Editorial 119 
and deduce, since a:, = — .r^, 
'-■''vsHy-Ni'-Nj 
TABLE II. Accuracij of Deteriniiiatiun of a fruiii Ranking symmetricully. 
nJN 
Kange '2jj 
(To- from (r.jj.j 
Values from Mean and S.D. 
4 0 
1 
TSff 
3-92007 <T 
3-66787 (T 
5-2737 o-/v^2^ 
4-7514 ff/v/2i^ 
1-3453 rr/V -iiV 
1-2954 o-/v'2iV 
The 8.1). of the ranges are 
the coettieieiit.s of 2nd 
(T 
1 
120 
3-28975 o- 
4-1137 o-/V2A^ 
1-2505 o-/V2iV 
cohunn x / — z-. . The 
v2iV 
1 
1 5 
1 
14 
3-00219 a- 
2-93050 <T 
3-7178 0-/V27V 
3-6290 o-/v'2# 
1-2384 a/J^jV 
1-2383 <7/V2iV 
ratios are therefore the 
coefficients of the 4th 
column. The S.D. of o- 
1 
ly 
2-85219 0- 
3-5358 a/JiW 
1-2397 o-y2iV' 
is of course crj\^^2jy, and 
1 
] 2 
1 
10 
2-76602 0- 
2-56310 0- 
3-4378 o-/v/2A"^ 
3-2233 y/v'2^ 
1-2429 o-/V2i? 
1-2576 o-/V2J^ 
the ratios are again co- 
efficients of the 4th 
column 
2 
TC7 
1-68324 0- 
2-4747 (r/V'SiV 
1-4702 o-/n''2A' 
1 
1 
1-34898(7 
2-2252 o-/n^2A^ 
1-6495 al^M 
ih 
1-04880 0- 
1-9926 o-/v/27\^ 
1-8999 a/s/2T^ 
4 
10 
-50669 0- 
1-4642 a/-J2N 
2-8897 (T/v^2iV 
It will be clear from this table that the quartiles are not the best ranks from 
which to find the standard deviation with least error. The range should be that 
corresponding to yij to from each end of the series in the ranking, say jL. This 
quatuordecimal range will provide cr with about 24 greater probable error than 
the moment method. The quartiles give it with 65 greater inaccuracy. The first 
and last deciles as a convenient divisor give only 26 less accuracy. The use of 
the quartiles was undoubtedly adopted because of their relation in the normal 
curve to the probable error. But theoretically they are very inferior to the 
quatuordecimals. 
(6) The table on the following page provides the data for the probable errors 
of a when found from ranges whose terminals are not symmetrical. 
It will be clear that the inaccuracy of the determination of cr fn^m ranking 
increases with great rapidity as we cause our asymmetrical ranks to approach each 
other. Further the range taken on one side of the median gives a worse result 
than the same range placed symmetrically about the median. Or again, looked at 
in another way, the range corresponding to of the frequency on one side the 
median gives a„= l-9Sa/'^2N, but the same frequency symmetrically placed about 
the median gives a„ = l-47cr/v'2iV". It might be supposed at first that our Table II 
could be derived from Table III by simply supposing that in doubling the 
frequency from which our determinations are made we have increased the accuracy 
