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I have the following problem:

  1. There is 1 group of students (student 1, student 2, ..., student n)
  2. This group of students takes several tests (math, French, Biology) - all scored as percentages (e.g.: student 1 has 60% on math, 70% on French, etc.)

Finding: the average score for math is 55% while for Biology it is 76 %.

Question: I want to know whether these average scores are statistically different

My answer would be: paired samples t-test

However: in most explanations and examples I get in textbooks as well as online about the paired samples t-test a time element is involved (eg: comparing test scores for the same group before and after introducing a new education method).

In case I am right and it is the paired samples t-test, what is my independent variable? (second part of my question)

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You are right. The situation calls for a paired t-test. The grades for math and for biology are not independent because they come from the same people. It is true that some people will be better at math than they are at biology and vice-versa, but you will have some people who score higher on both (due to being more intelligent generally, being better students, being more persistent and organized, etc.), and some who score worse on both.

Any time you can establish a correspondence between a score in one group and a score in another group, you should be using a paired t-test. It doesn't necessarily have anything to do with before and after—that is just a convenient example for Stats 101. Here are some additional non-temporal examples where a paired t-test is appropriate:

  • Testing sets of identical twins where one twin is randomized to one group and the other sibling is assigned to the other group.
  • Testing the same person twice simultaneously, such as testing each arm with an allergy skin test and one arm at random is treated with an experimental salve.

To answer your second question, your independent variable is course content.

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Yes, a paired sample test would be appropriate for comparing the various subject results as long as you have the same students taking the various subject exams. Time is a common way to set up paired t-tests in textbooks (as you mentioned, something like pre-test versus post-test), but it isn't the only application for paired testing.

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