Pearson correlation coefficient

Covariance

This is a measure of how two random variables change together, or the strength of their correlation.

Consider two random variables, $X$ and $Y$, each with $n$ values (i.e., $x_1$, $x_2$, $...$, $x_n$ and $y_1$, $y_2$, $...$, $y_n$). The covariance of $X$ and $Y$ can be found using either of the following equivalent formulas:

$$cov(X,Y)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})\cdot(y_i-\bar{y})$$

$$cov(X,Y)=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x_i-x_j)\cdot(y_i-y_j))$$

$$cov(X,Y)=\frac{1}{n^2}\sum_{i}\sum_{j\gt i}^{n}(x_i-x_j)\cdot(y_i-y_j)$$

where, $\bar{x}$ is the mean of $X$ (or $\mu_X$) and $\bar{y}$ is the mean of $Y$ (or $\mu_Y$)

Pearson correlation coefficient

The pearson correlation coefficient, $\rho_{X,Y}$, is given by :

$$\rho_{X,Y}=\frac{cov(X,Y)}{\sigma_X\sigma_Y}=\frac{\sum_{i}(x_i-\bar{x})(y_i-\bar{y})}{n\sigma_X\sigma_Y}$$

Here, $\sigma_X$ is the standard deviation of $X$ and $\sigma_Y$ is the standard deviation of $Y$. You may also see $\rho_{X,Y}$ written as $r_{X,Y}$.

The pearson correlation coefficient is a measure of the linear correlation between two variables X and Y.