# Math Madness #3: Jump Shots and Expectation

For those of you who do not embrace, much less embody, stereotypical geek indifference to athletics, you may have noticed that it is NCAA basketball tournament time. In fact, the final game between the Kansas and Kentucky is just about to tip-off. Living in the UK, I haven’t watched much college basketball this season, but I’m picking Kentucky in a close one.

Let me tell you why.

You win basketball games by scoring more points than your opponent. That much is obvious. But how do you make that happen? The math is relatively simple. You win basketball games by maximizing the number of points you score and minimizing the number of points your opponent scores with each possession.  Again, how do you make that happen?

A team’s offensive efficiency represents the average number of points they score with each possession. Similarly, a team’s defensive efficiency represents the average number of points their opponent scores with each possession. These statistics are dependent on both strategy and ability. Depending on a team’s shooting accuracy, different shots are expected to score different amounts of points per attempt (PPA; Figure 1).

While basketball commentators may mourn the death of the mid-range jump shot amongst the players of today, we can see that the two point jump shot is the least efficient shot1 on the court.

To maximize offensive efficiency, the strategy employed by a successful team should favor the most efficient shots and avoid inefficient shots. That means avoiding the lamented mid-range jump shot. The relative frequency of mid-range jump shots as a fraction of total shot attempts is inversely correlated with a team’s winning percentage (R=-0.42; Figure 2).

Efficient offensive teams favor efficient shots. Based on what we now know about expected points per attempt, we would expect an efficient team to follow a strategy like this:

1. Drive to the basket, get fouled (shooting foul = 1.38 PPA) or shoot a layup (~1.20 PPA).
2. Pass to an open three-point shooter (1.03 PPA).
3. If time is running out on the shot clock, shoot a two point jump shot (0.685 PPA).

Efficient defensive teams apply this to their strategy too. They limit their opponent’s foul shots (opponent foul shot attempts vs winning percentage: R=-0.36), deny layups (opponent layup attempt frequency vs winning percentage: R=-0.30), and prevent open three-point shots (opponent 3FG% vs winning percentage: R=-0.41).

With efficient defensive teams, you will frequently see a defender run hard at a three-point shooter. The shooter fakes, dribbles past the defender, and pulls up for a mid-range, two point jump shot. If the shot is made, the TV commentary will invariably praise the shooter and criticize the defense for being out of control. The numbers tell us that the shooter took the shot the defense was probably designed to give him.

The mid-range jump shot is a lost art. It should stay lost.

I’m picking Kentucky because they are more efficient overall than Kansas. Kentucky normally scores 0.22 more points per possession than their opponents. Kansas scores 0.18 more. But, it should be close. How confident am I in this pick? Not very. Maybe for next year I’ll build a model that will tell you precisely how uncertain I am.

I wrote the original version of this back in 2009. For this year, I’m breaking it into more digestible chunks, with a lot of rewriting and up to date statistics, when possible.

NOTES

Raw data for all NCAA Division I men’s basketball teams was obtained from DonBest.com

1. The raw data do not include breakdowns of two point shot selection. Therefore, I have inferred the frequency of dunks/layups and jump shots. To do so, I assumed teams successfully made jump shots at the same rate as three-point shots, based on prior observations of shooting frequencies. I also assumed that teams made dunks/layups with ~60% success rate based on limited data from specific teams in multiple years. The results are not highly dependent on the precision of this estimation.

## Author: Josh Witten

http://www.thefinchandpea.com