OG test 2, p. 457, problem 4 / Help on statistics

[irs031]

The median income of a group of College C graduates six months after graduation was $3000 higher than the median income of a group of college D graduates six months after graduation.

Q A: The 75 th percentile of the incomes of the group of College C graduates six months after graduation
Q B: The 75th percentile of the incomes of the group of college D graduates six months after graduation

I chose D, which turned out to be the correct solution. However, my logic is different from the logic presented in the explanation to the problem.

The official explanation states: "Recall that the median of a group of numbers is the middle number when numbers are listed from least to greatest. The median is also equal to the 50th percentile. The median does not indicate anything about the spread of the numbers of the group. In particular, for each group of incomes you do not know how much greater than the median the 75th percentile of the group of income is, nor do you know the relationship between the 75th percentiles of the two groups"

My logic: The problem however mentions nowhere that both sets are normally distributed. Given that median can be higher or lower than the mean, we can not answer the question.

My main question and the thing that I find most confusing in the statistics portion of the exam: Should a problem explicitly state that a given data set is normally distributed, before we can conclude that Median = mean or median is equal to the 50th percentile? The problem mentions nowhere anything about normal distribution, however the explanation of the problem states "The median is also equal to the 50th percentile"?

My second question: We have two data sets that are normally distributed and have equal means. However we don't know n of each set. Could we conclude that the 75th percentile of the first set is equal to the 75th percentile of the second set? The first set may have only 10 observations, while the second a 1000?

Thanks for the help.

[tommywallach]

Hey IRS,

This question has nothing to do with distribution (standard/normal), so I'm not sure your explanation makes sense, except insofar as it represents a sort of version of the actual answer. Even if they had a normal distribution, that tells us about the general distribution of the data, not the individual percentile points (i.e. where the 75th percentile is). I don't see how the fact that we don't know the relationship between median and mean is relevant here. The issue is the relationship between the median and the 75th percentile.

To your other questions, yes, a question MUST say it's normally distributed for you to assume that. What you're getting wrong is that 50th percentile IS NOT the mean. It's the median. So the median is always the 50th percentile (by definition).

To your next question, no, we could not know if two sets that are normally distributed around the same mean would have the same 75th percentile.

Hope that helps!

-t