APPENDIX 709 



For instance, he asks, Of how many ears of corn taken at random must 

 I measure the length in order to obtain, to a certain desired degree of 

 accuracy, the variability of the corn from which selection is made ? Must 

 I measure fifty, a hundred, or a thousand ears ? Again, how many variates 

 must I take to give a reliable determination of the mean ? 



Similarly, in any correlation study he will be concerned with the num- 

 ber of variates he must take in order to present a trustworthy determina- 

 tion of the correlation coefficients. 



While these questions cannot be answered in advance for all kinds of 

 populations, it is the object of this section to give some assistance to the 

 inquiring investigator in forming a judgment in this matter. The best 

 measure thus far devised upon which to base a judgment is the so-called 

 " probable error." 



So far as the mean is concerned, it has been seen that the probable 

 error of a single variate may be obtained approximately by counting, and 

 that the probable error in the mean is obtained from that of a single vari- 

 ate by dividing by the square root of the number of variates. This process 

 can often be applied in a rough way before much labor has been put on a 

 problem, and it becomes a useful guide where the mean alone is in question. 

 It should be remembered that the probable error in any result is, in gen- 

 eral, inversely proportional to the number of observations. 



A method similar to that just explained for the mean can be used to 

 find the approximate value of the probable error of the standard deviation, 

 since the probable error of _the standard deviation is obtained from that of 

 the mean by dividing by VJ. 



As for the coefficients of variability and correlation, the following tables 

 show the probable errors corresponding to values of the coefficient of varia- 

 bility from i per cent to 25 per cent, with numbers of variates from 25 to 

 1000, and the probable errors of the correlation coefficient for values from 

 o to i, with numbers of variates from 25 to 1000. 



If, then, we have an approximate notion as to the value of one of these 

 coefficients, we. can find from the table the probable error corresponding 

 to a certain number of variates. 



To illustrate the use of these tables, suppose that we know in advance 

 that the coefficient of variability is in the neighborhood of 20 per cent ; 

 then with a hundred variates we see from the tables that the probable error 

 would be approximately i per cent, while with five hundred variates it would 

 be only 0.44 per cent. We thus decide upon the number of variates by the 

 magnitude of the probable error and the degree of accuracy desired in our 

 results. 



Probable error in estimate of probability from a limited number of obser- 

 vations. While it has been said that, in a general way, the accuracy of a 

 statistical result increases as the square root of the number of observations, 

 this rule is often difficult to apply, and is an inadequate test in many 

 important cases. 



