  The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a
unit vector $h$ is defined by $D_h(a)=\|ah\|^2-|<ah,h>|^2$. We show that two
operators $a$ and $b$ have the same variance at all vectors $h\in H$ if and
only if there exist scalars $\sigma,\lambda$ with $|\sigma|=1$ such that
$b=\sigma a+\lambda1$ or $a$ is normal and $b=\sigma a^*+\lambda1$. Further, if
$a$ is normal, then the inequality $D_h(b)\leq\kappa D_h(a)$ holds for some
constant $\kappa$ and all unit vectors $h$ if and only if $b=f(a)$ for a
Lipschitz function $f$ on the spectrum of $a$. Variants of these results for
C$^*$-algebras are also proved.
  We also study the related, but more restrictive inequalities $\|bx-xb\|\leq
\|ax-xa\|$ supposed to hold for all $x\in B(H)$ or for all $x\in B(H^n)$ and
all positive integers $n$. We consider the connection between such inequalities
and the range inclusion $d_b(B(H))\subseteq d_a(B(H))$, where $d_a$ and $d_b$
are the derivations on $B(H)$ induced by $a$ and $b$. If $a$ is subnormal, we
study these conditions in particular in the case when $b$ is of the form
$b=f(a)$ for a function $f$.
