# Hot Hands, a Paradox, and one reason why it’s bad to combine within-subject data

The ‘Hot Hand’ phenomenon is a popular belief (applicable to many domains from sports to gambling) that players who were successful in their most recent attempt(s) have increased odds of being successful in their next attempt — they are on a so-called ‘hot streak’ or have a ‘hot hand’. The statistical validity of this belief can be investigated using actual data. Indeed, it has been. For example, Tversky and Gilovich (1989) investigated the hot hand belief in basketball.

Here however, we are not scrutinizing the hot hand belief, but rather using this framework and dataset to reveal the presence of a Simpsonian Paradox. The definition of this paradox will precipitate from the following example…

In the 1996/97 NBA jam seasons Michael Jordan shot a pair of free throws on 338 occasions. MJ made both 251 times, missed both 5 times, made only the first 34 times, and made only the second 48 times. These data are presented in the table above, as are the same data for Dennis Rodman, and also their combined numbers.

Let Phit and Pmiss denote the proportion of first shot hits followed by a hit, and the proportion of first shot misses followed by a hit, respectively. These proportions for Jordan and Rodman, along with their combined numbers are:

Jordan
Phit = 251 / 285 = .881
Pmiss = 48 / 53 = .906

Rodman
Phit = 54 / 91 = .593
Pmiss = 49 / 80 = .612

Combined
Phit = 305 / 376 = .811
Pmiss = 97 / 133 = .729

Notice that, contrary to the hot hand phenomenon, MJ actually shot worse after he made the first of two freethrow shots. The same goes for Rodman.

So both players are actually worse on their second freethrow, on attempts when they’ve made their first shot.

Combining MJ and Rodman’s freethrow data together, the opposite is true. This is the Simpsonian paradox.