ANTHROPOLOGY OF EASTERN EUROPEAN JEWS 173 



vals of one centimeter. Thus twelve persons were observed in 

 the class between 152 and 152.9 cm. in height ; 62 in the class 

 between 169 and 169.9 ^^-^ ^^^- The average stature being 

 164.5 cm., it is seen that in the first case " ,r " = — 12.5 cm, ; 

 ^',r-" = I 56.25 and ",i--/"= 1875. In the second class '' x" 

 = + 4.5, '' x^" = 20.25, and '' x^-f" = 1,255.5. This was done 

 for each deviation (grouped by centimeters), beginning with the 

 smallest and ending with the largest value. The products of 

 " x'^ ■/" were then added together and the sum of these divided 

 by " ;/ " = the number of individuals measured = 1,528, and the 

 square root extracted from the quotient. By this process it was 

 found that the standard deviation of 1,528 Jews was for their 

 stature d= 6.58. Within the limits of =b 6.58 it is theoretically 

 expected that about 6S percent of the number of variates should 

 lie. Empirically this was confirmed. We found that within 

 the limits of the standard deviation, i. e., 164.5 + 6- 5^ 

 (=171.08), and 164.5 —6.58 (= 157.92) were 1027 individuals 

 = 67.21 percent. 



As is well known, the determination of the mean or average 

 is never perfect, it is always only an approximation to the true 

 average. This is due to inevitable errors of observation and 

 calculation. These errors may be diminished by careful atten- 

 tion to details w4iile taking measurements, and calculating the 

 results, or by taking measurements on a very large number of 

 people, but they can never be entirely eliminated. The finding 

 of the ''probable error'' is a good method of determination of 

 the accuracy of the average value. This is determined by 

 multiplying the standard deviation by the constant 0.6745 and 

 dividing the product by the square root of the number of in- 

 dividuals measured thus : 



Standard Deviation a 



± 0.6745 X —-= ^ , — = ± 0.6745 "^, 



V number of observations v ;/ 



The probable error gives the closeness of the approximation 

 to truth. In the case of the Jews under consideration we have 

 calculated that the probable error is o. 1 133 cm. With the aid 

 of this figure we can say that there is an even chance that the 



