📂Statistical Analysis

Independence Does Not Imply No Correlation

Description

If variables are independent, it means there is no correlation, but lack of correlation does not necessarily imply independence.

The case when variables are independent if there is no correlation, that is when it is a necessary and sufficient condition, is when the random variables follow a normal distribution.

In the case on the left, there is positive correlation, and in the case on the right, there is negative correlation. The term cor in the diagram refers to correlation coefficient, which is an indicator of how linearly related the two variables are. If they are independent, the correlation coefficient is calculated to be very low, showing that the two variables do not have a correlation. What’s important here is that the correlation coefficient is an indicator of ’linear correlation.'

In cases like the one above, the correlation coefficient might be very low, but one can infer that there definitely is some relationship¹.

In other words, it means they are not independent. The correlation coefficient is an indicator of linear correlation only, so it fails to capture non-linear relationships. Therefore, when checking if there is a relationship between variables, one should not just trust the correlation coefficient, but should actually look at the data.