  This paper provides a connection between the concentration of a random
variable and the distribution of the roots of its probability generating
function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with
$\mathbb{P}(X = 0)\mathbb{P}(X = n) > 0$ and with probability generating
function $f_X$. We show that if all of the zeros $\zeta$ of $f_X$ satisfy
$|\arg(\zeta)| \geq \delta$ and $R^{-1} \leq |\zeta| \leq R$ then \[
\operatorname{Var}(X) \geq c R^{-2\pi/\delta}n, \] where $c > 0$ is a absolute
constant. We show that this result is sharp, up to the factor $2$ in the
exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord
type theorem for random variables that are not necessarily sums of i.i.d.\
random variables.
