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The clustering illusion is the intuition that random events which occur in clusters are not really random events. The illusion is due to selective thinking based on a counterintuitive but false assumption regarding statistical odds. For example, it strikes most people as unexpected if heads comes up four times in a row during a series of coin flips. However, in a series of 20 flips, there is a 50% chance of getting four heads in a row (Gilovich). It may seem unexpected, but the chances are better than even that a given neighborhood in California will have a statistically significant cluster of cancer cases (Gawande).
What would be rare, unexpected, and unlikely due to chance would be to flip a coin twenty times and have each result be the alternate of the previous flip. In any series of such random flips, it is more unlikely than likely that short runs of 2, 4, 6, 8, etc., will yield what we know logically is predicted by chance. In the long run, a coin flip will yield 50% heads and 50% tails (assuming a fair flip and a fair coin). But in any short run, a wide variety of probabilities are expected, including some runs that seem highly improbable.
Finding a statistically unusual number of cancers in a given neighborhood--such as six or seven times greater than the average--is not rare or unexpected. Much depends on where you draw the boundaries of the neighborhood. Clusters of cancers that are seven thousand times higher than expected, such as the incidence of mesothelioma in Karian, Turkey, are very rare and unexpected. The incidence of thyroid cancer in children near Chernobyl was one hundred times higher after the disaster (Gawande).
Sometimes a subject in an ESP experiment or a dowser might be correct at a higher than chance rate. However, such results do not indicate that an event is not a chance event. In fact, such results are predictable by the laws of chance. Rather than being signs of non-randomness, they are actually signs of randomness. ESP researchers are especially prone to take streaks of "hits" by their subjects as evidence that psychic power varies from time to time. Their use of optional starting and stopping is based on the presumption of psychic variation and an apparent ignorance of the probabilities of random events. Combining the clustering illusion with confirmation bias is a formula for self-deception and delusion. For example, if you are convinced that your husband's death at age 72 of pancreatic cancer was due to his having worked in a mill when he was younger, you may start looking for proof and run the danger of ignoring any piece of evidence that contradicts your belief.
A classic study was done on the clustering illusion regarding the belief in the "hot hand" in basketball (Gilovich, Vallone, and Tversky). It is commonly believed by basketball players, coaches and fans that players have "hot streaks" and "cold streaks." A detailed analysis was done of the Philadelphia 76ers shooters during the 1980-81 season. It failed to show that players hit or miss shots in clusters at anything other than what would be expected by chance. They also analyzed free throws by the Boston Celtics over two seasons and found that when a player made his first shot, he made the second shot 75% of the time and when he missed the first shot he made the second shot 75% of the time. Basketball players do shoot in streaks, but within the bounds of chance. It is an illusion that players are 'hot' or 'cold'. When presented with this evidence, believers in the "hot hand" are likely to reject it because they "know better" from experience. And cancers do occur in clusters, but most clusters do not occur at odds that are statistically alarming and indicative of a local environmental cause.
In epidemiology, the clustering illusion is known as the Texas-sharpshooter fallacy (Gawande 1999: 37). Khaneman and Tversky called it "belief in the Law of Small Numbers" because they identified the clustering illusion with the fallacy of assuming that the pattern of a large population will be replicated in all of its subsets. In logic, this fallacy is known as the fallacy of division, assuming that the parts must be exactly like the whole.
Gawande, Atul. "The Cancer-Cluster Myth," The New Yorker, February 8, 1999, pp. 34-37.
Tversky, A. and D. Khaneman (1971). "Belief in the law of small numbers," Psychological Bulletin, 76, 105-110.