STATISTICAL METHODS IN PROTOZOOLOGY 453 



Whenever we measure a group of individuals and compute 

 averages from such measurements, we are interested not in the 

 particular sample under consideration but in the general universe 

 of like organisms from which this sample was drawn, and we 

 hope to use this sample as an indication of what will be found in 

 the universe. If, however, as we have seen to be the case, the 

 individuals of the universe at hand are themselves variable as to 

 thus raise the following question: If many samples, each sample 

 drawn will not necessarily be equal to the mean value of the 

 universe, but will vary from the mean value of the universe by 

 some amount that is a function of the individual variability. We 

 thus raise the following question — If many samples, each sample 

 of size A^, were drawn from a given universe and for each of 

 these samples a mean value was computed, how variable w^ould 

 these means be? In an attempt to answer this question, the first 

 important result brought out by the statistical method is that 

 means of samples of the type described above will tend to dis- 

 tribute themselves in a normal form regardless of the form of dis- 

 tribution of the individuals, and the standard deviation of the 

 normal curve under which these means tend to fall will constitute 

 a natural measure of their variability. This standard deviation of 

 the means {o,n), is, as would be expected, a function of two factors, 

 namely the variability of the individuals, 0, and the size of the 

 sample, A^' the formula expressing the standard deviation of the 

 mean is 



^"'" VAT 

 As in the case of the individuals, the probable error of the mean 

 can be found by multiplying the standard deviation of the mean 

 by .67449, so that the formula for probable error of the mean 

 will read 



P. E.^ = .67449 ^-^ 



The value obtained by using this formula will set up limits such 

 that it is an even chance that the mean value of the universe 

 from which this sample is taken lies within the interval laid ofif 

 by adding this probable error to and subtracting this probable 

 error from the mean value of the sample. The likelihood that the 

 mean value of the universe lies within limits set up by any multiple 

 of the probable error will be found by referring to Table 2. Thus, 



