Using the model independent average neutron rms radius of $^{133}\text{Cs}$ and $^{127}\text{I}$ obtained from the analysis of the coherent elastic neutrino-nucleus scattering data of the COHERENT experiment, we remove the long-standing $1.5\,��$ tension between the Standard Model prediction and the weak mixing angle measurement from the atomic parity violation (APV) in caesium. The updated APV result becomes $\sin^2 \vartheta_{\textrm{W}}=0.239{}^{+0.006}_{-0.007}$, to be compared with the Standard Model prediction at low momentum transfer, $\sin^2 \vartheta_{\text{W}}^{\textrm{SM}} = 0.23857(5)$. Moreover, exploiting the fact that the APV result is highly sensitive to the caesium neutron rms radius, $R_{n}$, and assuming that the Standard Model is correct, we combine the APV and the COHERENT measurements in order to get a better determination of $R_{n}$. The value of $R_{n}=5.42\pm 0.31\,\textrm{fm}$ is obtained, improving significantly the current uncertainty. This result allows to infer a meaningful value of the caesium neutron skin, the difference between the neutron and proton distribution radii, equal to $��R_{np}=0.62\pm 0.31\,\text{fm}$, showing for the first time a $2\, ��$ deviation from zero.

5 pages, 2 figures