Does correlation tell us anything about standard deviation?

Question

I was wondering whether correlation between $2$ variables inform us about their standard deviation.

Let's say $X$ has a standard deviation of $\sigma_1$ and $Y$ has a standard deviation of $\sigma_2$.

Group 1 - Made of equal elements of $X$ and $Y$ being negatively correlated with each other.

Group 2 - Made of equal elements of $X$ and $Y$ positively correlated with each other.

Is is possible to claim which group has a lower standard deviation or variance?

Answer 1

By definition

$$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{\sigma_X\sigma_Y}$$

We cannot conclude anything about the standard deviations from the correlation since correlation depends on covariance $and$ standard deviation.

In fact, the standard deviations are only used to scale the covariance so that the correlation will lie between $-1$ and $+1$, improving interprebility.

Here is an example in R statistical software:

Suppose we multiply $X$ by a scalar.

x <- rnorm(10^6, mean=5, sd=3)
y <- rexp(10^6, rate=1)
cor(x, y)
cor(5*x, y)  
sd(x)
sd(5*x)

This gives

$$\text{Corr}(X,Y)=7.83\cdot10^{-5}$$

$$\text{Corr}(10\cdot X,Y)=7.83\cdot10^{-5}$$

but

$$\sigma_X\approx3$$

$$\sigma_{10X}\approx15$$

so increasing the standard deviation did not change the correlation!