These relations must be borne in mind if we wish to understand the results of statistical methods. Since the work of Quetelet, Galton, and others the statistical examination of individual differences in animals and plants has become a special science, which is primarily based on the consideration that the application of the theory of probability renders possible mathematical statement and control of the results. The facts show that any character, size of leaf, length of stem, the number of members in a flower, etc. do not vary haphazard but in a very regular manner. In most cases it is found that there is a value which occurs most commonly, the average or medium value, from which the larger and smaller deviations, the so-called plus and minus variations fall away in a continuous series and end in a limiting value. In the simpler cases a falling off occurs equally on both sides of the curve; the curve constructed from such data agrees very closely with the Gaussian curve of error. In more complicated cases irregular curves of different kinds are obtained which may be calculated on certain suppositions. The regular fluctuations about a mean according to the rule of probability is often attributed to some law underlying variability. (de Vries, "Mutationstheorie", Vol. I. page 35, Leipzig, 1901.) But there is no such law which compels a plant to vary in a particular manner. Every experimental investigation shows, as we have already remarked, that the fluctuation of characters depends on fluctuation in the external factors. The applicability of the method of probability follows from the fact that the numerous individuals of a species are influenced by a limited number of variable conditions. (Klebs, "Willkurl. Ent." Jena, 1903, page 141.) As each of these conditions includes within certain limits all possible values and exhibits all possible combinations, it follows that, according to the rules of probability, there must be a mean value, about which the larger and smaller deviations are distributed. Any character will be found to have the mean value which corresponds with that combination of determining factors which occurs most frequently. Deviations towards plus and minus values will be correspondingly produced by rarer conditions.