# Solved – The meaning of scale and location in the Pearson correlation context

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

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

Thanks.

Contents

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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