  We perform simultaneous analysis of (1) matter radii, (2) $B(E2; 0^+
\rightarrow 2^+ )$ transition probabilities, and (3) excitation energies,
$E(2^+)$ and $E(4^+)$, for $^{24-40}$Mg by using the beyond mean-field (BMF)
framework with angular-momentum-projected configuration mixing with respect to
the axially symmetric $\beta_2$ deformation with infinitesimal cranking. The
BMF calculations successfully reproduce all of the data for $r_{\rm m}$,
$B(E2)$, and $E(2^+)$ and $E(4^+)$, indicating that it is quite useful for data
analysis, particularly for low-lying states. We also discuss the absolute value
of the deformation parameter $\beta_2$ deduced from measured values of $B(E2)$
and $r_{\rm m}$. This framework makes it possible to investigate the effects of
$\beta_2$ deformation, the change in $\beta_2$ due to restoration of rotational
symmetry, $\beta_2$ configuration mixing, and the inclusion of time-odd
components by infinitesimal cranking. Under the assumption of axial deformation
and parity conservation, we clarify which effect is important for each of the
three measurements, and propose the kinds of BMF calculations that are
practical for each of the three kinds of observables.
