Law of Error and Correlated Averages. 433 



that the new octiles, deciles, &c. are approximately 



(m 2 m* \ , (m 2 , rri- \ . _ 

 T±vT>' (T±7r) ; ' &c - 



(r, s, &c. having the meaning assigned in the last paragraph). 

 Thus the translated curve of distribution is such that its quar- 



tiles are approximately ( — + q —j= J i 2 , its octiles ( — + r— i= \i 2 , 



-r- +s—y=\i 2 ; that is, the new distribution is 



approximately a probability-curve whose centre is -j- i 2 , and 



modulus —7= i 2 . 



In the preceding example the conformity of H to a proba- 

 bility-curve is known a priori by the usual theory, referred 

 to in our second section*. Let us take another example in 

 which this conformity is known a posteriori by actually 

 observing the measurements of a group : for instance, heights 

 of men. The annexed numerals give the number of men per 

 thousand of each particular height in inches ; as ascertained 

 by Mr. Elliott from the measurement of some 25,000 Ame- 

 rican recruits (International Statistical Congress, 1863). Thus 

 there are 121 men (per thousand) of the height 70 inches, that 

 is, as I understand, between the heights 69*5 and 70'5 inches. 



1 1 2 20 48 75 117 134 



57 58 59 60 61 62 63 64 65 66 67 



157 140 121 80 57 26 13 5 2 1 



68 69 70 71 72 73 74 75 76 77 78 



Now let the curve or locus of distribution thus constituted be 

 translated by squaring each of the measurements ; while each 

 compartment thus dislocated carries with it, so to speak, the 

 men appertaining thereto. E. g. the 121 men who were 

 originally found between 69*5 and 70'5 are now distributed 

 between 69'5 2 and 70'5 2 . It will be found that the trans- 

 lated observations fulfil the law of error in the same sense as 

 the original ones ; that is, in the sense in which a planet is 

 proved a posteriori to move in an ellipse. The parameters 

 being calculated from some of the observations, other obser- 

 vations are found to tally with the curve thus determined. 

 Let us adopt a uniform method of calculating the para- 



* The case might be brought under our first section by a mere change 

 of unit. 



