The Normal Logistic Curve, Its Generalization and Its Use as a Probability Function

In dynamical statistics we are very often confronted with the problem of finding the best analytical function for a particular time series. Use can be made of the parabola of higher degree or the method of orthogonal polynomials, thus fitting the function to empirical data in order to derive an analytical expression. We study periodical cases by application of a Fourier Series. However, in the social sciences as also in the natural sciences, as in chemistry and in physics, we have cases which repeat for which we are able to find a law of development. This law of development has such properties that we do not need a very long equidistant time series of observations, as we would in applying the Fourier series, and yet we have the possibilities of extrapolation, which is not possible in the use of the parabola. One of the most important functions expressing this law of development is the normal logistic law, which play in dynamical statistics the same role as does the Gauss function in statical statistics. The application of the normal logistic curve is but a special case of the generalized logistic curve. It has been discovered that the properties of the first derivative of a normal logistic curve is of such a nature that it can be used as a probability function in a theory of statical statistics, especially for some cases of a sampling theory. The higher moments show us that for some data of frequency tables, it will be better to use the normal logistic curve as a probability function; this is true also in the case of some asymmetrical or bimodal empirical functions. By this method we have discovered how to find the parameters, or constants, of a normal logistic law, using the method of moments directly