According to wikipedia, pearson correlation is scale and location invariant.

Does scale refer to "variance" and location refer to "mean" ?

Thanks.

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#### Best Answer

In this case, scale and location are more general. Given two random variables $X$ and $Y$, the correlation is scale and location invariant in the sense that $cor(X,Y) = cor(X_{T},Y_{T})$, if $X_{T} = a + bX$, and $Y_{T} = c + dY$, and $b$ and $d$ have the same sign (either both positive or both negative). Note that if $b > 0$ and $d < 0$ (and vice versa), $cor(X,Y) neq cor(X_{T},Y_{T})$ because the sign of the correlation between the transformed random variables will be inverted.

### Example:

$$X = 1,2,3,4,5$$ $$Y = 1,2,3,4,5$$ $$cor(X,Y) = 1$$

If $X_{T} = 1 + 2 X$ and $Y_{T} = 2 + 3 Y$ , then $$X_{T} = 3,5,7,9,11$$ $$Y_{T} = 5 ,8, 11, 14, 17$$ $$ cor(X_{T},{Y_{T}}) = 1 $$

But, if $X_{T} = 1 + 2 X$ and $Y_{T} = 2 – 3 Y$, then $$X_{T} = 3,5,7,9,11$$ $$Y_{T} = -1, -4, -7, -10, -13$$ $$cor(X_{T},{Y_{T}}) = – 1 $$

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