The meaning of scale and location in the Pearson correlation context

Question

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

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

Thanks.

Answer 1

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 $$X_{T} = 3,5,7,9,11$$ $$Y_{T} = -1, -4, -7, -10, -13$$ $$cor(X_{T},{Y_{T}}) = - 1 $$

Answer 2

No, scale and location are more general in this case. A scale and location transformation of a variable X is a deterministic function of X defined as Y=f (X)=aX+b. For the correlation coefficient to be scale and location invariant is the same as saying that for a and b real, the correlation coefficient of X and Y will be the same. If you look at the definition of the correlation coefficient, and use definitions related to expectation and variance of a linear transformation of a variable, you can work to that result.