Karl Pearson 
257 
the standard deviation (mean square error) of an indefinitely great number of 
measurements which he might make of A for the given value of B. He attributes 
these variations in A to "errors of observation," and he usually supposes such 
errors to obey the Gaussian law. This attribution is somewhat dogmatic. There 
is very little definite proof that errors of observation actually do obey the Gaussian 
law and secondly his " errors " are not in all probability solely due to observation. 
It is impossible to repeat each experimental measurement under precisely the same 
physical conditions for other variates C, D, E, ... and changes in these variates may 
be as influential as personal errors of observation. Further it is far from certain that 
the value of B has remained without some variation and this alone would tend to 
cause some variation in A. The physicist aims at a constant value of B — it is by 
no means certain that he always reaches it. Without much more investigation 
than is easy, or is at all likely to be made, we probably can at present do no better 
than assume the distribution of ^'s for constant B to be Gaussian. Thus the distri- 
bution of means determined by the physicist will have for its distribution surface 
where = S 
z = Z(,e 
t2- 
and the corresponding value of the probability to be taken from the " goodness 
of fit " tables will be found by taking out P for the given value of under the argu- 
ment n' + 1, where n' is the number of arrays, for there are now n' independent 
variables, i.e. mj,'s. 
In the above value for x^, m,^, is the theoretical value of A corresponding to 
the given B, nij, is the observed value and n^, is the number of observations on which 
it depends. The real difficulty arises in determining a^^^ which is the standard 
deviation of the array for an indefinitely large number of observations and cannot 
be determined properly from the few observations made to determine the A 
corresponding to a given B. 
If there are a considerable number of observations in an arrav and cr„ be the 
standard deviation of the array found from the observations themselves, then 
it is well known that the "best" or most probable value of o^, is given by 
a2 
In this case = S - 1) i^h - rn, 
a form which shows us that if we have arrays with only a. single individual and 
use the observed a^^'s, will be indeterminate, for -1 = 0 and cr\^ = 0. But 
the observed a^^^ would be very risky for any system of small arrays, and this 
method of approaching the difficulty must I think be dropped unless the physicist 
be incUned to increase very much the number of measurements he makes of A for 
a given value of B. But the above method of approaching the subject indicates 
Biometrika xi I7 
