Early-Season 3-Point Percentages: How Many Shots Are Enough?

A new season opens, and within a week someone is shooting 70% from three. Fans call it a breakout, analysts call it small sample size, and both are partly right. Statistics can say precisely how much a handful of shots tells us, and how fast that grows.

Seven out of ten

A player who starts 7 for 10 shot 70%, but a shooter with a true 36% rate would make seven or more of ten about 3% of the time. That is rare for one player, though across hundreds of players in the first week, a few such starts are expected. The binomial distribution calculator gives the tail probability, and the confidence interval for a proportion gives a plausible range of about 42% to 98% with the simple method, so the true rate could be anywhere from ordinary to elite.

How the margin shrinks

Precision grows with the square root of the number of attempts. At a 36% rate, the 95% margin of error is about ±9 points after 100 attempts, ±5 after 300, and ±4 after 600. The sample size calculator shows that ±10 points needs roughly 90 attempts, ±5 needs about 350, and ±3 needs nearly 1,000.

Some analysts measure stability differently. One analysis of league data found that team three-point percentage varied by under 10% after around 160 attempts and under 5% after roughly 300 (Basketball Analytics Lab).

Shrinking toward the average

A practical fix is padding, or shrinkage: add some average attempts to a player's record. Suppose we add 100 attempts at 36%. A 7-for-10 start becomes (7 + 36) / (10 + 100), about 39%, a believable estimate. Writers have used similar methods to suggest that after a couple of hundred attempts a player's percentage becomes more informative than the league average (The F5).

One example shows the point. A rookie who shot 42% on 62 attempts looked like a veteran who shot 42% on 677, but the veteran's larger sample was far more predictive the next season (BBall Index).

Hot starts and regression to the mean

Extreme early results tend to move back toward the average, because part of any extreme result is luck. That is regression to the mean, and it applies to shooting, as it does to the rest of life. Our article on football goals and streaks shows the same pattern.

A quick checklist

Not all threes are equal

Raw percentages also hide shot quality. An open corner three and a contested pull-up are different shots, so two players with identical percentages may have very different skill. Analysts adjust for this with expected percentages based on shot type and defender distance, and the padding idea above is a simple version of shrinkage methods used in more formal Bayesian models.

None of this makes percentages useless. It just means a number needs a sample size and a context before it deserves a headline.

When you see an early shooting percentage, ask how many attempts it rests on, what range an interval gives, what the player did last season, and what the shot quality was. A 70% start is a conversation starter. A 40% season on 600 attempts is a fact. Early in the season, patience is a better analytic tool than excitement.