We use the Magnetospheric Multiscale mission (MMS) to study electron acceleration at Earth's quasi-perpendicular bow shock to address the long-standing electron injection problem. The observations are compared to the predictions of the stochastic shock drift acceleration (SSDA) theory. Recent studies based on SSDA predict electron distribution being a power law with a cutoff energy that scales with upstream parameters. This scaling law has been successfully tested for a single Earth's bow shock crossing by MMS. Here we extend this study and test the prediction of the scaling law for seven MMS Earth's bow shock crossings with different upstream parameters. A goodness-of-fit test shows good agreement between observations and SSDA theoretical predictions, thus supporting SSDA as one of the most promising candidates for solving the electron injection problem.