Miscellanea 
183 
where w is a small number. But n being small the first and last terms give : 
22 
nV n)-nY sJn^'jn^' 
u being a small number. But uj-J N will then be very small. Accordingly if « be small, the 
last term in the radical is sensibly smaller than the probable error of the first and wc may read 
for the probable error oi in — M the expression : 
2o-2 
Further the probable error of the difference or sum of 2^ and a'' is of the order of and 
v 2n 
thus to a first approximation we might put in the smaller term or first term a-'^ = 'S,'^. There 
results : 
In other words, when the number of a sub-sample is very small, the probable error of m-M 
2^ /(i^ 2^ 
approaches -67449 \J ~- ~^ ^"'^^ "'^^ '67449 \/ ^ + 0*^'' '^"^^ excuse for using the latter 
form would be the negligibility altogether of the term iP'lN. In which case it would be V>etter 
a priori to adopt the value '67449 v^o-^/^i. It will be clear therefore that the value frequently 
adopted is not justified when a sub-sample is tested against a general sample. The proper 
method seems to be to compare : m-M with •& 1 449 a, / i? — ■ - ,t-tt7 r • 
^ V w N N{N-n) 
Now let it be reasonable to suppose a quantity significant when it is /3 times its standard 
deviation, or /3/'67449 times its probable error, then we have for significance test : 
,^ ^ /iT^^ 20-2-2'' n(M- 
N N{N-n)' 
Or: m-M>^^- ^/Vl+TO 
n 
and this is true whatever be the magnitudes of N and n. If it be said that the right-hand side is 
yo^ 2^ 
■ — ^" ' that accordingly significance cannot have been asserted to 
exist, where it is not existent, this is perfectly true. But there is another side to this fact, too 
often forgotten. No samples suf3fice to demonstrate the absolute absence of differentiation ; the 
statistician can only say : Relatively to the size of my samples, I find no significant differentia- 
tion. It may after all be there and would be demonstrated if the samples were tenfold as large. 
The absence of significance relatively to the size of the samples is too often interpreted by the 
casual reader as a denial of all differentiation, and this may be disastrous. Hence if the 
statistician using too large a value of the probable error errs on the side of safety, when he 
asserts significant differentiation for certain cases J, C, but that he has not found it for 
E, F, G, this may strengthen his demonstration in the first cases, but it weakens any 
influence as to non-significance in the latter cases. 
Using the above formula it may be that a considerable number of cases, for which no proof 
of significant differentiation has been given, — and which have been taken accordingly as having 
no differentiation, — can now be demonstrated to have significant differentiation. And this 
appears of some importance. 
Several other cases of probable error tests of significance deserve reconsideration, and I hope 
to find time to publish my notes on them shortly. 
