Correlation#

Correlation is a measure of the strength of a relationship that exists between two observable variables.

Introduction#

Preliminaries#

Before we can begin our study of correlation, let’s make some preliminary defintions that will help us keep everything clear and precise.

Univariate Statistics#

In order to differentiate between the statistics relationing to the x and y variables, we introduce some notation.

$\bar{x}$ and $\bar{y}$ are defined as the univariate sample means of the $x$ and $y$ variables. In other words, $\bar{y}$ is the sample mean of the $y$ variable, as if we were observing the $y$ variable in isolation. Similarly for $\bar{x}$ .

$s_x$ and $s_y$ are defined as the univariate standard deviations of the $x$ and $y$ variables. In other words, $s_x$ is the standard deviation of the $x$ variable, as if we were observing the $x$ variable in isolation. Similarly, for $s_y$ .