17. Mean, Standard Deviation, and Mode.—Suppose that the categories of a frequency-distribution correspond to values Yo, Yi, Y2 . . Yk of a variable Y; thus, in the case we have just been considering, the Y's are the numbers n, n—i, . o of A 's or the numbers o, 1, 2 . . .n of a's. Then the mean value of Y is found by multiplying . Yk by the numbers . in the corresponding categories, adding the The square root of this quantity is called the standard deviation or dispersion: in the cases in which the Y is an error, it was for merly called the error of mean square. The above definitions ap ply, and (17.3) holds good, whether we are dealing with a rep resentative distribution or with an actual distribution.
In some cases the numbers in the categories of a representative distribution gradually increase up to a value and then decrease. In such a case nf is the maximum frequency; and the corresponding value of Y, namely Y1, is called the modal value or mode. Thus in the example in sec. 16 the modal number of A's is 4. There might in some cases be two or more modes; in other cases there might be none—if, e.g., the frequen cies decreased throughout the whole range.
18. Mean, etc., for Binomial Distribution.—For a repre sentative binomial distribution. (sec. 16) in which „K„, is the probability that in n trials there will be m A's and n—m a's, it is not difficult to show that, the probability of A at each trial being (a) the mean value of m is np, that of n—m being n—np=nq; (b) the mean square of m is and therefore (c) the standard deviation (dispersion) of m is -sInpq.
(d) To find the modal value of m, we have > I, and the second < I, if (n+ Op lies between m and m+ i, i.e., if m is the integral part of (n+ Op. Taking m to have this value, is greater than and also greater than „K i.e., it is the maximum frequency. Thus in the example in sec. 16 we have p= n=6, and therefore the modal value of m is 4. If (n+i)p is an integer, and we take m= (n+ Op, then and „K. are equal, each having the maximum value.
19. Errors of Frequencies of Error.—In Sec. 16 we really began with a representative distribution, namely, that in which the number of cases of "A" was np out of a total number n; and we considered the probabilities of the various possible de viations from this distribution, i.e., we considered the entries in a representative frequency-distribution of the errors. We found that the frequencies, for N sets of n trials, would be given by the terms of a certain binomial expansion, each multiplied by N.
Thus the frequency of the cases of 3 A's and 3 a's, p being 0.6, would be N X•276480. But it is to be observed that this is a representative distribution: we should not expect, in any par ticular group of N sets of n trials, to find exactly N X-27648o of these cases, just as in n of the original trials we should not ex pect to get exactly np A's and nq a's. If we took N groups of N sets of n trials, we should again get errors of random sampling. But these errors become relatively less as the numbers involved become greater.
20. Frequency for Continuous Variation.-0.) We have hitherto been considering the frequency of occurrence of some definite number of events, e.g., of in events out of n. This is a case of discontinuous variation, and the graph of variation is composed of a limited number of ordinates. We shall presently have to consider the class of cases in which Y is a quantity which varies continuously, the number of its possible values being indefinitely great, and the probability of occurrence of any par ticular value indefinitely small. For these cases we require a different treatment of frequency. Let the probability that the value of the variable lies within limits 1dY, i.e., lies tween Y-1dY and Y+ 2dY, be U'dY; and let U be the limit of U' when dY is made indefinitely small. Then the ex pression of U in terms of Y is the equation to the frequency distribution of Y. The graph of U with respect to Y will then be a plane figure with a continuous boundary, which can be re garded as drawn on such a scale that its area is 1. The area of the portion of this figure which lies between ordinates corre sponding to any values and of Y will be proportional to the probability that Y lies between and This figure is the figure of frequency of Y.
To represent an actual distribution graphically, suppose that we have observed the number of cases in which Y lies between and Let 31, and Mr+1 be points on the base-line Oy corresponding to Y,. and Then the ratio of this number of cases to the total number can be represented by a rectangle whose base is Mr+i and whose area is this ratio. The ag gregate of these rectangles is a figure called a histogram: its area is I, i.e., is the same as that of the figure of frequency. This latter figure may be regarded as the limit of the histogram of a representative distribution when the intervals r - ,.+i are made indefinitely small.