I 
Miscellanea 179 
IV. On the Classification of Frequency-ratios f. 
By D. M. Y. SOMMERVILLE, D.Sc. 
In statistical work which deals with integral variates, the data frequently appear in the form 
of ratios, or inireduced proper fractions ; and to facilitate comparison these are arranged in 
classes according to magnitude, all the ratios falling within the same class being considered as 
equivalent. The problem then arises to find the best distribution of the fractions so that there 
may be approximately the same number in each class ; or, if the fractions with various 
denominators do not all occur with the same frequency so that it is necessary to assign to them 
certain weights, to find the distribution which will make the total weight of each class 
approximately the same. 
I. Let *lp denote any proper fraction with the denominator ju, and *l:p-n the assemblage of 
all the proper fractions whose denominators do not exceed n. The following theorem is then 
established : 
If the fractions *li>n are distributed into « classes, - to - to -, - — ^ to -, and 
^ n n 71 n 11 n 
any fraction which falls between two classes is counted ^ in each of these two classes, each of 
the others being counted 1 in the class in which it occurs, then in each class there will fall 
fractions, except in the extremes which contain n + ^. 
rif the fractions at the extremes, v, ^, - ; \, ^, are also counted ^ thei-e will 
I 2 n I '2 n 
be Kw + l) fi-actions in the extreme classes also.] 
This is the normal distribution (n.d.). 
There are three other " even " distributions : 
(1) w — 1 classes, + 2) in each, extremes ?i-hf. 
(2) 7i-\- \ classes, in each, extremes n. 
(3) ji + 2 classes, ^ (?!. — 1) in each, extremes n. 
These are obtained from the n.d. for n— 1, w + 2 respectively. 
Then by making pairs of classes coalesce, from the second onwards, we get the following 
evenest distributions : 
11 3 
(1) n even: + l classes, 0 to - to - , + l in each, extremes n + h. 
71 71 n 
11 3 
(2) ?i odd : i(?i + 3) clas.ses, 0 to -, to r, w in each. 
The N.D. can be easily written down. To find the classes in which */ p occurs, divide 
»i, 2m, 3«, ... by p ; let q^i <lii qz, ••• be the quotients and /u/^j/s, ... the remainders. Then if 
/a=^0, - lies iu the (og+nth class, but if L=0, f = 2f and - lies A in the o-.th and h in the 
P • p 71 p 3» _ 
(5's+l)th class. Each class must contain either *jp or ^jn-p, and if any class contains both. 
t Abstract by the Author of a paper "On the Distribution of the Proper Fractions," by 
D. M. Y. Sommerville, D.Sc, Proc. Roy. Soc. Edlii. Vol. XXXVI. (1906), pp. 116 — 129. 
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