The author examines the eftect of the ‘ log p ’ transformation (where p varies) on the stochastic dependence between the mean and variance. If the stabilization of variance is obtained with such a transformation, then the relation variance-mean, in the non-transformed numbers, is represented in the log-log scale by a curve, the slope of which run progessively from 1 for the small values of the mean, to 2 for large ones. Many series of replicated samples give scatter diagrams of points ( m, s 2) which are approximated quite well by one of the theoretical curves, for p equal 1 to 2. The stability of variance and the normality are then obtained by the mean of a transformation between 'log and ‘log 2’. A ‘ log p ’ distribution makes a transition between a Poisson distribution for the small numbers, and lognormal for the large ones, and is locally identical to a Taylor's law, with parameters between 1 and 2. Increasing heterogeneity of the sampling conditions, p tends towards 1, i.e., the distribution tends towards a lognormal one. Composing a Poisson distribution, with a lognormal distribution of its parameter, we obtain a near negative-binomial distribiution, the parameter K of which is equal to the inverse of the variance of the Naperian logarithm of the initial values. The model does not fit: the explanation of the observed regularities is probably more complex. The dependence upon the spatial scale suggests that a part of the phenomenon is linked to hydrodynamic turbulence.