Frequency-Distribution of Errors 14

(ii.) But instead of expressing U in terms of Y, we might ex press Y terms of certain total probabilities. The usual method of doing this is to regard the figure of frequency as divided into too equal parts by ordinates: the values of Y corresponding to these ordinates, with the two extreme values of Y, are called the percentile values. The important values, on this system, are the median and the quartile values. If MI. Pi, 312 M3 are ordinates dividing the figure of frequency into four equal parts, the value of Y which corresponds to is the median value, and the values which correspond to and are the quartile values.

21.

The Normal Law of Error.—(i.) The result obtained in sec. 16 gives the true law of frequency of error for the type of case considered, namely the law that „K„„ the probability of occurrence of in A's and n—m a's, is When n is small there is no difficulty in calculating any particular value of „K. or even all the values; thus in sec. 16' we have seen that, if p = 0.6, the probabilities of occurrence of the values 6, 5, 4, 3, 2, I, 0 of m are 0.047, 0.187, 0.311, 0.276, 0.138, 0.037, 0.004. But, when n is large, the single probabilities are very small, and we have to take them in groups. We therefore require a formula which will enable us to calculate the sum of the prob abilities in any particular group, i.e., the probability that the value of m will lie between any two stated values. An approx imate formula—less or more correct according as n is smaller or larger—is given by the normal law of error. This law is also known as the Gaussian law of error, though it was discovered. by Laplace : or as the law of large numbers. It would more cor rectly be called the normal law of frequency of error or the law of large frequencies.

(ii.) The method consists, essentially, in replacing a sum by an integral. To see how this happens, let us take p to have some definite value, and let us see how the form of the frequency polygon (sec. 14) for a representative distribution alters as increases. There are n+ I ordinates; as n increases, the number of ordinates increases, and their size decreases. If the interval

between successive ordinates is always h, the frequency-polygon becomes wider and flatter. If, on the other hand, we take the interval to be h/n, so that (n being large) the breadth of the polygon is practically constant, it becomes higher and narrower in the middle. We can obtain regularity by adopting a middle course. We have seen in sec. 18 that the standard deviation, which is a convenient measure of the dispersion of the values about their mean, is Alnpq, i.e., is proportional to -sln. For purposes of comparison let us keep the maximum ordinate fixed in position, and take the interval between consecutive ordinates to be h/Alnpq instead of h, the ordinates themselves being mul tiplied by Ainpq, so as to keep the area of the polygon unaltered. Then it will be found that the heights of the ordinates in any particular position, fairly near the maximum ordinate, alter comparatively slowly, and that the tops of the ordinates tend to lie on a certain curve, which is the limit of the upper boundary of the frequency-polygon. This curve is the normal curve of error; and the figure bounded by it, which is the limit of the frequency-polygon, is the normal figure of frequency.

The following short table (Table III.) illustrates this, so far as the maximum ordinate is concerned; p being taken to be 0.6.

The figure given by (21.2) and (21.3) is the normal figure of frequency.

(iv.) The figure is symmetrical about a central ordinate; and it admits of any value of m, positive or negative, so that its range is from to + . In these respects it differs from the (discontinuous) graph of ,,,K„,; and, as already stated, it is only for moderate values of in—up that a portion of its area represents the sum of a number of K's. It may, however, be noted that the figure agrees with the graph of „K„, in the following respects: (a) the mean value of m, for the figure, is np; (b) the standard deviation of m is -V npq; (c) the complete area of the figure is 1.

(v.) Since (sec. I 5) m—np is the " error" of m, we denote it by the symbol e; and (21.2) can then be written