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Paired data arise when an experimental or observational study design deliberately samples two units as a dyad to form a basis of comparison. Use this tag for any question about paired data or their analysis.
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Accepted
Difference in Means vs. Mean Difference
(I'm assuming you mean "sample" and not "population" in your first paragraph.)
The equivalence is easy to show mathematically. Start with two samples of equal size, $\{x_1,\dots,x_n\}$ and $\{y_1,\do …