# What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?

What is the meaning of the quantity:

$$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$

Is there, for example, a geometric explanation? Is there a term for it in statistics?

• if the answer ok for you @apadana, can you please accept and/or upvote it? – gunes Mar 26 at 20:50

## 1 Answer

It is the square root of the determinant of the covariance matrix (between $$X$$ and $$P$$). The determinant of the covariance matrix is called as Generalized Variance, which quantifies the co-variability of multivariate random variables to a scalar. What you write is the square root of it, so I believe it won't be too odd to call it as Generalized Deviation.

Edit: After some research, I found that, in some contexts, it's referred as Generalized Standard Deviation, or Wilk's standard deviation.

• Have you seen "generalized deviation" used anywhere, or did you come up with that term? – Dave Mar 11 at 18:22
• @Dave I've come up with it actually. – gunes Mar 11 at 21:22
• "Co-variability" is more accurate, than "variability". E.g. if the determinant of a covariance matrix is zero, there's linear dependence among the de-meaned random variables. (In the two by two case here, $\epsilon = 0$ if and only if $X$ lies in the linear span of $P$ and $1$.) – Michael Mar 12 at 1:28
• Right @Michael changed it – gunes Mar 12 at 7:32