Miscellanea 
181 
rjn to {r+\)jn is then the number of representative points confined between the lines rx=ni/, 
{r+\)x = ni/, le. i (>- + 2) - ^ (;• + 1) (w + 1) =i (w+ 1), counting 0/0 as i This fraction, 
which occurs in each class, disappears when we make the subtraction, and we have the result 
stated above. 
V. Note on the Significant or Non-significant character of a 
Sub-sample drawn from a Sample. 
By KARL PEARSON, F.R.S. 
If two independent samples be drawn from an indefinitely large group or pojuilation, and 
their means be m and M' and their sizes n and N', and their standard deviations cr and 2', then 
the usual test of significant and non-significant diflPerence in type is made by comparing the 
difference of mean m— 31' with the probable error of this difference -67449 ■J<T^l7i + 2"^lN'. This 
process may be considered as legitimate, if the samples are absolutely independent and drawn 
from an indefinitely large population. 
It has become not unusual to apply this test to cases of the following kind, where its 
application has yet to be justified : a population is described by a sample, say JV in size, M in 
type and 2 in variability. This sample is obtanied from p localities, or if in one locality by 
p methods or instances, or generally there is a p-fo\d heterogeneity in its collection. One of the 
p sub-groups of the sample is defined l)y n, m and o-. It is frequently assumed that the proper 
test for .significant or non-significant difference between the sub-sample and the general sample 
is the relative magnitude of m- M and •67449 ^ 'n^ 7f' '^'^'^ treatment is, I think, erroneous. 
To begin with it must be observed that as the sub-sample is made larger and larger, the value of 
its mean must approach closer and closer to that of the general sample, and thus the probable 
error of the difference ought to be less and less and ultimately vanish. Instead of this it 
approaches the finite value "67449 ^22^/^7. Clearly the above expression for the probable error 
of the difference of types in sub-sample and sample is not correct. We have yet to ascertain 
how far it is approximate, when N is large as compared with n. 
The sort of problem to which the above doubtful process is applied is of the following kind, 
for example : a general sample of the population is found to have q per cent, of its members 
