SECT. 23.] Averages. 463 



The difficulty felt by most students is in applying the formula to actual 

 statistics, in other words in putting in the correct units. To take an actual 

 numerical example, suppose that 1460 men have been measured in regard to 

 their height &quot;true to the nearest inch,&quot; and let it be known that the 

 modulus here is 3 6 inches. Then dx = l (inch); h~ l = 3 Q inches. Now 



= l; that is, the sum of all the consecutive possible values 



is equal to unity. When therefore we want the sum, as here, to be 1460, we 



1460 -^ V&quot; -(-Y 



must express the formula thus ; y=-, e Vs-6/ , or?/ = 228&amp;lt;? V3-6/ . 



tjir x 3-6 



Here x stands for the number of inches measured from the central or mean 

 height, and y stands for the number of men referred to that height in our 

 statistical table. (The values of e~&amp;lt; 2 for successive values of t are given in 

 the handbooks.) 



For illustration I give the calculated numbers by this formula for values 

 of x from to 8 inches, with the actual numbers observed in the Cambridge 

 measurements recently set on foot by Mr Galton. 



inches calculated observed 



a; = ?/ = 228 -231 



x = l y = 212 =218 



x = 2 = 166 =170 



x = 3 y = lll = 110 



.r = 4 y= 82 = 66 



x = 5 y= 32 = 31 



.r = 6 y= 11 - 10 



x=7 y= 4 6 



*=8 y= 1 =3 



Here the average height was 69 inches : dx, as stated, = 1 inch. By 

 saying, put x = 0, we mean, calculate the number of men who are assigned 

 to 69 inches; i.e. who fall between 68 5 and 69 5. By saying, put.r = 4, 

 we mean, calculate the number who are assigned to 65 or to 73 ; i.e. who lie 

 between 64-5 and 65-5, or between 72-5 and 73-5. The observed results, it 

 will be seen, keep pretty close to the calculated: in the case of the former 

 the means of equal and opposite divergences from the mean have been taken, 

 the actual results not being always the same in opposite directions. 



(2) The other point concerns the interpretation of the familiar pro 



bability integral, ~|V*d*. Every one who has calculated the chance 



Jrjo 



of an event, by the help of the tables of this integral given in so many 

 handbooks, knows that if we assign any numerical value to , the 

 corresponding value of the above expression assigns the chance that an 



