Notes on some Bushman Crania and Bones, 237 



in each group multiplied by the deviation from the centre are then 



calculated and added, being in this case 86 negative and 111 



positive, giving a total of + 25. This number, divided by the total 



number of skulls in the group, 79, yields a quotient + "32 which 



has to be added to the centre 75 to give the average 75'32. In 



any such calculation the sum of the first column ZX divided by 



the total number of observations made, the sum of column Z, gives 



the amount to be added ( + ) or subtracted ( — ) from the centre 



selected by inspection to yield the average. The product ZX is 



then once more multiplied by X, the amount of deviation from 



the mean, all the products being now positive. 



The amounts in the column ZX^ are now added up and the sum 



divided by the total number of observations Z, in this case 79, so 



ZX^ 879 

 that we have -w = „g - ll'lo. 



This gives thus the sum of the squares of the deviations from the 

 arbitrarily selected centre to correct this to the deviations from the 

 true centre or average ; for the whole group the square of the 

 deviation of the arbitrary centre from the average must be sub- 



—fy- ) which in this case 



is (-32)2 or -1024. 



ZX^ /ZX^ 



^ - { -tt) = 11-13 - -1024 = 11-0276. 



- m - 



Therefore (t = 3-3211. 



From the value of o- the standard deviation, the range within 

 which the averages of other observations of the same group would 

 fall can be calculated. Thus the chances that they would lie 



between the average M and + —ttj, the standard deviation divided 

 by the number of observations, are even. Between M + 2 — ^ are 



49 to 1, M + 3 -^ are 1,000 to 1, and M + 4 -^ are 33,300 to 1. A 



knowledge of the standard deviations of two groups enables these to 

 be compared with a knowledge of the chances against any difference 

 being due to random sampling. 

 The formula to apply is — 



where Mi and M2 are the averages, Ni and N2 the number of 



