A general model for averaging the acoustic target strength functions of fish is stated in calculable form. It accounts for the influences of the distribution of generally coupled spatial and orientation states of fish, geometric perspective, and beam patterns on observations of target strength. The model is developed and applied to observation of fish by directional, downward?looking sonars. A particular example is considered in which the sonar is represented by an ideal circular piston, the spatial distribution of fish is homogeneous, and the orientation distribution is spatially homogeneous and characterized by a uniformily distributed azimuthal variable and an independent, essentially normally distributed tilt angle variable. Averaged and averaged?squared backscattering cross sections are computed from high quality gadoid target strength functions measured at two ultrasonic frequencies. Results for a sonar half?beamwidth of 2.5 deg for three different realizations of the tilt angle distribution are expressed in the logarithmic domain and regressed linearly on fish length. The significance of species, frequency, and orientation distribution differences among the regressions is noted. Estimates of the mean ratio of averaged?squared backscattering cross section and squared?averaged backscattering cross section are presented.