# 4.7: Variance Sum Law II - Correlated Variables

- Page ID
- 2331

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:

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