Covariance helps you understand how variables are linearly related.

Would it be possible to have two pairs of variables in a deterministic relationship (i.e. linearly correlated variables) that have different values for the covariance?

My guess would be that if you had one pair with low sample variances and the other pair with high sample variances, you could have the same relationship but a different covariance. Am I on the right path?

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    $\begingroup$ You are not only on the right path but you've nailed it! This is why correlation is considered a "standardized" covariance. $\endgroup$ – TrynnaDoStat Feb 26 '15 at 19:23
  • $\begingroup$ @TrynnaDoStat I think you could make your comment into an answer. $\endgroup$ – Patrick Coulombe Feb 26 '15 at 19:28

You are not only on the right path but you've nailed it! Correlation is considered a "standardized" covariance.

One of the reasons covariance is not a good way to measure the strength of a linear relationship is because it is not invariant to deterministic linear transformations. Let $X$ and $Y$ be random variables and let $a$ and $b$ be real numbers. Covariance has the property that if $X,Y$ are random variables then $Cov(aX,bY)=abCov(X,Y)$. Multiplying two random variables by a constant changes the covariance and this is an undesirable property for measuring the strength of a linear transformation. As an example of why this is undesirable, consider a scenario where you want to measure the linear relationship between height and weight. If you measure height in inches, you would want your measurement for strength of linear relationship to be the same as it would be if you measured height in feet (feet multiplies inches by the constant 1/12).

However, the correlation coefficient is invariant to deterministic linear transformations. That is, $Corr(aX,bY) = Corr(X,Y)$ and $Corr(X+a,Y+b) = Corr(X,Y)^*$.

*Covariance has this final property as well.

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