# 9.7: Variance Sum Law II - Correlated Variables

• • David Lane
• Rice University
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Learning Objectives

• State the variance sum law when $$X$$ and $$Y$$ are not assumed to be independent
• Compute the variance of the sum of two variables if the variance of each and their correlation is known
• Compute the variance of the difference between two variables if the variance of each and their correlation is known

Recall that when the variables $$X$$ and $$Y$$ are independent, the variance of the sum or difference between $$X$$ and $$Y$$ can be written as follows:

$\sigma_{X \pm Y}^2 = \sigma_{X }^2 + \sigma_{Y}^2 \label{eq1}$

which is read: "The variance of $$X$$ plus or minus $$Y$$ is equal to the variance of $$X$$ plus the variance of $$Y$$."

When $$X$$ and $$Y$$ are correlated, the following formula should be used:

$\sigma_{X \pm Y}^2 = \sigma_{X }^2 + \sigma_{Y}^2 \pm 2 \rho \sigma_X \sigma_Y \label{eq2}$

where $$\rho$$ is the correlation between $$X$$ and $$Y$$ in the population.

Example $$\PageIndex{1}$$

If the variance of verbal SAT were $$10,000$$, the variance of quantitative SAT were $$11,000$$ and the correlation between these two tests were $$0.50$$, what is the variance of total SAT (verbal + quantitative) and the difference (verbal - quantitative)?

Solution

Since the two variables are correlated, we use Equation \ref{eq2} instead of Equation \ref{eq1} for uncorrelated (independent) variables. Hence, the variance of the sum is

$\sigma^2_{verbal + quant} = 10,000 + 11,000 + 2\times 0.5\times \sqrt{10,000} \times \sqrt{11,000}$

which is equal to $$31,488$$. The variance of the difference is also determined by Equation \ref{eq2}:

$\sigma^2_{verbal - quant} = 10,000 + 11,000 - 2\times 0.5\times \sqrt{10,000} \times \sqrt{11,000}$

which is equal to $$10,512$$.

If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law:

$s_{X \pm Y}^2 = s_{X }^2 + s_{Y}^2 \pm 2 r\, s_X \, s_Y$

This page titled 9.7: Variance Sum Law II - Correlated Variables is shared under a Public Domain license and was authored, remixed, and/or curated by David Lane via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.