The 'whistler critical Mach number', M crit w , is one of the dimensionless parameters that characterizes collisionless shocks. Originally, it was introduced to indicate the critical point above which whistler waves do not propagate upstream. Indeed our analysis of Geotail data at the Earth's bow shock shows intense whistler waves in the sub-critical regime, M A < M crit w , but not in the super-critical regime. In this paper, we further report that M crit w seems to regulate the electron acceleration efficiency at the shocks. At the shock transition layer it is found that the spectral index Γ of electron energy spectra defined by f(E) $propto$ E -Γ is distributed between 3.5 and 5.0 in the sub-critical regime, while the hardest energy spectra with Γ = 3--3.5 are detected in the super-critical regime. We discuss a possible relationship between M crit w and the electron acceleration.