  The discrepancy between the Standard Model theory and experimental
measurement of the muon magnetic moment anomaly,
$a_{\mu}=\left(g_{\mu}-2\right)/2$, is connected to precision electroweak (EW)
predictions via their common dependence on hadronic vacuum polarization
effects. The same data for the total $e^+e^- \rightarrow \text{hadrons}$ cross
section, $\sigma_{\rm had}(s)$, are used as input into dispersion relations to
estimate the hadronic vacuum polarization contributions, $a_{\mu}^{\rm
had,\,VP}$, as well as the five-flavor hadronic contribution to the running QED
coupling at the $Z$-pole, $\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2)$, which enters
natural relations and global EW fits. The EW fit prediction of
$\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02722(41)$ agrees well with
$\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02761(11)$ obtained from the
dispersion relation approach, but exhibits a smaller central value suggestive
of a larger discrepancy $\Delta a_{\mu}=a_{\mu}^{\rm exp} - a_{\mu}^{\rm SM}$
than currently expected. Postulating that the $\Delta a_{\mu}$ difference may
be due to missing $\sigma_{\rm had}(s)$ contributions, implications for $M_W$,
$\sin^2 \! \theta^{\rm lep}_{\rm eff}$ and $M_H$ obtained from global EW fits
are investigated. Shifts in $\sigma_{\rm had}(s)$ needed to bridge $\Delta
a_{\mu}$ are found to be excluded above $\sqrt{s} \gtrsim 0.7$ GeV at the
95\%CL. Moreover, prospects for $\Delta a_{\mu}$ originating below that energy
are deemed improbable given the required increases in the cross section. Such
hypothetical changes to the hadronic data are also found to affect other
related observables, such as the electron anomaly, $a_e^{\rm SM}$, the ratio
$R_{e/\mu} = (m_\mu/m_e)^2 (a_{e}^{\rm had,\,LO\,VP}/a_{\mu}^{\rm
had,\,LO\,VP})$ and the running of the weak mixing angle at low energies,
although the consequences of these are currently less constraining.
