Tabulation of certain Frequency-Distributions. 397 



(ii.) When, on the other hand, the values of X are only 

 taken originally by very large intervals, there may be a good 

 deal of difficulty in determining the centile values towards 

 the extremities of the range. Thus, in the example given in 

 § 1, the last class (cloudiness = 10) comprises more than halt 

 the total number of observations ; and there are no direct 

 data for the subdivision of this class. All we can do in these 

 cases is to apply a suitable method of interpolation. Thus, 

 in the above example, if we assume that the eleven " degrees " 

 correspond to equal divisions of a scale, the range will be 

 from — \ to +IO5. Altering this, for convenience, to a 

 range to 11, and smoothing the observations (which are 

 obviously incorrect at the upper end of the scale), I obtain * 

 the following values of X : — 



orations. } ° 10 20 30 40 50 60 70 8,J 90 1Q 



doziness } -000 -112 -928 3-826 9-551 10-647 10-939 10-994 11000 11-000 11-000 



From these data the u frequency-constants " are easily 

 calculated by means of (6) or (8). 



(iii.). A further difficulty is often caused by ignorance of 

 the exact bounding value or values of X. Prof. Pearson has 

 already called attention f to this difficulty, especially in the 

 case of economic statistics : and he points out to me that in 

 the above example the first class includes all cases where 

 there is no cloudiness at all, as well as those in which there 

 is a cloudiness of less than ^jy, while the last class includes all 

 cases of total cloudiness, which cannot be graduated. Making- 

 no assumption as to the values of X and X p , I get the following- 

 arrangement : — 



Per cent, of j Q 1Q 2Q go 4Q 5Q m 7Q gf) gQ 



observations. J 

 Degree of 1 . g75 . g94 . 926 3 . g24 9 . 5?6 10 .g 23 10 .g 19 m . g76 1{) . 982 jq.^., 10 . 9 g o 

 cloudiness. J •" 



It will be seen that this gives a decidedly smaller range 

 of values of X. The discrepancy between the two results 

 illustrates the importance of making exact observations at the 

 ends of the scale. 



6. Finally, it may be observed that the adoption of this 

 method of calculating the mean square, mean cube, .... does 

 not affect the formulae for the probable errors in the frequencv- 



* The values of X are given to three decimal places, though the data 

 are not really sufficient to give them to more than two places. The 

 method of smoothing and interpolation is too complicated to be explained 

 here. 



t Op. cit. pp. 397, 398. 



