This paper presents a general analysis of dynamical models for scale-free ensembles of binary galaxies. Such models have constant but arbitrary orbital eccentricity and power-law relationships between pair separation and both number density and the scale of the relative velocity distribution. Their properties are easily simulated and analytic expressions are given both for the correction factors required to obtain unbiased estimates of the mean mass and mean mass-to-light ratio of a binary sample, and for the uncertainty in such estimates caused by sampling fluctuations and by observational errors. The practical application of these models is discussed, and appropriate methods are suggested for estimating the intrinsic density distribution, the pair interaction potential, the orbital eccentricity and a representative M/L ratio. A simple correction factor method of obtaining mean masses and mass-to-light ratios is shown to be subject to considerably less systematic error and to give more reliable results than apparently more sophisticated distribution fitting techniques. A reanalysis of Turner's (1976a) data sample illustrates the use of scale-free models, and brings to light a number of errors and inconsistencies in earlier work. The sample shows no evidence for intrinsic luminosity–separation correlations nor for any positive correlation of luminosity and M/L ratio; it gives at most only very weak evidence for predominantly circular rather than predominantly radial orbits. The mass-to-light ratio obtained by the correction factor method for Turner's primary data sample has a formal error equal to its nominal value. This situation can be remedied by the unbiased rejection of a few pairs which contribute heavily to the variance of the estimate. For such culled samples the derived M/L ratios have a 20 per cent uncertainty due to sampling fluctuations, compounded with a somewhat larger systematic uncertainty arising from poor knowledge of certain model parameters; they are a factor of 2 smaller than the values given by Turner (1976b). Monte Carlo simulations of the dynamics of Turner's sample are constructed on the hypotheses of Keplerian orbits and a dispersion-free relation between mass and luminosity. Such models agree well with the data if observational errors in the velocities are rather larger than those quoted by Turner.