Covariance vs. Covariation

Attribute	Covariance	Covariation
Definition	Measure of how two variables change together	Measure of the relationship between two variables
Range	[-∞, ∞]	[-1, 1]
Interpretation	Positive covariance indicates that the variables increase or decrease together, negative covariance indicates that one variable increases as the other decreases, and zero covariance indicates no relationship	Positive covariation indicates a positive relationship between the variables, negative covariation indicates a negative relationship, and zero covariation indicates no relationship
Formula	cov(X, Y) = E[(X - μX)(Y - μY)]	covar(X, Y) = E[(X - μX)(Y - μY)] / (σX * σY)

Definition

Covariance and covariation are two statistical terms that are often used interchangeably, but they actually have distinct meanings. Covariance is a measure of how two variables change together, while covariation is a more general term that refers to the relationship between two variables, whether it is positive, negative, or neutral.

Calculation

When calculating covariance, you multiply the difference between each data point and the mean of the first variable by the difference between the corresponding data point and the mean of the second variable, and then take the average of these products. Covariance can be positive, negative, or zero, depending on the relationship between the two variables. On the other hand, covariation is a more general term that simply describes the relationship between two variables, without specifying the exact calculation method.

Interpretation

Covariance is often used to determine whether two variables move in the same direction (positive covariance), opposite directions (negative covariance), or independently (zero covariance). A positive covariance indicates that when one variable increases, the other variable tends to increase as well. Conversely, a negative covariance suggests that when one variable increases, the other variable tends to decrease. Covariation, on the other hand, is a broader term that encompasses all types of relationships between variables, not just the direction of change.

Application

Covariance is commonly used in finance to measure the relationship between the returns of different assets in a portfolio. A high positive covariance between two assets indicates that they tend to move in the same direction, which may increase the overall risk of the portfolio. On the other hand, a negative covariance between assets can help diversify the portfolio and reduce risk. Covariation, on the other hand, can be used in a variety of fields, such as psychology, sociology, and biology, to study the relationships between different variables without focusing on the specific direction of change.

Limitations

One limitation of covariance is that it is not standardized, meaning that the magnitude of the covariance depends on the units of measurement of the variables. This can make it difficult to compare covariances across different datasets. Covariation, on the other hand, is a more general term that does not have this limitation, as it simply describes the relationship between variables without specifying the exact calculation method.

Conclusion

In conclusion, while covariance and covariation are related concepts, they have distinct meanings and applications. Covariance specifically measures how two variables change together, while covariation is a more general term that describes the relationship between variables without focusing on the direction of change. Understanding the differences between covariance and covariation can help researchers and analysts make more informed decisions when analyzing data and relationships between variables.