248 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



If, then, we always had a series of samples for each place 

 and time of sampling, we might be fairly certain whether or not 

 the population had changed in its character. But we have, 

 as a rule, only one sample for each place and time of sampling, 

 and that is generally all that is practicable in routine work. 



Thus, considering only the frequency of occurrence of 

 character D for the range of values 51-25 to 51-50, we find that 

 it is 24, or 7-3 % of the whole, for June, and 12, or 6-3 % of the 

 whole, for July. The problem is this : does the difference 

 between 7-3 % and 6-3 % represent a " real " difference in the 

 populations of June and July, or is it only a difference due to 

 random sampling ? No answer, admitting of no doubt 

 whatever, can be given to this question, but we can say yes or 

 no, and assign a rough degree of probability to the answer. 



Fig. c represents the curve that might be calculated 

 giving a large number of samples taken at the same time in June 

 and at the same place. The central ordinate gives the observed 

 frequency of our single sample, that is 7-3 %, and we assume 

 that this is the mean of the theoretical series that would be 

 given by our hypothetical series of samples. Ordinates drawn 

 on either side of this central one give the divergences of the 

 frequencies from the mean one that are due to errors of random 

 sampling. We now take the observed frequency for the 

 July sample : it is 6-3 %, or 1 % less than the mean of June, 

 and there will be another ordinate 8-3 %, which is 1 % greater 

 than the mean and which we must regard as equally probable of 

 occurrence. We draw these two ordinates in the curve, one on 

 either side of the mean. There is a block of frequencies greater 

 than 8-3, and another block less than 6-3, and we calculate their 

 combined area. Then we divide this by the whole area of the 

 curve and the quotient represents the probability that diver- 

 gences from the mean greater and less than 1 % on either side of 

 the mean will occur as the result of the error of random 

 sampling. 



