Correlation = Slope of the Linear Relationship

Question

Does $Corr(X, Y)$ give the slope of the line representing the linear relationhip between the two?

i.e.

total trials $=100$

$N_R = 100 - N_B$

$N_B = 100 - N_R$

Thus, $Corr(X, Y) = -1$

If so, can the same be said for $Cov(X, Y)$?

Answer 1

Correlation between two variables is a measure of the strength of a linear relationship. It will always be a value between $-1$ and $1$ inclusive, or in rare cases it can be undefined (ex. all the points lie on a perfectly straight horizontal or vertical line, in which case one of our variables is constant.)

The slope of a line of best fit, or regression line, is different from correlation, but they do share the same sign, which can be seen if you look at the formulas for both.

Answer 2

No, the correlation is a measure of the alignment of the points. The closer to $\pm1$ (which are the extreme values), the better the alignment. The correlation, the covariance and the slope have the same sign, that's all.

Answer 3

Since it is possible to have a slope that is greater than $1$, it is not true.

In terms of simple linear regression, the slope is given by $\frac{Cov(X,Y)}{Var(X)}=r_{xy}\frac{s_y}{s_x}$ which is not equal to the correlation.

Answer 4

Nope! It's just a measure of how linear your data are -- if the regression line fits the data very well then $Cor(X,Y) \approx 1$ or $-1$.

One way to see this is to consider a dataset where your second feature is perfectly determined by the first, e.g. some data like $\{(x_i,2x_i)\}_{i=1}^n$. Clearly these data are perfectly correlated -- they all lie on the line $y = 2x$, and so

$$ Cor(X,Y) = \frac{Cov(X,2X)}{\sqrt{Var(X)Var(2X)}} = \frac{2Var(X)}{\sqrt{4Var(X)Var(X)}} = 1 $$

But the regression line (also $y = 2x$) has slope 2, and so the correlation of your covariates doesn't equal the slope of your regression line.