Mathematics/Statistics/Correlation Coefficients

Tip: This is often represented as r.

The Correlation Coefficient is a value that describes "how well can a straight line fit this data". It is similar to covariance except that a correlation coefficient will always be between [-1, 1].

A value of exactly 1 indicates that there is a strong positive correlation between x and y. That is, as x increases, so does y. And as y increases, so does x.

A value of exactly -1 indicates that there is a strong negative correlation between x and y. That is, as x increases, y decreases. And as y increases, x decreases.

As values approach 0, it indicates a weaker and weaker correlation, with 0 indicating that there is absolutely no correlation between x and y.

There are a few ways to calculate a correlation coefficient.

Pearson's Correlation Coefficient

This is one of the most popular forms of calculating a correlation coefficient.

The equation to calculate the Pearson Correlation Coefficient is

$r={\frac {n(\sum xy)-(\sum x)(\sum y)}{\sqrt {(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}}}$

Where

• $\sum x$ is the sum of all our original x values in our dataset.
• $\sum y$ is the sum of all our original y values in our dataset.
• $\sum x^{2}$ is the sum of all our x values, after squaring them first.
• $\sum y^{2}$ is the sum of all our y values, after squaring them first.
• $\sum xy$ is the sum of all our x and y pairs, after multiplying together first.

For additional explanation, see this youtube video.

Sample Correlation Coefficient

The equation to calculate the Sample Correlation Coefficient is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r={\frac {1}{n-1}}\sum _{i=1}{n}({\frac {x_{i}-{\bar {x}}}{S_{x}}})({\frac {y_{i}-{\bar {y}}}{S_{y}}})}

Where

• $x$ is the mean of our x.
• $y$ is the mean of our y.
• $S_{y}$ is the standard deviation of our x.
• $S_{x}$ is the standard deviation of our y.

For further explanation, see this Khan Academy video.