Mensuration

MENSURATION).

(ii.) If we want to find the most probable values of the con stants, the problem becomes one of inverse probability, and we have to make some assumption as to the a priori probabilities of different values of the constants. When the number of indi viduals is large, this presents no difficulty. If, for instance, the distribution is of heights of men, and we require the most probable value of their mean, the range of practically possible values is small, and we can assume that within this range all values are equally likely. But caution is necessary when we are dealing with small numbers. In the case of a Gaussian distri bution, the most probable values, on the above assumption, are obtained by taking first and second moments; subject to some qualification in view of the fact that measurements are grouped.

(iii.) When a value

g has been found for a parameter, note should be made of the " probable error" of g. This is a quantity E such that it is an even chance that the true value of the para meter lies between g—e and g-Fe.

3o.

Test of Closeness of Fit.—When we have chosen a form ula, we have still to satisfy ourselves that it is a suitable one. The method of doing this belongs properly to correlation (sec 33), since it deals with the probability of joint occurrence of a number of variations. There are two classes of cases. In the one class the formula is completely settled a priori, and there are no constants to be determined. In the case of throwing a die, for instance, our hypothesis may be that all faces are equally likely to be uppermost; we test the hypothesis by comparing the actual numbers of throws of I, 2. . . 6 with the theoretical num

bers, namely 1/6.N for each face, and see whether the dis crepancies between hypothesis and fact are such as might be due to random sampling. In the other class of cases there are un known constants. Whatever values are taken for these con stants, there will be discrepancies between the theoretical fre quencies calculated from them and the actual frequencies; and there will be the same enquiry as to these discrepancies. Both cases are usually dealt with by a method due to K. Pearson, and known as the method.

31. Exclusion of Extreme Values.—Occasionally, in a set of observations of a group supposed to be homogeneous, extreme variations—giants or dwarfs—will occur. If the probability of their occurrence as the result of random sampling is extremely small, regard being had to the total number n, are they to be excluded? Their inclusion or exclusion may make a good deal of difference not only in the deduced values of the constants but also in the measure of closeness of fit. Several forms of criterion for exclusion have been proposed. But these relate to exclusion from a particular distribution. If we deal statistically with a number of distributions of the same kind, the extent to which these extreme variations occur may be an important element in our enquiries.