test actuary Info

The law of small numbers?

The mean IQ of the population of eighth graders in a city is known to be 100. You have selected a random sample of 50 children for a study of educational achievements. The first child to be tested has an IQ of 150. What do you expect the mean IQ to be for the whole sample?

• 100
• 150
• 101

The majority of people incorrectly respond 100. If the first child has an IQ of 150 and we can expect the remainder to have the mean IQ of 100 each we have a total of 5050 IQ points, which when divided by the 50 children gives us an average expected IQ of 101. People who respond 100 assume that there would be some low IQ scores to balance out the high ones. In short people believe that chance is self correcting. People behave as though they have a belief in a non-existent 'law of small numbers' : Small random samples of a population will resemble each other more closely than statistical sampling theory would predict. In fact the opposite 'law of large numbers' is correct : The larger the sample you draw from a population, the closer its average will be to the population average. People who respond 150 are victim of a logical fallacy of faulty generalization that's called Hasty generalization. They base a broad conclusion upon the statistics of a survey of a small group that fails to sufficiently to represent the whole population. Another nice example of the law of small numbers is : "Every odd number is prime" Proof: It is true for 3,5 and 7. So it must be true for all odd numbers. Watch out with the law of small numbers! source

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