A note on the relations of certain parameters following a logarithmic transformation

Abstract

The logarithmic transformation has been used in the statistical analysis of certain marine biological data. Parameters calculated from the transformed distributions have been used in the description of the observations, but it appears that some of the mathematical procedures adopted have not been fully understood.Winsor & Clarke (1940) have analysed data from plankton hauls. The problem considered by them was the determination of the variability in numbers of animals caught by repeated hauls through the same body of water. The raw data were characterized by a constant coefficient of variation, i.e. the variability in catch was proportional to the size of the catch. By transformation from the actual numbers caught to their logarithmic values, it was possible to equalize the variances and to apply the method of analysis of variance to esti-mate the variability due to the different sources. Finally, an estimate of the coefficient of variation was obtained from the logarithmic values. Their method of estimation was quoted by Snedecor (1946, p. 451) and was employed by Barnes & Bagenal (1951) in their study of repeated trawl hauls. Barnes & Bagenal also calculated confidence limits for comparison of observations following the method of Silliman (1946) in his work on pilchard eggs. These methods seem to be based on a misunderstanding of the nature of a trans-formation. The formal relation between the variance of such transformed data and the coefficient of variation of the untransformed data follows from the moments of the ‘log-normal’ distribution, given first by Wicksell (1917). From these moments it will be shown that the method given by Winsor & Clarke is mathematically unsound.