Is the covariance (1,2) the same as the covariance (2,1)? Or is there a sign change or something?
The covariance between two variables represents the measure of how they vary together. The covariance of (1,2) is the same as the covariance of (2,1). Covariance is a symmetric measure, meaning that the order of the variables does not affect the value of the covariance. So, there is no sign change or difference in the covariance between (1,2) and (2,1).
The covariance between two variables measures the relationship between them, specifically how they vary together. It measures the direction and strength of the linear relationship between the variables.
To calculate the covariance between two variables X and Y, you need a dataset that contains paired observations of X and Y. Let's assume you have a dataset with n observations.
The formula to calculate the covariance between two variables X and Y is as follows:
cov(X, Y) = Σ((xᵢ - μₓ)(yᵢ - μᵧ)) / (n - 1)
where xᵢ and yᵢ are individual observations of X and Y, and μₓ and μᵧ are the means of X and Y, respectively.
Now, coming back to your question, the order of the variables does not affect the covariance. This means that the covariance between (1,2) and (2,1) will be the same. The covariances will be calculated using the same formula mentioned above, substituting the individual observations accordingly.
Therefore, cov((1,2)) = cov((2,1)). In this case, the order of the variables within the covariance calculation does not matter.