  Let $X$ be a random variable with finite second moment. We investigate the
inequality: $P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}\ge P\{|Z|\le 1\}$, where $Z$
is a standard normal random variable. We prove that this inequality holds for
many familiar infinitely divisible continuous distributions including the
Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, log-normal,
student's $t$ and inverse Gaussian distributions. Numerical results are given
to show that the inequality with continuity correction also holds for some
infinitely divisible discrete distributions.
