Difference of sums or sum of differences this is the question
If you have a bunch of paired measurements, let's say you measured peoples weights before and after holidays (AND you are assuming these are samples from an underlaying Gaussian distribution) what is the correct thing to do, to calculate the difference in weight between before and after holidays?! Difference of averages or average of differences?
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$\begingroup$If you have $x_1,…,x_n$ sample points (before holidays) and $y_1,…,y_n$ afterwards. If you use the usual arithmetic mean $\overline{x}:=\frac{x_1+…+x_n}{n}$ and $\overline{y}:=\frac{y_1+…+y_n}{n}$ then you have $\overline{x}-\overline{y}=\frac{x_1+…+x_n}{n}-\frac{y_1+…+y_n}{n}=\frac{x_1-y_1+…+x_n-y_n}{n}=\overline{x-y}.$
You see there is no difference between the two. On the other hand if you use the geometric mean.
$\overline{x}:=\sqrt[n]{x_1…x_n}$ and $\sqrt[n]{y_1…y_n}$ then $\overline{x}-\overline{y}\neq\overline{x-y}$ where the latter is defined as $\overline{x-y}:=\sqrt[n]{(x_1-y_1)…(x_n-y_n)}.$
hope to be helpful
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