mjd@pobox.com
Explain convariance

Covariance is a measure of the relationship between two random variables. It tells us how much the two variables change together. A positive covariance indicates that the variables are positively related, meaning that as one variable increases, the other variable also tends to increase. A negative covariance indicates that the variables are negatively related, meaning that as one variable increases, the other variable tends to decrease. If the covariance is zero, it indicates that there is no relationship between the two variables. In mathematical terms, covariance is defined as the expected value of the product of the deviations of two random variables from their respective means.
mjd@pobox.com
I thought there was something about the covariance only measuring the degree of _linear_ relation between the two variables.

Yes, you are correct. Covariance measures the degree of linear relationship between two variables. It does not give any information about the non-linear relationships between the variables. Also, it does not tell us the direction of the relationship, only the strength of the relationship. The value of covariance can be any real number, but it does not have any unit of measurement. To standardize the value of covariance, we use correlation coefficient which ranges between -1 and 1. A value of 1 means a perfect positive linear relationship, a value of -1 means a perfect negative linear relationship, and a value of 0 means no linear relationship.
mjd@pobox.com
What do you mean, it does not tell us the direction of the relationship.

I apologize for any confusion - what I meant is that covariance by itself does not indicate whether the relationship between the two variables is positive or negative. A positive covariance indicates that the variables are positively related, meaning that as one variable increases, the other variable also tends to increase. A negative covariance indicates that the variables are negatively related, meaning that as one variable increases, the other variable tends to decrease. However, the magnitude of the covariance value does not give any information about the direction of the relationship. For example, the covariance between two variables can be the same whether one variable increases as the other decreases or vice versa.