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Say I have variables $A, B\; \&\; C$, where $A + B = C$ for every observation in the population. This isn't a sample, so I know to use population calculations.

For my data set, $Var(A) + Var(B) \not= Var(C)$, so there is some covariance.

My question is about how to choose the correct way to calculate the standard deviation of $C$. Is it okay to just find the variance of my $C$ terms, and take the square root of that? Or do I need to perform $\sqrt{Var(A) + Var(B)}$, or do something else entirely?

In addition, I am interested in knowing the standard deviation of all three variables. Given that there is covariance, would that make some of them - either $Var(A)$ and $Var(B)$, or $Var(C)$ - not as meaningful?

Thanks!

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1 Answers 1

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$$\begin{align} \newcommand{\Var}{\operatorname{\sf Var}} \newcommand{\Cov}{\operatorname{\sf Cov}} \newcommand{\E}{\operatorname{\sf E}} \Var[C] & = \Var[A+B] \\[1ex] & = \E[(A+B)^2]-\E[A+B]^2 \\[1ex] & = \E[A^2]-\E[A]^2+\E[B^2]-\E[B]^2+2\E[AB]-2\E[A]\E[B] \\[1ex] & = \Var[A]+\Var[B]+2\Cov[A,B] \end{align}$$

So you can calculate standard deviation of $C$ using these three terms.