  For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$,
$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as
$\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and
$\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d
{\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of
commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A},
\mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We
give an expression for the maximal joint numerical range of a tuple of doubly
commuting matrices. Consequently, we obtain that the above distance formula
holds for tuples of doubly commuting matrices. We also discuss some general
conditions on the tuples of operators for this formula to hold. As a result, we
obtain that it holds for tuples of Toeplitz operators as well.
