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← Points Per Shot

Points Per Shot

Three short lessons on basketball numbers: expected value, the three-point revolution, and the long argument about the hot hand.

Spark & Whetstone · sparkwhetstone.org/learn/points-per-shot · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Expected points

A shot’s value per attempt is what it is worth times the chance it goes in. A two-pointer made half the time is worth 1.0 point per shot; a three made 35% of the time is worth 1.05.

That small difference, repeated over many shots, decides games.

Question. A free throw is worth 1 and a player makes 80%. A two-pointer goes in 45% of the time. Which is worth more per attempt?

  1. The free throw
  2. They are worth the same
  3. The two-pointer

Say it back. In your own words: how can a shot that misses more often be worth more?

Lesson 2. The three-point revolution

In professional basketball the share of shots taken from three has climbed for decades, as teams worked out that long twos are the worst value on the court: barely closer than a three but worth one point less.

The three-point line is also shorter in the corners, so corner threes are the best of the long shots.

Question. A shot just inside the arc goes in 40% of the time; one just outside goes in 34%. Which is worth more per attempt?

  1. They are equal, since the gaps in value and in chance cancel
  2. The one outside the arc
  3. The one inside the arc

Say it back. In your own words: why did teams start avoiding the long two?

Lesson 3. The hot hand

In 1985 three psychologists studied streak shooting and concluded that the "hot hand" was an illusion. In 2018 two economists showed a subtle bias in the way streaks had been counted: in short sequences, the expected share of hits after a hit is below the shooter’s true rate.

Corrected for that bias, the old data showed a small hot hand after all. How big it is in real games is still argued about.

Question. Flip a fair coin 3 times, many times over. In each set, look at flips that follow a head and take the share that are heads, then average those shares. What do you get?

  1. Less than a half
  2. More than a half
  3. Exactly a half

Say it back. In your own words: what did the 2018 paper say was wrong with the hot-hand test?

At the table: Paper-ball shooting

Players: Two or more, ages 6 and up. You'll need: A bin, scrap paper, tape for two lines on the floor.

  1. Mark a close line worth 2 points and a far line worth 3.
  2. Everyone takes ten shots from each line and keeps a tally of makes.
  3. Work out points per shot for each line: makes × value ÷ 10.
  4. Decide together which line is the better choice, and whether that is the same for everyone.

Answers

  1. The two-pointer — 1 × 0.8 = 0.8 and 2 × 0.45 = 0.9, so the two-pointer is worth a little more per attempt.
  2. The one outside the arc — 2 × 0.40 = 0.80 and 3 × 0.34 = 1.02. The slightly harder shot is worth a quarter of a point more.
  3. Less than a half — Averaging over the six sets with a flip after a head gives 5/12, about 42%. That is the selection bias the 2018 paper found.