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This is a relatively easy question but I am new to statistics so I hope I could get a better idea. Me and my friend were discussing the following:

My friend said that if someone wants to find the average monthly family income of a country and goes about finding this by saying: if the average family size is 4, I just multiply this by the average monthly income per person.

However, I think think reasoning is incorrect but I can't really explain why.

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Well, for one thing, the average income probably does not include children, while the average family size does. Also, is that the average income of people who work, not including unemployed? In that case, if it were "the average number of working people per household", then that reasoning would be sound. – Macro May 13 '12 at 19:26
Averages can be confusing. A frequently used technique to cope with them is to reason about totals instead. Here, you would want to express total country monthly income in two ways: in terms of the number of families (multiply that by the mean family income) and in terms of the number of people (multiply that by the mean per capita income). At this point, the assumptions needed to justify your friend's assertion should become apparent. – whuber♦ May 14 '12 at 16:05
Aha. So is it correct then to say that you can’t take the average of averages and hope it will match the average. Is that his big mistake here? Or is it that he is working with totals? – Valerie May 14 '12 at 16:21

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The United States government does not track "family" income, per se. Rather, the measurement is referred to as "household" income, and it includes the incomes of all individuals in the household over the age of 18. These data are primarily collected by the Census Bureau. Marco makes an important point, namely, that "family" size includes children. Therefore multiplying the average monthly income per person in the United States (or any other country) by four would not provide an accurate estimate of median household income. This would also ignore the fact that men are paid more than women. The Census Bureau also takes race, size of household, and geographic location into account when computing their estimates.

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Do you also think that if you do it the way my friend does the number obtained would be too high? Since usually families consist of children that don't have a (large) income? – Valerie May 13 '12 at 21:54
That's correct. As I mentioned, the use of "household" income, which includes the incomes of all individuals in the household over the age of 18 (i.e., the "breadwinners"), controls for this problem. – Alexander May 13 '12 at 22:13
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Macro is right that it depends on how you define income per person. Let's assume for simplicity that the income per person in a family is gotten by taking the family income adn dividing by the family size. Call ci the per person income for the ith family and ni the number of family members in the ith family. Suppose we have a total of M families. How would we define the average family income? It would be (∑ni ci)/ (∑ni) . Each sum goes from 1 to M.

Now what your friend suggests is the average family size which is Fs=(∑ni)/M multiplied by the average monthly income per person which is Fs (∑ci/M) = (∑ni)(∑ci)/M^2 Then compare (∑ni ci)/ (∑ni) to (∑ni)(∑ci)/M^2 They are not the same even algebraically.

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Aha. Because my friend claims that the average monthly family income in a certain country is for example 5000 euro. He argues as follows. The average family size is 4. First we compute the average income per person, and multiply this number by 4. This gives the average income per family. – Valerie May 13 '12 at 21:53
It would work if all families were size 4 but not necessarily otherwise. – Michael Chernick May 13 '12 at 22:21

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