388 
Prohcible Errors of Frequency Types 
Then when q = 10, = 6"3006 and = 23 approximately, 
„ ? = 12, = 7-5609 „ p = ^ 
Hence to find the seniiaxes of ellipses containing 22/23 or 43/44 of the whole 
frequency; take the numbers from Tables VII. and VIII. and multiply by the 
factor from Gibson's Tables {Biometrika, Vol. iv. p. 385) corresponding to popula- 
tions of iV/10 and iV/12 respectively. 
To facilitate the construction of the ellipse, a series of ellipses has been drawn 
on Diagram C of different dimensions and varying in ratio of Si/Sa from "1 to "3. 
In practice the values of the axes should be taken from Tables VII. and VIII. 
and multiplied by the appropriate factor. Then their ratio gives Si/Sg. By means 
of a piece of tracing paper applied to the corresponding series of ellipses, the 
required curve can be approximately drawn and transferred to the diagram. It 
will then be possible to determine whether the frequency really lies with a definite 
probability within the chosen type, as indicated by the /3i, 
The series of tables has been extended by the inclusion of Table X. which 
gives the probable error of 
The s.D. was determined thus : 
/c/'''^-^, + 3''^= + /3/^>- 4/3. -3A ~(2^3A-6) 
_r 2 4 2 1 
L(/3, + 3) (4/3, -3A) (2^.-3A-6)J^^^ 
3 3 
whence 
where 
+ 7r + 
(4^,-3/30 (2/3,-3^,-1 
= c- {A^(T% + 2ABRp^p^ ap^ap^ + B-'a-p^]^, 
2(4^,-3/30(2^-3/3,-6)' 
^ =8/3,(/3,-3)-9A^ 
R = 1 S/3 t±Ml^±3- + 4) 
(/3. + 3) 
Erratum. In the printing of my former paper a slip of sign escaped my 
notice. On p. 129, Biometrika, Vol. vil., 1. 7, for a = (2;So - 3/3i - 6)/(/32 - 3) read 
a = (2^,-3y8i-6)/(/32+3), which was, of course, the value used in the calculation 
of the tables and diagrams. 
