I think that assumption valid from a problem solving point of view, but not from a real-world point of view.

Don't forget the population is defined as 8th graders. Broadly speaking about 1% of the US population are 8th graders. So a city of 50,000 would have only 500 8th graders.

There is no real definition of a "city" in the US just an urbanised area, and many of them are very very small. For comparison: In the UK (where "city" is strictly defined) about 1/3rd of cities have less than 50,000 people, some way way less.

So I would say that it is more likely that for most cities the population size is extremely relevant, but if you picked a 8th grade class at random it is almost certain they exist in a city where the population size is irrelevant.

I guess that both parties (also the one that conceded) can be right depending on what we mean by "mean". Disclaimer: I'm not a native English speaker, so it's very likely that my interpretation is wrong or that I can't properly express what I intend. If that's the case, I apologize.

In IQ testing, the average score is defined to be 100. It's also well known that IQ scoring distributes like (more or less) a normal distribution. Both the facts above are not totally relevant to the question by themselves; they just mean that despite the fact that not every human has been IQ tested, you are safe in saying that a random guy's expected IQ is 100.

Back to the point: what is the "mean" stated in the question? If by it we intend:

mean = (Sum_i^N IQ_i)/N

where N is the number of guys in the city, than David Lyons is certainly correct. On the other hand, if by "mean" we intend the expected value of the IQ distribution, than things are not much clear. Remember, 100 is defined to be the average. A city population is of course a population, but from a statistical point of view can also be a sample of a larger population (that can also be modeled as infinite, when modelled with random variables). It can actually be that summing the IQs of all the people in the city yields a different number from 100*N; and even in this case, the question wouldn't be totally wrong in saying that the mean is 100. Also, IQ scoring from different people are often modelled as iid. Knowing a single score doesn't tell anything about the score of others.

So, to summarize:

- is the mean stated in the question the total sum of the IQs of the people in the city divided by the number of people? In that case, you need to know the population size to properly answer;

- are the IQ scores of the sample modeled as iid's with 100 as average? In that case, the correct answer is the same to the one we would give to the following question: "suppose you draw 50 iid random variables with 100 expected value. Now you know that one of that variable has the value of 150. Which is the expected mean of the 50 variables?" and in that case, the correct answer is 101.

If we leave out the words, "of eighth graders in a city" from the question, there would be no dispute in this thread that the answer is exactly 101. I also think Kahneman just treated those words as superfluous when he answered, since the given mean is the same as the whole universe mean. He's pretty smart.

Nevertheless, as discussed already, if we are considering a specific population, that is NOT a random sample of all people, and that is closer in size to 50 than it is to the whole world, size matters.

These days it is actually the median, not the mean, that is defined as 100, but as you say that doesn't make a difference to the OPs question.

More or less.. basically each SD from the median is defined as 15 IQ points.

I think that would be 100% clear, as I said in an earlier post.... saying "the average is 100 for the population" is NOT the same as saying "the expected IQ for a person in the population is 100"

The distinction is an important one, but the OP was not ambiguous.

The reason I'm drawing the distinction (I'm not a life nit, promise!) is that the people who do this sort of calculation for a living (statisticians, actuaries) OFTEN make these errors in the real world, not because it's "close enough" but because they miss the significance. If the OPs question had been in an actuarial exam then 101 would have most definitely been wrong, unless the answerer explained explicitly he was assuming the population size was infinite and the impact of that assumption on the answer. The fact that the population was specified as 8th graders would have actually been a hint in an exam that the absolute population size was potentially relevant - as NoG alluded to in his post above.


A final analogy: to most people the number of decks in a blackjack shoe is irrelevant, and it can be treated as an infinite shoe. But for those whom it is relevant, it is sometimes EXTREMELY relevant.

Unless it's infinite 101 is always incorrect.

If it's infinite, then 101 is correct. If it's less than infinite, then I recommend you actually read this thread.

The population is N and has avg m=100. The population minus the 150 person is N-1 and has avg m'=(N*m-150)/(N-1).

The sample is size n and is now made of a 150 person and n-1 randomly selected people from the N-1 part of the population that has avg m'

So the sample in fact has avg that is (49*m'+150)/50 or

(49*(N*100-150)/(N-1)+150)/50

If N=50 then the avg is 100.

If N=100 the avg is 100.505...

If N=1000 the avg is 100.951

If N=1 mil the avg is 100.99995

It tends to 101 in the large N limit.

In real life the way the sample was collected matters and the avg itself may be a function of time evolving so a sample taken later than the total population average was calculated may be different than in a totally static situation. The IQ of a person changes with time during a month for a number of reasons.

In real life also if you suspect the sample is atypical the probability of that skyrockets if the first tested is so big already. Far less than 1 in 50 people has iq 150 or higher. Probably 1 in 10000 or 1000 or around there. So that sample got real lucky there. So to get the first one to be 150 already is a problem for the sample of 50 being a properly selected random sample. Of course if it is properly selected accept what just happened without any worry and do the above.

Just realize that for example if the sample was a random class in a school selected randomly somewhere in the country/city(well ok say 2 random classes in a school) the fact the class has a 150 member in it raises the probability its not a random class after all as interactions between students take place that may elevate the IQ of others when a very smart person is around them to challenge or inspire them and the 150 itself may be a function of better teachers this class had or better nutrition, better income or educational level of parents in that part of the population that the school is etc. So the way the sample was created will matter in general when you get such deviation already. Eg a random class selected out of all classes in the country doesnt make the sample, exclusively produced from that class and another say at most, a good random sample of the population (the students are correlated).

Correct. Unless the number of people in the city is infinite, 101 is the incorrect answer.

Sorry, i misread you as having the opposite viewpoint

All of the children are above average.
-- Garrison Keillor

"Pretty much everyone has more than the average number of legs"

That's a pretty good straw case to show mean, median and distribution. I'll have to remember that.

A nitpicker would wonder what % of the sampled children are from 8th grade but not me.