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To interpret and transmit biological signals, proteins use correlated motions. Experimental determination of these dynamics and the structural distributions they generate remains a key challenge. Here, using 1146 crystal structures of the main protease (M^pro) from SARS-CoV-2, we were able to infer a model of the enzyme's structural fluctuations. M^pro is regulated by concentration, becoming enzymatically active after forming a homodimer. To understand the structural changes that enable dimerization to activate catalysis, we employed our model, predicting which regions of the dimerization domain are structurally correlated with the active site. Mutations at these positions, expected to disrupt catalysis, resulted in a dramatic reduction in activity in one case, a mild effect in the second, and none in the third. Additional crystallography and biophysical experiments provide a mechanistic explanation for these results. Our work suggests that a statistical crystallography, in which numerous crystallographic datasets are analyzed, can reveal the structural fluctuations of protein native states and help uncover their biological function.