Clustering Illusion

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What is Clustering Illusion?

Clustering Illusion is the habit of treating a streak or a pile in a small random sample as evidence of a cause. Chance itself produces those clumps. A short sample is too short for an even, well-mixed picture to be the right picture of chance.

It contains too many alternations and not enough long runs.

That line is how Thomas Gilovich, Robert Vallone, and Amos Tversky describe a sequence built to look like chance in every small piece. The paper is “The hot hand in basketball: On the misperception of random sequences,” Cognitive Psychology 17 (1985), pages 295–314. A fair coin is half heads across a long run. People also want a short stretch to look half-and-half. That locally even stretch switches too often and runs too seldom. A real run of heads then looks like a biased coin, a hot shooter, or a street that was chosen. The same habit works on a map. Dots from a flat chance field leave empty squares and crowded squares. The crowded squares look selected. Representativeness Heuristic is the page for judging a sample by how well it matches the process you have in mind. This page is what happens when a real clump fails that match and receives a cause anyway.

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Clustering Illusion in 3 Depths

• Beginner: A run of heads, or a knot of dots, is ordinary in a short chance sample. Write how long the record is before you name a cause.
• Practitioner: Fix the squares, the week, or the rule before you look. Then compare the empty cells and the crowded cells with the count a flat rate would produce.
• Advanced: A clump can still be a real change in the rate. In a short record, successes that follow three successes fall below the base rate even when every trial has the same chance. A higher share has to clear that lower line. A region picked after the pile was seen answers a different question.

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Origin

The London map was counted in 1946. The misreading of short sequences was measured in 1985. The name Clustering Illusion is the label Gilovich uses for that family of misreadings in 1991. R. D. Clarke was a Fellow of the Institute of Actuaries at the Prudential Assurance Company. His note, “An application of the Poisson distribution,” is in the Journal of the Institute of Actuaries, volume 72 (1946), page 481. During the flying-bomb attack on London, he writes, frequent assertions said the impacts were grouped in clusters. Flying bombs are the V-1s. He tested the claim on 144 square kilometres of south London, chosen because the mean density did not vary materially inside that area. The squares and the chi-square are the case study below. His conclusion is that the fit lent no support to the clustering hypothesis. The psychology is Gilovich, Vallone, and Tversky (1985). Gilovich was at Cornell University. Vallone and Tversky were at Stanford University. A faculty research grant from Cornell’s College of Arts and Sciences supported Gilovich. Office of Naval Research grant NR 197-058 supported Tversky. The paper was accepted on 15 March 1985. They do not print the two-word name. They call the pattern a misconception of chance, and later a cognitive illusion. They tie the account to Amos Tversky and Daniel Kahneman, “Belief in the law of small numbers,” Psychological Bulletin 76 (1971), pages 105–110, and to the 1974 paper on judgment under uncertainty. People expect the marks of a chance process to show up in every short piece, as well as in the whole sequence. Gilovich, Vallone, and Tversky say that expectation creates two errors. One is the Gambler’s Fallacy: after a run of tails, the next toss feels due to be heads. The other is rejecting a sequence that has the runs chance actually produces, because those runs look too ordered. The second error is this page. Gilovich uses the name in How We Know What Isn’t So: The Fallibility of Human Reason in Everyday Life (The Free Press, 1991). The book gathers the basketball records and the habit of seeing order in random data. It is the naming source used here. It is not a new count of bombs or of shots.

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Key Points

Four checks keep a clump attached to the record that produced it. A felt streak, a grid fixed first, and a short sequence each answer a different question.

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Applications

Use a window you fixed first, the switch rate you expected, and the lower line a short streak is supposed to beat. Leave each printed figure inside its own study.

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Case Study

The counted picture is a map of flying bombs, divided before the test, then set beside a Poisson law. Clarke’s note is a single page. He says the example arose in a practical investigation. He does not name a client, a military order, or a theory about particular streets. He writes that frequent assertions were made that the points of impact tended to be grouped in clusters. He applied a statistical test to that allegation. The area was 144 square kilometres of south London. He selected it because the theoretical mean density was very nearly constant there. He divided it into 576 squares of a quarter of a square kilometre each. Over the period he considered, those squares held 537 flying bombs. The Poisson mean was 537 divided by 576. The expected number of squares, and the number he counted, line up as follows. No bomb: 226.74 expected, 229 counted. One bomb: 211.39 and 211. Two: 98.54 and 93. Three: 30.62 and 35. Four: 7.14 and 7. Five and over: 1.57 and 1. Each column sums to 576. The last row is a single observed square. The note does not print how many bombs that square held.

The closeness of fit which in fact appears lends no support to the clustering hypothesis.

He says what would have counted as clustering: an excess of squares with a high number of bombs or with none, and a deficiency in the intermediate classes. The chi-square on the comparison was 1.17. He reports 4 degrees of freedom. The probability of obtaining that value or a higher one was .88. A clustered map of the kind he defined would have produced a large chi-square and a small probability. This probability is large because the discrepancy is small. The lesson stays inside the grid. Once the area had been chosen for a nearly flat density, the empties and the piles match a Poisson field. The clusters the eye picks out were not an extra grouping of the kind the test was built to detect. The result leaves a wider map untouched. A rise and fall along the bombs’ line of approach, outside an area chosen for constant density, is a different question. The note also leaves aiming untouched. It says this grid did not support the extra clustering people asserted.

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Boundaries and Failure Modes

Clustering Illusion is the wrong summary in three places. A real difference in risk can pile events up. A short shooting record can hide a change in rate. The name can also be used to dismiss a cluster whose boundary was fixed first. The first boundary is the area. Clarke’s test starts after he has chosen a region of nearly constant mean density. The pattern that would have supported the clustering hypothesis is extra clumping inside that region. A chi-square of 1.17 did not support it. The same page says nothing about a gradient across a wider city, and nothing about a square he did not draw. Where the risk truly changes from block to block, one Poisson mean is the wrong model, and a pile can be the signal. The claim on this page is narrower. A flat field already looks lumpy. The second boundary is the basketball record. It is easy to over-read in both directions. The 1985 paper found no positive dependence of the size the fans described. Fans put a hypothetical 50 percent shooter at 61 percent after a make and 42 percent after a miss. The 76ers’ weighted means were 51 percent after a hit and 54 percent after a miss, across nine players and 48 home games. Serial correlations were negative for eight of the nine. Only Daryl Dawkins differed significantly from zero, and his correlation was negative. Paired comparisons across the nine ran against the hot-hand prediction. As the conditioning streak grew from one make to three, the tests were t = −2.79, t = −3.14, and t = −4.42. The printed probabilities were p < .05, p < .05, and p < .01. After a hot window of three or four hits in the last four shots, the weighted mean was 50 percent. After a cold window of zero or one hit in those four, it was 57 percent. Within single games, 727 player records with more than two runs gave t(726) < 1. For seven players with enough games, the Lexis ratio of game-to-game variation ran from 0.56 for Dawkins to 1.03 for Julius Erving, with a mean of 0.84. None was significantly above 1. An informal poll of 18 recreational players asked for five streak shooters in the league. Only two left out Andrew Toney, and half the respondents named him first. His runs did not depart from chance. Television records added seven New York Knicks across 22 games and seven New Jersey Nets across 13 games. Of 23 major players on the three teams, two had a significant serial correlation. Bill Cartwright’s was positive. Dawkins’s was negative. The interview at a 76ers practice fits the fans more than the shot chart. Seven players and the coach took part. Six of the eight said that after a few makes they sometimes know the next shot will go in. Five endorsed a better chance after two or three makes than after two or three misses. Two did not endorse that sentence, and one did not answer. Seven of the eight said they take more shots after a series of makes. The players all said the ball should go to someone who has just made several shots. Five players and the coach gave numbers. Five of those six put the percentage after a hit above the percentage after a miss. The means printed with those estimates are 62.5 percent and 49.5 percent. Miller and Sanjurjo do not treat the 76ers season as the study their bias reverses. They say in-game field goals are a poor test for a separate reason: the defense can tighten on a player who has just scored. They correct the controlled Cornell shoot. Fourteen men from the varsity and junior varsity, and twelve women from the varsity, shot from a distance set near 50 percent. Two 15-foot arcs were drawn, with their centers 60 inches out from the left and right sides of the basket. Players moved along the arc so two shots in a row did not come from the same spot. Each was asked for 100 shots, 50 from each arc, and three did not finish. The paper’s mean shooting percentage is 47 percent. The mean after a hit was 48 percent, and the mean after a miss was 47 percent. Serial correlations were negative for 14 of the 26 players. One player, labeled 9, showed a significant positive correlation. The runs test and the check on sets of four shots also singled out that player. Hot and cold windows of four shots averaged 46 percent and 47 percent. Paired tests on the one-shot, two-shot, and three-shot comparisons were all reported as t < 1. Miller and Sanjurjo reproduce the three-shot columns from that table. Players averaged 49 percent after a streak of three or more hits and 45 percent after a streak of three or more misses. The average gap is +3 percentage points. They replicate the paired test at p = .49. The baseline of a zero gap is the wrong baseline in a finite sequence. At the design target of 100 shots and a 50 percent rate, the expected gap is −8 percentage points. They also describe the observed average, under that shared target, as about +4 points, which sits +12 points above the −8 point expectation. The test they report at p < .01 uses each player’s own percentage. It moves the average from +3 to +13, with a standard error of 4.7 points. One player had no streak of three hits, so the sign count is 19 of 25 in the hot-hand direction, binomial p < .01. Five of those 25 were individually significant at p < .05, and that count of five is itself significant at p < .01 by a binomial test. A streak that starts at four hits gives them an adjusted average of 10 points, p = .07, one-sided, with a standard error of 6.9 points. They call the +13 conservative, because the downward pull is larger under some models in which a hot state is real. They also say the runs test, the serial correlation, and the one-shot comparison have little power to detect that state. The +13 is an average of adjusted differences on this task. The authors set it beside a 12-point gap between the median three-point shooter and the best one in the 2015–16 NBA season. The comparison comes from an ESPN table they accessed on 24 September 2016. That 12-point gap is a scale remark. The Cornell players were paid a base of 2 dollars, plus or minus bets. A high bet won 5 cents on a hit and lost 4 cents on a miss. A low bet won 2 cents or lost 1 cent. The average correlation of the shooter’s bet with the shot was .02. The observer’s was .04. Both bets tracked the previous shot, at .40 for the shooter and .42 for the observer. Five of the 26 shooter correlations were significant. Four were small and positive, between .20 and .22, and one was negative. The third boundary is misuse. A pile inside a region you named in advance can still beat the chance count, and the illusion is the wrong reason to ignore that excess. A search across streets, weeks, or metrics, kept only when the pile appears, is the other mistake. It paints the target after the holes. Illusory Correlation pairs two kinds of events and over-counts the pair. Availability Heuristic lets a memorable run stand in for the record. Gilovich, Vallone, and Tversky treat memory for long runs as a possible addition, then show the misreading with sequences laid on the page. Memory is not required. Illusion of Control turns a chance result into a personal skill. The Cornell bets are that picture. Shooter and observer agreed at .22, and at .05 once the previous shot was partialed out.

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Common Misconceptions

Three shortcuts travel with the name. Even spacing is treated as the look of chance. One basketball paper is treated as proof that streaks are imaginary. A different mistake is treated as a second name for this one.

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Related Concepts

Clustering Illusion sits next to a sample judged by resemblance, a run expected to reverse, and a target drawn after the holes. It also sits next to a memorable run, a paired coincidence, and a chance result read as skill.

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One-Line Takeaway