Basic properties of the normal distribution

The normal distribution is the continuous variate distribution which Gauss derived for the errors in making astronomical observations, as mentioned in Section 3.2. He considered that the total error in making an observation is the sum of a very large number of very small errors each of which might be positive or negative at random. He then proved mathematically that, if x is the total error, the population of total errors has a probability density function (defined in Section 3.2) of the form (4.1) $$p\left( x \right) = {1 \over {k\surd \left( {2\pi } \right)}}\exp \left( { - {{{x^2}} \over {2{k^2}}}} \right)$$ where k is large if the total errors are large, and small if they are small. Gauss showed further that k, in fact, is equal to the Standard deviation σ of this population; and, nowadays, we write directly (4.2) $$p\left( x \right) = {1 \over {\sigma \surd \left( {2\pi } \right)}}\exp \left( { - {{{x^2}} \over {2{\sigma ^2}}}} \right)$$ The reader should note that no attempt need be made to ascertain the nature of the individual causes of the small errors, their overall effect can be ascertained from the total error population itself. This basic idea has been extended to practically every type of Observation or measurement in science and technology and elsewhere. For, in many cases, it is too costly or the outcome too uncertain to investigate these sources of small errors but by using Gauss’s normal distribution we can, nevertheless, assess their overall effect. In fact, wherever there are a number of sources of small errors we can safely assume that the normal distribution applies, thus dealing with one of the basic problems of statistics namely how to classify the data.