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I am confused about the statistical statement I am making based on the following scenario:

Scenario: The membership of a professional organisation is only open to the top 10% of students in a district. There are 400 members, coming from (say) 45 schools.

Statement: The organisation represents the top 10% of students in the district.

My confusion is that School A may have lower quality students who make the top 10% than School B.

e.g.

  • School A has students who scored 41, 42, 43, 44 and 45 out of 50 (top 10%)
  • School B has students who scored 46, 47, 48, 49 and 50 out of 50 (top 10%)

In this case, can they be the top 10% of students (especially when School B has students who scored 41, 42, 43, 44 and 45)?

I am using just 2 schools for simplicity.

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You say membership "is only open to the top 10% of students in a district," and then question whether the organization represents the top 10% of students. There are (at least) two big issues here:

  1. It seems from the rest of your question that membership is open to the top 10% of students from each school in a district. Is that what you meant? If so, then as you notice the organization clearly is not exactly the top 10% of overall students. In the real world it's probably likely that some schools in a district have students of significantly higher "quality" than others (because the school is better, because of socioeconomic differences, because it has selective admissions...). If that's the case, then maybe the student in the 80th percentile at the better school is equivalent to the student at the 95th percentile at the bad school. Then it is clearly not true that the organization represents the top 10% overall. (Even if the schools are all of roughly equal quality, this is still something of an issue, because of random variation for the individual students.)

  2. You say membership is "open to" students from the top 10%. If that means that students in the top 10% then have the choice themselves of whether to join the organization, there may be additional biases that would mess up the representativeness of the sample. For example, students on the lower end of the top 10% who are trying to pad their resumes might be more likely to join the organization, while those at the very top see it as a waste of time from their more valuable pursuits. Or the opposite.

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