10 - The Law of Small Numbers

Описание: We tend to draw strong conclusions from small samples, as if they were representative. Kahneman explains why this produces errors in research, business, and daily life: we see patterns where there is chance. The chapter reinforces basic statistical intuitions about variability and sample size.

Kahneman introduces the "law of small numbers"—a tongue-in-cheek reference to the law of large numbers, which states that large samples are more representative of the population than small samples. The problem is that our intuitions violate this principle: we mistakenly treat small samples as if they were highly representative of their parent population. System 1 is not equipped to understand the role of sample size in determining the reliability of sample statistics.

The chapter opens with a striking example: the counties in the United States with the lowest incidence of kidney cancer are mostly rural, sparsely populated, and located in traditionally Republican states. Your System 1 immediately generates causal explanations—perhaps the clean air and healthy diet of rural life? But the counties with the highest incidence of kidney cancer are also mostly rural, sparsely populated, and located in traditionally Republican states. The explanation is statistical: small counties show extreme results simply because they have small populations, which leads to high variability.

Kahneman reveals how this statistical illusion has led to significant policy errors. For example, the Gates Foundation invested hundreds of millions in creating small schools based on research showing that the best schools tend to be small. However, the worst schools also tend to be small—not because school size affects quality, but because small samples produce extreme results more frequently. This is pure statistics, but our minds seek causal explanations instead.

The chapter emphasizes that System 1 is overly impressed by consistency and coherence in small samples. When you see a pattern in a few observations, System 1 automatically generates a causal story, even when the pattern is entirely due to chance. This creates an exaggerated faith in small samples—we treat them as highly informative when they are often highly variable and unreliable.

Kahneman confesses that he and his colleague Amos Tversky fell prey to the law of small numbers in their own research. They would run studies with absurdly small samples and believe the results, only to find that replication studies produced completely different findings. The lesson is profound: even experts who understand statistics intellectually can fail to apply that understanding intuitively. The law of small numbers explains why many published research findings fail to replicate, why business strategies based on small samples often fail, and why we see meaningful patterns in random sequences. Awareness of this bias is the first step toward better judgment.