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Consider the following facts:

  • Person A's salary is in the 90th percentile in Company A.
  • Person B's salary is in the 70th percentile in Company B.

The question is: What piece of information is enough to ensure that Person A's salary is greater than person B's salary? Check all that apply:

  • The mean salary in Company A is greater than the mean salary in Company B
  • The median salary in Company A is equal to the median salary in Company B
  • The 80th percentile of Company A is greater than the 70th percentile of Company B

At least one of the above options, but possibly more than one, must be true. To me, the question seems faulty because I don't think any of them is enough to ensure Person A's salary is greater than Person B's. It seems to me that just because Person B is in the 70th percentile (which means his salary is greater than 70% of other salaries) in his company doesn't mean he can't be in the 99th percentile as well; after all, 99>70 so it seems to meet the definition of being in the 70th percentile. If that were the case, and if Person A's salary lay somewhere in the 90%-99% range, none of the above conditions would prevent Person B's salary from being greater than Person A's. So it seems like none of the conditions ensure that Person A's salary is greater than Person B's. However, at least one of the above must be true, possibly more than one.

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  • $\begingroup$ The third bullet gives you $B_{SAL}<80_A<A_{SAL}$ $\endgroup$
    – it's a hire car baby
    Aug 11, 2020 at 14:21

1 Answer 1

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Clearly the first two fail. Company A could have all the salaries in a narrow range and Company B could have a wide range. You are expected to accept the third, with the idea that the 90th percentile at A is greater than the 80th percentile at A is greater than the 70th percentile at B. The question is whether the statement that Person B is in the 70th percentile at Company B, we know s/he is not also in the 71st percentile. I think for this question you are expected to read it that way.

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  • $\begingroup$ From the third answer, it seems like the question means that being in the 70th percentile means being in one of the 70-79th percentiles, not just the 70th. $\endgroup$
    – Kirill
    Nov 11, 2014 at 2:58
  • $\begingroup$ @Kirill: If so, the third statement also does not guarantee that Person A's salary is greater than Person B's. We need Person B's salary to be exactly what is mentioned in the third statement or the third statement fails. I agree it could be written more precisely, which is why I phrased my answer in terms of what was expected. $\endgroup$
    – Ross Millikan
    Nov 11, 2014 at 3:01
  • $\begingroup$ I think the question would be clearer if you replaced percentiles with deciles, because then the question of whether 70th percentile includes the 71th percentile drops off; 7th decile doesn't overlap with 8th, and so on. Maybe decile is the intended word here? $\endgroup$
    – Kirill
    Nov 11, 2014 at 3:08
  • $\begingroup$ @Kirill: If that is true, the third can fail. Let Person B be exactly at the 75th percentile of Company B. Let Company A's salary structure have all employees between 1 and 2. Then Person A's salary will be less than 2. Company B can have 73% of the employees below 1 and 27% above 2. The third statement would be true, but Person B's salary is above Person A's $\endgroup$
    – Ross Millikan
    Nov 11, 2014 at 3:22
  • $\begingroup$ I would read "8th decile of A" $>$ "7th decile of B" as saying that 80th percentile of A $>$ 79th percentile of B (not in the sense of averages, but in the sense of intervals), in which case Person A $>$ 90th percentile of A $>$ 80th-percentile of A $>$ 79th percentile of B $>$ Person B, which in that reading makes 3 the correct answer, so I think it works. $\endgroup$
    – Kirill
    Nov 11, 2014 at 3:26

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