Gaussian distribution geometric mean

An aerosol rarely consists of particles that are the same size, and usually a distribution of sizes around a mean is observed. The observed data may be fitted by statistical approximation to a distribution. The number of particles in a size range when plotted against the logarithm of the particle diameters frequently exhibit a normal (Gaussian) distribution. This is known as a log-normal distribution and is described by a parameter known as the geometric standard deviation. Theoretically, a monodisperse aerosol will exhibit a geometric standard deviation of 1 in practice, however, an accepted limit is 1.2 [6]. 

Here, the curve is symmetric about uXm- The median of the curve is not identical with the number average X (see below). The function corresponds to the error distribution about the geometric mean. The ratio of the degrees of polymerization is therefore important with the logarithmic normal distribution, in contrast to the Gaussian distribution, where the difference is important. 

Local geometrical features are essential for understanding binding properties, catalytic behavior, and molecular recognition. Many of the descriptors used for global analysis can be adapted to study local features. For instance, mean and Gaussian curvature distributions of a surface, curvature and torsion of a molecular space curves, and the variation of the fractal index Df(r) over a molecular model serve this purpose. 

Most frequently, an aerosol is characterized by its particle size distribution. Usually this distribution is reasonably well approximated by a log-normal frequency function (Fig. 4A). If the distribution is based on the logarithm of the particle size, the skewed log-normal distribution is transferred into the bell-shaped, gaussian error curve (see Fig. 4B). Consequently, two parameters are required to describe the particle size distribution of an aerosol the median particle diameter (MD), and an index of dispersion, the geometric standard deviation (Og). The MD of the log-normal frequency distribution is equivalent to the logarithmic mean and represents the 50% size cut of the distribution. The geometric standard deviation is derived from the cumulative distribution (see Fig. 4C) by... 

---