Last week I was giving a talk on decision making and biases and we had some robust debate on the bias called the “Law of small numbers”. Below is the slide from the talk.
Some confusion existed because of ambiguous wording, in particular the first phrase “Mean IQ is 100” was seen to apply to the subsequent sample, calculated post-hoc.
However they appeared to be a few in the room who still didn’t believe the maths, so here it is explained.
To get the actual average IQ of the group we add the total IQ and divide by the number of people, in this case 50.
The first persons IQ is 150 so the running total is now 150. For the next 49 people we don’t know the exact score so the best we can do is assume they represent the average person, i.e. someone with an IQ of 100. The total IQ of the group of 50 is now 150 + 49 x 100 = 5050
We divide this by 50 to get an answer of 101.
So the improved wording would be “The mean IQ of all people is 100. From this group we select 50 people and the first person selected has an IQ of 150”
This example is showing that people will generally expect a run of lower than 100 IQ’s for the remaining 49 people to “balance” the average so it becomes 100. Of course this does not occur, on average.
The same thinking is also behind the gamblers fallacy where someone will see a run of, for example, black numbers on roulette and then expect a run of red numbers to balance the average.