Example of the law of small numbers

The law of small numbers says that people underestimate the variability in small samples. Said another way, people overestimate what can be accomplished with a small study. Here’s a simple example. Suppose a drug is effective in 80% of patients. If five patients are treated, how many will respond?

Many people reason that 80% means 4 out of 5, so if 5 people are treated, exactly 4 will respond. Always.

Others understand that things are not guaranteed to work out so neatly, but they still believe that it is highly likely that 4 people would respond. Maybe a 90% chance.

In fact, there’s only a 41% chance that exactly 4 would respond out of a sample of 5.

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