Correlation Matrices

Learn about bivariate analysis and how correlation matrices can assist in analyzing the relation between two continuous variables.

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Correlation matrix

The Pearsonâ€™s correlation coefficient (also known as Pearsonâ€™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 â€“ 0.19 very weak relationship
r = 0.20 â€“ 0.39 weak relationship
r = 0.40 â€“ 0.59 moderate relationship
r = 0.60 â€“ 0.79 strong relationship
r = 0.80 â€“ 1. very strong relationship

Here is the formula for correlation, however, we donâ€™t need to compute it from scratch:

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

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