128 
Prohahle Errors of Frequency Constants 
magnitude in order that we may appreciate whether a certain quantity is or is 
not really significant. Accordingly it had been the practice of the Biometric 
Laboratory, if a frequency was only moderately a-normal, to use the values of the 
higher /3's in determining probable errors, which would flow from the assumption 
of normality; if on the other hand the distribution had values of /3i and ySa — 3 
differing a good deal from zero, to assume that the higher moments might be 
obtained from a skew distribution of Type III. (i.e. a distribution for which 
2/3.2 — 3/3i — 6 = 0). The justification of such hypotheses lies in the fact that if our 
data are to be of much value, the probable errors must themselves be small, hence 
in calculating these errors it is legitimate to insert into the formulae for them 
values for the /3's that only differ from their true values by small quantities. 
Such insertion can only introduce second order, and therefore for our purposes 
usually unimportant, changes in the probable error. This point is emphasised when 
we remember the large percentage errors of the high ^'s. For example, it is usual 
to take the probable error of a standard deviation cr = '67449 cr/V'2n, but its true 
value = •67449^^(1 + 2'^)'; the former really results from assuming the kurtosis, 
w2n 
»7 = yS.,— 3, to have the value zero of the normal curve. There are very few argu- 
ments made from probable error which would be seriously affected if the probable 
error were altered by 25 per cent, of its value, or if tj took values from - "875 to 
1"125, i.e. we might give /So any value between 2"125 and 4"12.5 to get in practice a 
sufficiently close result. 
Now the object of the present paper is to extend this idea by applying a method 
of determining /Sa, /3.i, ySg and still more exact than the methods indicated above. 
It is well-known that the frequency curves in common use are deduced from the 
integral of 
1 dy _ ax + b 
y dx Co -I- CiX + CiX^ ' 
and that this really assumes the condition that the coefficients of higher terms in 
the denominator on the right, e.g. Cj, c^, etc., are all zero. These conditions involve 
a finite difference relation between the successive moments, first published in 
1903*, and enable us to determine any higher moment from the first three fx,.2, fi^ 
and yu,4. Such a finite difference momenta! equation actually exists for all probability 
frequency distributions of the hypergeometrical series type, which cover so wide 
a range of chance problems. 
It is practically impossible to determine in a large percentage of cases whether 
the higher moments do or do not within their probable errors obey this finite 
difference relation, for the reason above stated, i.e. the high values of their 
probable errors. The present tables assume that they do ; in other words /Sj, fi^. 
* Biometrika, Vol. ii., p. 281. 
