Size-frequency distribution normal-probability curve

The equation of the normal-probability curve, as applied to size-frequency distributions, is... 

Hatch extended his method of analysis to size-distribution curves ranging from coarse-screen analysis through fine particles measured microscopically. While excellent results were obtained by using this technique on laboratory samples, the method cannot be generalized to cover all types of distributions encountered in practice. As already explained in Chapter 3, size-frequency distributions may assume a variety of shapes. The Hatch development applies only to distributions which follow the normal or log-probability law. When size-distributions are hyperbolic in the lower extremes and follow normal log-probability laws in the upper extremes, the Hatch analysis must necessarily fail. Nevertheless, the relationships developed by Hatch have a far-reaching practical importance... 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the curve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical bell-shaped curve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

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