Brownian Moton in the Stock Market 1924 to 1956, and intervals of one day to 12 years, the sgr does indeed Increase with the square root of the time, corresponding to a slope of n Even the data beginning in July 1929 Is not exceptional in this regard Continuity of the data from the two figures may be obtained by noting the shift from a trading- to calendar-time scale, which occurs mostly between a day and a week Note that thus measure of the dispersionof r, 0 6745 or(r) from Figs 9 and 10, Is from a sample of N for some single interval The years 1924-1956 were the hmits of readly accessIble data, pub lished by F w Stephens and the Securties Research Cor sampling was achieved by paging through the publications and flipping a coin to decide which page and common stock, of the NYSe, to select The short range data of Fig 10 were taken directly from the Wall Street Journal In a similar fashion In order to clarfy the next step, let us anticipate the results of an Inductive analysis of the data by describing some models which have man of the features of our market-the ensemble of 1000 or more logarithms of indIvIdual stock prIces, as functions of the time MoDEL I Let us first Imagine 2000 pennies grouped in 1000 pairs All 1000 pairs are tossed simultaneously at intervals 8, or 1/ 8 tosses per unit time Heads count +1, taIls -1, and we record the payoff, 0, or-2 y,(a)of the yth pair (=1 1000)of the ith toss We also record the ean of a sample N in number for each toss, m(i)=1/N a-i v, (i) evidently this will be very close to zero for large n We also record the cumulated sum after k tosses or after an nterval r of each pair, and of the mean,Ie,Y(r)and M(r) where t=h8 We also ascrIbe an arbitrary starting point for each random walk Y(r), which, to preserve the analogy Ith our previous notation, we shall call log P, (t) for the yth pair We deliberately add and subtract this arbitrary constant, to emphasize that Y, (r)is the deviation from some arbitrary starting point (ef Web Fechner law) Y(r)=ogP()+∑ P3(t), (5) M(r)=∑:m()=(1N)∑=∑:-() Evidently the dispersion of the y's, or(n) can be computed theoretically, and also estamated from the data, among other methods, by a method similar to that above on stock prices The dispersion of M(r),ov(n since M(r) Is a single random walk, can also be computed theoretically but must be expermentally obtained from samples of nonoverlapping Intervals of duration T With the above model, It is not difficult to see that o m()or and both o m( and or() Increase with the square root of h, or the sq