Editorial 
523 
with mean x-=m, the mean of the population, and with standard deviation 
= o-jhjn, a well-known result. 
On the other hand the distribution of samples of standard deviations is 
-i — 
2/ = 2/„S"-^e (v). 
This curve was first reached by " Student " as a highly probable result 
following from the relations he had obtained from the moments of S^*. 
Mr Fisher's work thus enables us to justify " Student's " assumption. 
" Student " has discussed at some length the distribution curve for He 
has obtained the values of the moment coefficients fi.^, /xj and /X4 and the 
general expressions for the means when n is even and odd. The whole problem 
is of such importance that it seems worth reconsidering, and providing tables 
showing the approach of the distribution curve to normality as n rises from 
4 to 100. 
The following investigation largely repeats work given by " Student," but it 
expresses the values for /Xj, /A4, and /3i and /S2 in a different formf. We shall not 
use approximate expressions for the constants, for the order of terms in \jn 
depends so largely on the relative magnitude of their coefficients, that such 
expressions become unreliable for values of n under 100. 
Clearly (v) is a skew curve with range limited at one end, ^ = 0, and not at 
the other, S = 00 . See Figure p. 524. 
We shall write the standard deviation of o-j, and the moments of the 
frequency about the end of the range 0 as if/, M^, etc., while the moment- 
coefficients about Q will be as usual /ctj (= 0), /u.2, etc. Obviously ix., = a-^. It is 
desirable to ascertain 2, %, 0-3 and the skewness as well as /3i and /Sg for the 
distribution. We do this to show the rapidity of change to a normal distribution. 
It is well, however, to notice a priori that for n large the distribution does become 
normal. 
* "Student's" approximate values for jSi and ^2 (^oc. cit. p. 10) are, we fear, erroneous. He gives 
3 1 1 
D2= ?i - - + , but it is needful to have a further term in — , in order to obtain Bi and 82 correctly to 
i on 
the second approximation in ^. If this further term hepjri^, then: 
''■ = ffi('+ -fr^)' ""S"""' "Sludenfs" 5(1-5), 
ft ■ K'-i)- 
An examination of our table (p. 529) shows that "Student's" corrections are not of the right sign to 
agree with the facts, and that further no constant value of p would give good results even for fairly 
high values of n, i.e. it is probable that the term in i in is of equal importance with that in — 
t "The Probable Error of a Mean," Biometrika, Vol. vi. pp. 1 — 25, more especially pp. 4, 6, and 
8 to 10. 
Biometrika x 67 
