The geometry of log‐normal and related distributions and an application to tracer‐dilution curves

Summary Most of a typical log-normal curve lies very close to two straight lines, namely the tangents at its points of inflection. This holds good for nearly all the rising part of the curve and for most of the top half beyond the maximum. When loge (t – to) is normally distributed with standard deviation s, whether the whole curve can be observed or not, s, to, the mean, the mode and the area under the whole curve can be derived from the triangle formed by these two tangents and the base line. The triangle also provides a simple test of goodness-of-fit. The corresponding inflection triangles of two other skew distributions are investigated in outline namely a chi2 (or gamma) distribution and one arising in Brownian movement or random walk in one dimension with drift. These are compared with the log-normal one. The normal distribution and its inflection triangle are given as a limiting case of the log-normal one and it is shown that the curve on the cover of Statistica Neerlandica cannot be a normal one. The application is to the extrapolation problem in heart dye-dilution curves, which record the distribution of passage times of an indicator through the lungs and heart. The usual semi-log extrapolation for determining the area under the complete curve is discussed critically; it is shown how the inflection triangle provides a simpler procedure that is probably more accurate. The general formulae, an outline of the application and numerical tables for 0 < s 0.8 are given in the main text; the mathematical derivations of all the basic formulae are in an appendix (sections 11 to 16). The information needed for practical applications without studying the theory is contained in sections 3 to 6 and the associated tables and figures.