ALFRED COWLES BRD 311 average performance, the record of each service was reduced to an ef. fective annual rate which was then weighted in accordance with the length of the period represented. The average annual effective rate of all the services, thus arrived at, is.43 per cent TABLE I SULTS OF COMMITMENTB IN STOCKS RECOMMENDED BY 16 FINANCIAL SERVICES(RELATED TO MARKET AVERAGES) Service Weel Per cent 234 123456 234 17.2 34 234 234 462 860 104 0.5 234 52 o123456 020 2284 04 33.0 PROBABILITY TESTS In an attempt to determine whether the service having the best rec- ord achieved its result through skill or chance, we resorted to the the- ories of compound and inverse probability. Our conclusion is thus rendered consistent by obtaining approximately the same answer in two different ways. With the aid of various checks, involving 1250 computations of the action of individual stocks selected at random we derived a formula A, D.(0)=5.42+1.5t(A D,=average deviation, t, in units of 4 weeks 21), representing the deviation, for all periods from one month up to one year, of the a verage individual stock from the average of all stocks ce Number 1, for the 9 six months' periods from January 1 July 1, 1932, was successful 7 times and unsuccessful 2 times with the aid of the table referred to, the averages of "chances in 1000 to do worse''for the 7 periods in which it was successful and the 2 pe- riods in which it was unsuccessful were found to be 842 and 66 re spectively. By the theory of direct probabilities, the probability of a single service being right at least 7 times in 9 is equal to the sum of the first3 terms of the binomial(}+号)° p=1/2+9/29+36/20=46/512=090 The probability that a single service could in 9 predictions be 7 times