Key takeaways
- Before calling a run of results momentum, ask how long the runs produced by pure chance would be over the same number of attempts.
- Distrust streak statistics computed inside short sequences: the standard measure is biased against finding a hot hand.
- Expect genuine hot-hand effects to be brief, and to be partly hidden in games where opponents and shot choices adjust.
- Keep how momentum feels separate from what the scoreboard shows, because perception can swing faster than performance.
Momentum is real in some settings, but it is narrower than the word suggests. In the controlled shooting experiment at the centre of a famous 1985 study, a 2018 statistical correction found players hitting about 13 percentage points more often after three makes than after three misses, and a large study of professional darts found strong but short-lived runs of form. Outside controlled settings the evidence is thinner: in games, opponents and shot choices adjust, and much of what gets called momentum is the ordinary clustering that random sequences produce.
The history matters, because for more than thirty years the accepted answer was the opposite. That 1985 study called the hot hand a fallacy, and it took a subtle flaw in how streaks are measured to reverse the verdict.
Two claims hiding in one word
"Momentum" is used for two different things, and the evidence for each is different.
| Claim | What would show it | Evidence discussed below |
|---|---|---|
| Performance momentum (the "hot hand") | Success more likely after success than after failure, beyond what chance produces | Bias-corrected controlled shooting; professional darts |
| Psychological momentum | A competitor's sense of control and confidence moving with the score | A table tennis scenario experiment |
| Momentum as a story | A run of results that looks meaningful in hindsight | Usually explained by chance clustering |
The first two can come apart. A player can feel unstoppable without performing any better, and a real hot spell can pass unnoticed.
Why chance alone makes streaks look suspicious
Take a shooter who makes exactly half her shots, every shot independent of the last. She takes three shots. Write H for a make and M for a miss, and for each possible sequence look only at the shots that come straight after a make.
| Three-shot sequence | Shots that follow a make | Share of those shots that were makes |
|---|---|---|
| MMM | none | no value |
| MMH | none | no value |
| MHM | the third | 0 |
| HMM | the second | 0 |
| MHH | the third | 1 |
| HMH | the second | 0 |
| HHM | the second and third | 1/2 |
| HHH | the second and third | 1 |
All eight sequences are equally likely. Averaging the six that produce a value gives (0 + 0 + 1 + 0 + ½ + 1) ÷ 6 = 2.5 ÷ 6 ≈ 0.417. A 50% shooter appears to make only 41.7% of her shots after a make.
Nothing about her shooting changed. The distortion comes from the method: each sequence gets one vote, however many qualifying shots it contains, so the sequences rich in back-to-back makes (HHM and HHH) contribute several shots but count no more than a sequence with a single qualifying miss. Miller and Sanjurjo proved that this streak selection bias exists in any finite sequence of independent outcomes; it shrinks as sequences get longer, grows with the length of the streak being studied, and remains substantial at sequence lengths often used in empirical work.
The 1985 study that named the fallacy
Gilovich, Vallone and Tversky set out to test a belief almost everyone in basketball held. They surveyed fans, analysed field-goal and free-throw records from professional basketball, and ran a controlled shooting experiment designed to remove shot selection and defensive pressure.
That experiment was the heart of the paper. Twenty-six players from Cornell University's men's (14) and women's (12) teams each took 100 shots from a distance at which they were expected to make about half, changing position after every shot. After three makes they hit 49%; after three misses, 45%. The four-point difference was not statistically significant. The authors described shooting as analogous to coin tossing and, as Miller and Sanjurjo quote them, called the belief in the hot hand "a powerful and widely shared cognitive illusion". For decades the result was treated as a stark example of people misreading randomness, and the belief it targeted became known as the hot hand fallacy.
The 2018 correction
The 1985 comparison is exactly the measurement the three-shot table shows to be biased. Miller and Sanjurjo calculated that, for a shooter taking 100 shots with a 50% success rate, comparing performance after three makes with performance after three misses should produce a gap of about −8 percentage points if shots were truly independent.
Against that benchmark, the observed +4 points is roughly 12 points better than an independent shooter would produce. A more careful player-by-player correction put the average difference at +13 percentage points, statistically significant. Their abstract states that once the bias is corrected, "the long-standing conclusions of the canonical study are reversed."
What the correction establishes, and what it does not:
- It does show that the original data contain evidence of hot-hand shooting, and the authors report moderate to large bias-adjusted effects across the controlled shooting studies they re-examined. They conclude that "belief in the hot hand is not a cognitive illusion."
- It does not show that people judge streaks accurately. The same authors say players and coaches may still see a hot hand where none exists, overreact to a real one, or miss one.
- It does not settle in-game behaviour. In games, defences adjust to a player who is scoring and shot choices change. The authors note that the drop in shooting percentage after makes seen in game data disappears once even partial controls for defensive pressure and shot location are added.
Beyond the basketball court
Darts. Marius Ötting and colleagues studied almost a year of professional tournament data, 167,492 throws, in a sport with minimal interaction between players. Using models in which a hidden state stands in for a player's underlying ability, they found "a strong but short-lived serial dependence" in that state: evidence for a hot hand that fades quickly.
Golf. Clark examined hole-to-hole scores from 747 tournaments played by 35 randomly chosen professionals in 1997. Players were just as likely to make a birdie or better after a par or worse as after a birdie or better, and the study found no evidence of a hot hand. That is one season of one kind of data, and a null result is not proof that no effect exists: a small or brief one could sit inside the hole-to-hole noise.
Table tennis, and the feeling of momentum. Den Hartigh and Gernigon showed 80 players a scenario in which they either came back from 1–7 to 7–7 in a deciding game or watched an opponent do it. Perceived momentum rose during the comeback and fell quickly during the collapse. A time-out at 7–7 cut the positive momentum and helped players recover from the negative, while the instructions given during it made no significant difference. The study measured perceptions in a staged scenario, not shots played, which is precisely the distinction the table above draws.
Putting momentum into a simulation
A simulation has to decide whether consecutive outcomes are linked, and the research offers three design lessons:
- Streaks arrive for free. A model with no momentum term at all still produces long runs, because chance clusters outcomes. Seeing streaks in simulated results is not evidence that the model contains momentum.
- If momentum is included, keep it brief. The darts result describes a dependence that is strong while it lasts and decays quickly, which argues for a term that fades within a session rather than one that carries across a season.
- Separate the feeling from the probability. A pressure or confidence mechanic changes how outcomes are distributed; it should not be presented as the game reading a player's mind.
Golf Career Simulator Pro Tour builds momentum into its model: momentum, confidence, fatigue and pressure move within an event, so a strong front nine is not a guarantee of anything. Shots are decided by how you choose to play them, weighted by your golfer's skills and the conditions, and every tour, event and golfer in the game is fictional. A hot round in it is a property of the game's model, not evidence about real golfers. How the same game approaches pressure is covered in modelling pressure in a golf career sim.
When you read your own simulated results, the three-shot table is the warning. Compare a run with the runs chance alone would produce over the same number of attempts, which is the exercise understanding sports sim probability walks through. For tennis, where single points can swing a match, playing the big points looks at what research says about deciding under pressure.
Five questions to ask of any streak claim
- Was the streak chosen after the fact? Pick the most striking run from a whole season and it will almost always look meaningful, because it was selected for looking that way.
- How long is the sequence? In short sequences, the after-a-streak comparison is biased against finding a hot hand, so a null result there is weaker evidence than it looks.
- Did the difficulty change? A player on a run may take different shots, and opponents adjust to stop them. Rate measures that ignore context mislead, which is the theme of basketball advanced stats explained.
- Is the claim about performance or about feeling? "We had momentum" usually describes the second.
- What is the baseline? Compare a player's run with that player's own long-run rate, not with the league average.
More explainers arranged by sport are on the sports hub, and the studio's simulation titles are listed under sports GM.
Good to know
Frequently asked questions
Is momentum in sports real?
What is the hot hand fallacy?
Was the hot hand fallacy debunked?
Is there a hot hand in golf?
What is psychological momentum?
Sources
- Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers (Econometrica, 2018) (opens in a new tab)The Econometric Society — accessed
- Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers (full text) (opens in a new tab)arXiv (Miller and Sanjurjo) — accessed
- Hot hand (opens in a new tab)Wikipedia — accessed
- The Hot Hand in Professional Darts (opens in a new tab)arXiv (Ötting, Langrock, Deutscher and Leos-Barajas) — accessed
- An examination of the "hot hand" in professional golfers (Clark, 2005), abstract (opens in a new tab)Perceptual and Motor Skills, via NCBI PubMed E-utilities — accessed
- Time-out! How psychological momentum builds up and breaks down in table tennis (Den Hartigh and Gernigon, 2018), abstract (opens in a new tab)Journal of Sports Sciences, via NCBI PubMed E-utilities — accessed
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