Correlation Matrices

Correlation matrix

The Pearson’s correlation coefficient (also known as Pearson’s $r$) is a statistic that measures the degree to which two variables move together in a linear fashion. Correlation takes on a value between -1 and 1, in which $r=-1$ implies perfect negative correlation (points move together in a perfect straight line with a negative gradient) and $r=1$ implies perfect positive correlation (points move together in a perfect straight line with a positive gradient).

The image below details how correlation can be judged as a rule of thumb from Straightforward Statistics for the Behavioral Sciences. (Evans JD, 1996). To judge a negative correlation, just place a minus sign before each number in the table below:

---

Correlation Value	Description
r = 0 – 0.19	very weak relationship
r = 0.20 – 0.39	weak relationship
r = 0.40 – 0.59	moderate relationship
r = 0.60 – 0.79	strong relationship
r = 0.80 – 1.	very strong relationship

Here is the formula for correlation, however, we don’t need to compute it from scratch:

$$r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}$$