What does the multiplication of standard deviation of two variables gives?

Question

If we need to find the correlation between two variables it is given by the formula - co variance of two variables divided by the multiplication of Standard deviation of the two variables. My questions is why we multiply standard deviation of two variables? What I can interpret from this?

Answer 1

To save typing, let's assume that $X$ and $Y$ each have mean $0$, and have positive finite standard deviations $\sigma_x$ and $\sigma_y$. Then their covariance is $\operatorname{cov}(X,Y)=E[XY]$.

Suppose they are perfectly and positively correlated, so $X=kY$ with probability $1$ for some positive $k$. Then $\sigma_x = k \sigma_y$ and $$\operatorname{cov}(X,Y)=E[XY]=E[kY^2]=k\sigma_y^2=\sigma_x\sigma_y$$

You want the correlation coefficient to be $1$ in such as situation, so you divide the covariance by $\sigma_x\sigma_y$ to ensure it is. So the correlation coefficient is defined to be $\frac{cov(X,Y)}{\sigma_x\sigma_y}$.

If they are uncorrelated then the covariance is $0$ and so too is the correlation coefficient. If they are perfectly and negative correlated, so $X=-kY$ with probability $1$ for some positive $k$ then $\sigma_x = k \sigma_y$ and $$\frac{\operatorname{cov}(X,Y)}{\sigma_x\sigma_y}=\frac{E[XY]}{\sigma_x\sigma_y}=\frac{-k\sigma_y^2}{\,k\sigma_y^2}=-1$$ as you might hope.