-

ble indicates that a success is more likely than a failure

to be followed by a success. Thus, there

is

a pattern in

the data that are reasonably available to me and, I conjec-

ture, in the data that are reasonably available to Gilovich

and Tversky’s

100

basketball fans. It seems reasonable

to suggest to basketball fans that the mental equivalent of

Simpson’s paradox could lead to a cognitive statistical il-

lusion that results in their “seeing patterns in the data that

do not exist.”

3.

STATIONARITY

Tversky and Gilovich correctly concluded that there is

no evidence of the hot hand phenomenon in the free throw

data. In this section, it is demonstrated, however, that the

simple model of Bernoulli trials is

also

inappropriate. In

particular, it

is

shown that several of the Celtics players

shot significantly better on their second free throw, perhaps

as a result of the practice afforded by the first shot.

Look at Table

1

again. Larry Bird made 84.3% (285 of

338) of his first shots compared to 88.5% (299 of 338)

of his second shots. Thus, there is evidence that he im-

proved on his second shot. The null hypothesis that his

probability of success was constant can be investigated

with McNemar’s test, which uses the fact that the null

distribution of

(4)

b-c

2,

=

___

can be approximated by the standard normal curve. (Re-

call that

b

and

c

are defined in Table 2.) For Larry Bird,

b

=

34 and

c

=

48, giving

Jb?-c

34

-

48

21

=

~~

=

-1.55.

The same analysis can be performed for the other eight

Celtics; the results are given in Table 5. The first col-

umn of the table lists the player’s names. The second and

third columns list, respectively, the relative frequencies

of successes on the first and second shots. The remain-

ing columns list the values of

b

and

c

from each player’s

2

x

2 table and the value of

z1

computed from Equation

(4). The players are listed according to the difference in

relative frequencies between the first and second shots.

Table

5.

Selected Statistics for Comparing the Success Rates on

the First and Second Free Throws for Nine Members of Boston

Celtics

Player

as,)

i;cS2,

b

c

Zl

Cedric Maxwell

.TO

.ao

57 97

-

3.22

Robert Parish

.67 .75 49 76

-

2.41

Nate Archibald

.76 .83 42 62

-

1.96

Rick Robey

.53 .60 37 49

-

1.29

Larry Bird

.84

.aa

34 48

-

1.55

Gerald Henderson

.73 .77 24 29 -.69

Chris Ford

.70

.73 15 17

-

.35

Kevin McHale

.72 .69 35 29 .75

Total

-

M.

L. Carr

.a

.72

ia

21

-

.4a

-

311

428

z2=

-

4.30

Thus, Maxwell, who shot ten percentage points better on

the second shot than on the first, is listed first, and McHale,

who shot three percentage points better on the first shot,

is listed last. Note the following features of the data.

(1)

Eight of nine players had a higher success rate on

their second shots.

(2) Three players had one-sided approximate

P

val-

ues below

.05:

Maxwell

(.0006),

Parish

(.0080),

and

Archibald (.0250). The interpretation of these

P

values

should take into account that nine tests were performed.

If, in fact, each player had a constant success rate on

his two shots, the approximate probability of obtaining

at least one

P

value equal to or smaller than

.0006

is:

1

-

(1

-

.0006)9

=

.0054.

Similarly, the approximate

probability of obtaining at least two

P

values equal to

or smaller than

.0080

is .0022. Finally, the approximate

probability of obtaining at least three

P

values equal to or

smaller than .0250 is .0012. Thus, the three statistically

significant results do not seem to be attributable to the

execution of many tests.

(3) McNemar’s test can be viewed as testing that a

Bernoulli trial success probability equals

.5

based on a

sample of size

b

+

c.

Thus, several of the analyses

of

in-

dividual pIayers presented in Table

5

are based on very

little data and, hence, have very low power. To combat

this difficulty, it is instructive to combine the data across

the nine players. In particular, if the null hypothesis of

constant success probability is true for all nine players,

then the observed value of

where the

sum

is taken over the nine tables, can be viewed

as an observation from a distribution that is approximately

the standard normal curve. The observed value of

Z,

is

-4.30, given in the bottom row of Table 5. This value

indicates that there is overwhelming evidence against the

assumption that all nine null hypotheses are true.

4.

SUMMARY

This article puts forth an argument to reconcile what

avid basketball fans believe and what Tversky and

Gilovich found. It is argued that the fans and the re-

searchers were analyzing different sets of data. While

the researcher’s data had no pattern, the fan’s data had

a pattern. This pattern, however, was due to the effects

of aggregation and not the hot hand phenomenon. This

finding indicates that researchers should take care to con-

sider what data are available to laypersons. In addition,

this finding underscores the importance of increasing the

awareness of statistical fallacies among the general public.

This article also demonstrates that several Celtics play-

ers showed a significant improvement in their shooting

ability on the second free throw. Thus, while the hot hand

phenomenon is not supported by these free throw data,

neither is the simple model of Bernoulli trials.

[Received March

1992.

Revised November

1993.1

The American Statistician, February

1995,

Vol.

49,

No.

I

27