People are quite familiar with the colloquial usage of the term ‘correlation’: that it tends to resemble a phenomena where ‘things’ move together. If a pair of variables are said to be correlated then if one variable goes up, it’s quite likely that variable two will also go up as well.

Pearson’s Correlation is the simplest form of the Mathematical definition in that uses the covariance between two variables to discern a statistical, and albeit, a linear relationship. It looks at the dot product between the two vectors of data and normalises this summation: the resulting metric is a statistic which is bound towards +/- 1. A positive (negative) correlation indicates that the variables move in the same (different) direction with +1 at the extreme, indicating that the variables are moving in perfect harmony. For reference:

Pearson’s Correlation is the covariance between x and y, over the standard deviation of x multiplied by the standard deviation of y.

Where x and y are the two variables, ⍴ is the correlation statistic, and σ is the covariance metric.

It’s also interesting to note that the OLS Beta and Pearson’s Correlation are intrinsically linked. Beta is mathematically defined as the covariance between two variables over the variance of the first: it attempts to uncover the linear relationship between a set of variables.

OLS Beta is intrinsically linked to Pearson’s Correlation

However, the only difference between the two metrics is the ratio that scales the correlation based on the standard deviation of each variable: (sigma x / sigma y). This normalises the boundaries of the beta coefficient to +/- 1 and thereby giving us the correlation metric.

Let’s now move onto the Sampling Distribution of Pearson’s Correlation

#statistics #data analysis