** Correlation vs Covariance
**

Correlation and covariance are closely related concepts in theoretical statistics. They are important in determining the relationship between two random variables.

**What is Correlation?**

Correlation is a measure of the strength of the relationship between two variables. The correlation coefficient quantifies the degree of change of one variable based on the change of the other variable. In statistics, correlation is connected to the concept of dependence, which is the statistical relationship between two variables

The Pearson’s correlation coefficient or just the correlation coefficient r is a value between -1 and 1 (-1≤r≤+1). It is the most commonly used correlation coefficient and valid only for a linear relationship between the variables. If r=0 no relationship exist, and if r≥0 the relation is directly proportional; the value of one variable increases with the increase of the other. If r≤0 the relationship is inversely proportional; one variable decrease as the other increases.

Because of the linearity condition, correlation coefficient r can also be used to establish the presence of a linear relationship between the variables.

**What is Covariance?**

In statistical theory, covariance is a measure of how much two random variables change together. In other words, covariance is a measure of the strength of the correlation between two random variables.

In another perspective, it can be seen that correlation is just the normalized version of covariance, where the covariance is divided by the product of the standard deviations of the two random variables. The range of covariance can be large; therefore it is not easy to compare. This difficulty is overcome by bringing the covariance values to a range where it can be compared by normalizing it (kind of like what z-score does). Although the covariance and variance are linked to each other in the above manner, their probability distributions are not attached to each other in a simple manner and have to be dealt separately.

**What is the difference between Correlation and Covariance?**

• Both correlation and covariance are measures of relation between two random variables. Correlation is the measure of strength of the linearity of the two variables and covariance is a measure of the strength of the correlation.

• Correlation coefficient values are a value between -1 and +1, whereas the range of covariance is not constant, but can either be positive or negative. But if the random variables are standardized before calculating the covariance then covariance is equal to the correlation and has a value between -1 and +1.

Julie says

“The correlation coefficient quantifies the degree of change of one variable based on the change of the other variable.”

This sentence could be misunderstood in that the magnitude of the slope (when variable A is plotted versus variable B) is represented by the correlation coefficient, which it is not.

Furthermore I would not use the term “proportional” here, because that would mean that A = 0 when B = 0 and vice versa, which is not necessarily so. There could be a linear relationship with an offset.