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I'm still in awe of whuber's crayon-depiction of covariance: How would you explain covariance to someone who understands only the mean?.

I get that in order to represent variance of $x$ with CDF $F$ using this crayon method, we would need to draw 3 i.i.d random variables $(x_1,x_2,x_3)$, and compute the amount of "red" when one corner of the rectangle is $(x_1,x_1)$ while the other is $(x_2,x_3)$. Is it obvious that this procedure would lead to a net positive amount of red area?

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Your interpretation of the corners is not correct. You'd have three squares*, formed from the corners $(x_1,x_1)$, $(x_2,x_2)$ and $(x_3,x_3)$ taken in pairs. Each square is red, so the net sum of coloured areas must be red.

Illustration of the three pairs of squares

* quite literally the variance is a scaled version of this "sum of squares", since it corresponds to the average-sum-of-squared-pairwise-differences version of the variance formula.

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