|STUDIES IN STATISTICAL REPRESENTATION. 83 



or the ending of any month be shifted towards higher aver- 

 age values the average for the month in question will be 

 increased: vice versa if the position be changed in the 

 direction of a lower average value. Again if M is the 

 number of days in the average month the relative weight 

 of the correction is only t/M, where t is the number of 

 days of the shift. Thus a new value of the average is 

 given, viz : — 

 (7) v =v + d.j- = v + c 



It may be noted that for all months except January and 

 December the correction d(t/M) must be applied twice, 

 once each for the beginning and end of the month, and, as 

 before, the signs of the different values of d will require 

 careful attention. 



By way of example we may note that the totals of lines 

 (1), (2), and (4) in Section 2, are respectively 15,788; 15,789; 

 and 15*781 : hence these divided by 12 give for a the follow- 

 ing values :— (1) 1*3157; (2)1*3157; (3)1*3151; only the 

 last of which is satisfactory; that is, the proper mean is 

 given only when the values for the equalised months are 

 used. 



4. Corrections for unequal lengths of quarters and half 

 years.— The corrections for the intervals between the end 

 of the uniform quarters and the ends of March, June, 

 September, and December, can be found by the application 

 of the same principles. They are given by the following 

 equations, viz : — 



(8) c = -y- 1 for the 1st quarter. 



(8a) c = DlTl ± D2Tg for the 2nd quarter. 



(8b) c ^Ml+M for the 3rd quarter. 



Q 



(8c) c = 5^ for the 4th quarter. 



