ON THE PROBABLE EBRORS OF FREQUENCY CONSTANTS. 
PART III. 
EDITORIAL. 
(1) The two previous parts* of this resume of the theory of probable errors 
dealt with the probable errors of moments, and deduced the probable error of any 
frequency constant by considering that constant as a function of the moments, the 
probable errors of which had earlier been investigated. This third section proceeds 
from a different standpoint; it treats of the probable errors of constants supposed to 
be determined by a knowledge of the ranges in which certain proportions of the 
frequency lie. The fundamental papers on the subject are EdgeVorth's " Exercises 
in the Calculation of Errors " {Phil. Mag. Vol. 36, pp. 98 et seq. 1893) and Sheppard's 
" Ori the Application of the Theory of Error to Cases of Normal Disti'ibution and 
Correlation" {Fhil. Trans. Vol. 192, pp. 130 et seq. 1898). We are unable to follow 
Edgeworth's reasoning, particularly that portion of it in the footnote on p. 99, where 
he states that the displacements of the two quartiles and the median are independent. 
They have, as we shall see, considerable correlation. With Dr Sheppard's numerical 
conclusions we are generally in agreement. 
(2) We start from a population distributed according to any law of frequency, 
the ordinate at being ?//, and iih giving the total frequency beyond on the right, 
say. The total population M is supposed very large, and a sample N is taken from 
it, small as compared to M, but such that 1/ViV is small as compared to unity. 
Let «i = «<i X iV/if be the mean quantity that would be found beyond y/ in many 
samples, then in any individual sample we shall not find Hj but Hi = «i + Bih beyond 
/ /■ 
// 
1 / 
0' 
or of the sampled population. Accordingly the ordinate which cuts off n^\N 
of the sample frequency will not be at the point x-^ but at some distance hx^ from it 
and the area of the dotted curve which represents the sample on 'hx^ will be Shj. 
Thus if y^ = iji Nj ^[ be the ordinate at x^ of the sample we shall have to 
* Bioinetrika, Vol. n. pp. 273 — 281 and Vol. ix. pp. 1- 
been reproduced from Lecture Notes. 
Biometrika xiii 
-10. As in the earlier papers, this paper has 
