# Bound on variance of sum of variables

Suppose I have two finite sets of data A and B, with equal length n.

What's the best upper and lower bound I can make on var(A+B), in terms of var(A) and var(B)?

• This question simplifies greatly when recast in terms of the variances, due to the simple formula for var(A+B). That version is addressed by the Cauchy-Schwarz Inequality.
– whuber
Commented Oct 3, 2011 at 17:32
• Okay, I've now found $var(A+B) \leq var(A) + var(B) + 2 \sqrt{var(A) var(B)}$, but is this bound tight? Commented Oct 3, 2011 at 17:39
• @Greg Snow's example (where $A=B$) demonstrates tightness.
– whuber
Commented Oct 3, 2011 at 18:46
• The upper bound can be written a bit more cleanly in terms of standard deviations $sd(A + B) \le sd(A) + sd(B)$. Commented Jan 27, 2023 at 21:06

The lower bound will occur when $B = -A$ so that $A + B = 0$ and the variance is $0$. This also occurs when $B$ is shifted from $A$, but in any case you cannot have a variance less than $0$ and this shows a case where it can equal $0$.

The upper bound comes when $B = A$ or $B = c1 + c2 \times A$ which gives a correlation of $1$. In this case you get the variance is the comments above.