The normal distribution function and the normal probability laws have played a significant role in probability theory since the early eighteenth century, and it is important to understand from what this signifiance derives.

To begin with, there are random phenomena that obey a normal probability law precisely. One example of such a phenomenon is the velocity in any given direction of a molecule (with mass \(M\) ) in a gas at absolute temperature \(T\) (which, according to Maxwell’s law of velocities, obeys a normal probability law with parameters \(m=0\) and \(\sigma^{2}=M / k T\) , where \(k\) is the physical constant called Boltzmann’s constant). However, with the exception of certain physical phenomena, there are not many random phenomena that obey a normal probability law precisely. Rather, the normal probability laws derive their importance from the fact that under various conditions they closely approximate many other probability laws.

The normal distribution function was first encountered (in the work of de Moivre, 1733) as a means of giving an approximate evaluation of the distribution function of the binomial probability law with parameters \(n\) and \(p\) for large values of \(n\) . This fact is a special case of the famed central limit theorem of probability theory (discussed in Chapters 8 and 10) which describes a very general class of random phenomena whose distribution functions may be approximated by the normal distribution function.

A normal probability law has many properties that make it easy to manipulate. Consequently, for mathematical convenience one may often, in practice, assume that a random phenomenon obeys a normal probability law if its true probability law is specified by a probability density function of a shape similar to that of the normal density function, in the sense that it possesses a single peak about which it is approximately symmetrical. For example, the height of a human being appears to obey a probability law possessing an approximately bell-shaped probability density function. Consequently, one might assume that this quantity obeys a normal probability law in certain respects. However, care must be taken in using this approximation; for example, it is conceivable for a normally distributed random quantity to take values between \(-10^{6}\) and \(-10^{100}\) , although the probability of its doing so may be exceedingly small. On the other hand, no man’s height can assume such a large negative value. In this sense, it is incorrect to state that a man’s height is approximately distributed in accordance with a normal probability law. One may, nevertheless, insist on regarding a man’s height as obeying approximately a normal probability law, in order to take advantage of the computational simplicity of the normal distribution. As long as the justification of this approximation is kept clearly in mind, there does not seem to be too much danger in employing it.

There is another sense in which a random phenomenon may approximately obey a normal probability law. It may happen that the random phenomenon, which as measured does not obey a normal probability law, can, by a numerical transformation of the measurement, be cast into a random phenomenon that does obey a normal probability law. For example, the cube root of the weight of an animal may obey a normal probability law (since the cube root of weight may be proportional to height) in a case in which the weight does not.

Finally, the study of the normal density function is important even for the study of a random phenomenon that does not obey a normal probability law, for under certain conditions its probability density function may be expanded in an infinite series of functions whose terms involve the successive derivatives of the normal density function.