X52 Pearl and Surface. 



What would be the mean aud standard deviation of such a distribution 

 of means? It is clear that such a distribution is exactly analogous to 

 the distribution of mean quintile position given in table 16. 



If we have a series of n frequency distributions with means 

 ai, &i, Us an the mean, in, of the sum of these distributions is 



_ ai-|-a8 + a3-|- — a^ 



~ n 



In the present instance ai ^ a2 = as = a„ = 3'0 



3n 



in = = 3-0. 



n 



This it will be seen corresponds with the value of the means 

 given in table 16. 



Now the standard deviation, ös, of the sum of a series of n fre- 

 quency distributions whose standard deviations 01, 02, 03 on are 



measured from the same mean is 



' n 



But Ö1 = Ö2 = Ö3 = on = V 2. 



VI • V^ v^ 



Hence ös = = v;=r. 



n Vn 



For series A and B in table 16, n = 54 and for series C, 

 n = .55. 



Substituting these values 



ÖS = 0-1925 and 0-1907 

 respectively. 



These values are comparable with the standard deviations given 

 in the next to the last column of table 16. It is evident that they 

 are of a very different order of magnitude. Roughly these theoretical 

 standard deviations are only about one-eighth as large as the observed 

 values. Thus in the above theoretical distributions the means of the 

 separate distributions do not deviate very far from the mean of the 

 sum of the distributions. In other words the distribution of the theo- 

 retical means gives a very peaked curve while the observed means give 

 a very flat curve. 



In order to make the differences between these two curves clear 

 Fig. 16 has been prepared. This figure shows the graphs of two nor- 

 mal curves each having the same area and mean but different stan- 

 dard deviations. The flat topped curved has a standard deviation of 



