# Pearson correlation coefficient - Mathematics
Data Science for Political and Social Phenomena
Author: Guillaume Redoulès

# 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:

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

or

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

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.

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