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Background: I would like to offer readers a statistic that conveys the similarity of two sets of numbers. I thought that I had what I needed with correlation coefficient (indeed, I have a coefficient of determination/r2 of 98.7). But I've just realized that r and r2 conveys only that the two sets of numbers are correlated in their upward and downward movement; and what I want (also) to convey is that they are also very nearly identical. But yet I feel like regression analysis is more than I need (or, more precisely, more than I can do with Excel). To be a little more concrete, imagine a data set with monthly earnings for two stores in a strip mall. Correlation coefficients would capture whether the stores' earnings were covariate. But I want to demonstrate also that the monthly earnings also happen to be very nearly the same actual number. On a graph, they are visually one is on top of the other. But is there a statistical metric that captures that?

Question: What is a measure of association that conveys actual similarity of values between two variables and not just correlation?

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If you want to demonstrate that two sets of numbers are nearly identical, you don't need the correlation at all. You can just take the difference between them.

If store one has monthly earnings a1,a2,etc and the other store has monthly earnings b1,b2, etc. then you just take the difference to get: (a1-b1, a2-b2,....). If the numbers are close to identical, then a1-b1, a2-b2, etc should all be close to zero. Take the average of all (a1-b1, a2-b2,...) and that average number should be close to zero.

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  • $\begingroup$ You probably want to take the average of the absolute value of the differences - otherwise $(0, 10)$ and $(10, 0)$ are considered identical. This is known as the $\ell_1$ distance; you could also try the $\ell_2$, the square root of the mean of the squared distances. $\endgroup$ – Dougal Aug 7 '14 at 4:04
  • $\begingroup$ You're right about the absolute value, however I have to disagree about the root mean square error (RMSE) distance. RMSE has been shown to vary widely with very small modifications in the input. It may not be appropriate for what Thom is looking for, unless he wants a super-sensitive measure. $\endgroup$ – rocinante Aug 7 '14 at 10:25
  • $\begingroup$ RMSLE is useful in scenarios with outliers/noisy data. $\endgroup$ – Jessica Mick Aug 7 '14 at 12:16

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