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I have two groups and each group has four members. I assign each member equal score 5 in group A. I assign each member score 10 in group B.

If I want to argue that the members in these two groups are significantly different, how do I do that without running software to examine it? Is there any theory or literature for me to argue that?

Thanks in advance.

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    $\begingroup$ If you decreed that individuals in group A get assigned a 5 and group B get assigned a 10, then there is no randomness at play here, so there is no statistics to do. $\endgroup$
    – Matthew Drury
    Commented May 3, 2016 at 22:52
  • $\begingroup$ Thank you so much. I have another question. If there are only two people, John and Mary, their scores are 5 and 10. Can I say their score is significantly different to each other? $\endgroup$
    – Terence Tien
    Commented May 3, 2016 at 23:08
  • $\begingroup$ if the total score is 1000, I would say no difference; if the total score is 10, I would say the difference is significant based on my common sense. $\endgroup$
    – Deep North
    Commented May 4, 2016 at 1:55
  • $\begingroup$ Statistical inference would be based on some assumptions relating to a probability model (even if it's only exchangeability under the null). I see no indication of any assumptions we could make based on your post nor your comments. So the answer is we can't say anything until we know more about what your null is (at the least!) and what your assumptions are. $\endgroup$
    – Glen_b
    Commented May 5, 2016 at 4:42
  • $\begingroup$ Thank you for explaining the importance of the assumptions. $\endgroup$
    – Terence Tien
    Commented May 5, 2016 at 4:46

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Statistical significance is used to generalize sample data to the average mean of the population. If your population were those 8 people, then you don't need to worry about statistical significance; we know the groups are different.

What you may care about more in a situation like this is practical significance, which is context dependent.

For example lets say 5 and 10 signified the number of minutes it took the two groups to run a mile:

1) If you were a track coach, then there would be a great deal of practical significance.

2) If you were trying to predict if one of the groups will be unable to walk 3 miles, the practical significance would be low (10 minutes is a decent time to run a mile, so both groups are in shape).

3) If you were hiring typists, there is no practical significance (unless those typists run around a lot).

Practical significance always has to be answered in context of why we are analyzing the data.

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  • $\begingroup$ Ten minutes to walk three miles? I would say that the people are in superhuman shape! $\endgroup$
    – Dilip Sarwate
    Commented May 5, 2016 at 11:19
  • $\begingroup$ @DilipSarwate, I meant using the one-mile run time to predict if an individual would walk three miles at any speed. I am not in the best shape, but I am fairly certain that 3 miles in ten minutes isn't walking pace for me either :) $\endgroup$
    – Chris P
    Commented May 5, 2016 at 16:43

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