Test of Stationarity

Test of Stationarity

To detect temporary elevations in performance—periods in which a player’s hit rate is substantially above his overall average—the test of stationarity partitions a player’s entire shooting record into nonoverlapping sets of four consecutive shots. Within each set, performance is classified as high (three or four hits), moderate (two hits), or low (zero or one hit). If a player occasionally experiences “hot” streaks, his record should contain more high-performance sets than would be expected if the shots were independent and governed by a constant hit rate.

The expected proportions of high, moderate, and low sets under the constant-rate assumption are derived from the player’s overall shooting percentage. For a player with a 50% hit rate, for example, the expected proportions are 5/16 high, 6/16 moderate, and 5/16 low. A chi-square goodness-of-fit test is then used to compare the observed frequencies of the three categories to these expected values. To avoid any artifact introduced by an arbitrary starting point, the entire analysis is repeated four times—beginning the partition into quadruples at the first, second, third, and fourth shot of the player’s record—and the results are examined for consistency.

This approach was applied across multiple studies. In an initial investigation of nine major college basketball players, none of the chi-square values approached statistical significance, offering no support for the nonstationarity hypothesis. A subsequent study that tracked 100 shots per player yielded similar findings; for every player except one (Player 9), the test failed to detect any departure from stationarity. Even in that single exception, the evidence was not robust across all four starting-point partitions, so the reported tests did not provide consistent evidence of hot-hand streaks. Failure to detect a departure does not establish that the hit probability was constant.