What is the Variance Sum Law?
For example, suppose you ran a research project which involved sampling the weight of apples in New York orchards, and then you did a similar project on oranges in Southern California. Now, imagine you need to work with the collected data from both research projects and draw some conclusions on the weight of apples and oranges. But first, what would the variance of your new data set be?
The Variance Sum Law – Independent Case
If your two sets are independent, like the apples and oranges example, you can use the simplest version of the variance sum law.
Var(X ± Y) = Var(X) + Var(Y).
This just states that the combined variance (or the differences) is the sum of the individual variances.
Sum and Difference
Note that the variance of the sum of both sets and the difference of both sets is exactly the same. This may take you by surprise on first sight, but after thinking it through you’ll realize that since Var(X-Y)=Var(X+(-Y))= Var(X) + Var(-Y) and Var(Y)= Var(-Y), Var(X+Y) must be equivalent to Var (X-Y).
Variance Sum Law — Dependent Case
Cov(x,y) is the covariance of x and y.
Since the population correlation coefficient ρ is defined as = Cov(x,y) / σxσy, this can also be expressed as
Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.
Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.