We develop an effective field theory (EFT) for nuclear vibrations. The key ingredientsquadrupole degrees of freedom, rotational invariance, and a breakdown scale around the threephonon level -are taken from data. The EFT is developed for spectra and electromagnetic moments and transitions. We employ tools from Bayesian statistics for the quantification of theoretical uncertainties. The EFT consistently describes spectra and electromagnetic transitions for 62 Ni, 98,100 Ru, 106,108 Pd, 110,112,114 Cd, and 118,120,122 Te within the theoretical uncertainties. This suggests that these nuclei can be viewed as anharmonic vibrators.