394 Mr. W. F. Sheppard on the 



Degrees of Cloudiness at Breslau, INTO— 1885. 



Degree 012 3456789 10 



Frequency 751 179 107 69 46 9 21 71 194 117 2089 



In these cases the moments of the figure of frequency cannot 

 be conveniently expressed in terms of the areas bounded by 

 equidistant ordinate?, and some other method of tabulation 

 should therefore be adopted. 



2. The proper method, in all such cases, is that which is 

 known as the centile (or " percentile ") method. Instead of 

 dividing the range of values of X into equal portions, and 

 finding the corresponding areas of the figure of frequency, 

 the area of the figure of frequency is to be divided into equal 

 portions, and the corresponding values of X are to be found 

 and tabulated. A division into ten equal portions, giving the 

 "decile" values, will usually be sufficient ; but a division into 

 twenty or more is better. The values of X should be found, as 

 far as possible, by direct observation ; interpolation being 

 only resorted to when the original measurements were so 

 rough that several values of X fall into a class containing a 

 particular centile. 



When both the bounding ordinates are infinite, the centile 

 method of classification is all that is required. When only 

 one bounding ordinate is infinite, both methods must be 

 adopted ; the moment (of any order) of the figure of frequency 

 must be calculated in two portions, the centile method being 

 applied to the portion adjoining the infinite ordinate, and the 

 ordinary method to the remainder. 



3. The simplest case is that in which both bounding ordi- 

 nates are infinite. Let Z denote the ordinate of the figure of 

 frequency corresponding to abscissa X, the whole area of the 

 figure of frequency being unity. Then the ?nth moment is 



l 'ZWX (1) 



'X 



Now let A denote the area of the figure up to the ordinate Z. 

 Then we have 



Jx 



and therefore 

 Hence, by (1), 



A=f X ZrfX, (2) 



Jx 



Z = dA/dX. . . (3) 



M m = f* p X m .dA/dX.dX 

 Jx 



= f X m dA (4) 



_ 'n 



