The estimation of bacterial densities from dilution series

Abstract

A transformation devised by Mather is applied to give a new scheme for computing the maximum likelihood estimate of a bacterial density from the evidence of a dilution series. Tables are provided to expedite the application of the method; with their aid, the calculations take a from similar to, but much simpler than, those for probit analysis of quantal responses. Maximum likelihood estimation is compared with the method proposed by Fisher. The latter has the advantage of extreme simplicity, at least for series to which existing tables can be applied, and the loss of information involved in its use may often be compensated by the saving of time in calculation. An experimenter who has fairly reliable prior indications of an approximate value for the density, however, ought to concentrate his attention on dilutions that he believes will contain between 4 and 1/4 organisms per sample; he must not apply Fisher's method to his results, but the maximum likehood estimate will be so much more precise than any estimate from an experimental design using more extreme dilutions as to repay the small additional labour in computation.