328 
ME. W. CROOKES ON THE ATOMIC WEIGHT OF THALLIUM. 
The sum of the squares of the ten differences between a and each individual observa- 
tion is : — 
e. 
e 2 . 
+ •024 
•000576 
+ •008 
•000064 
+ •007 
•000049 
+ •002 
•000004 
•000 
•000000 
-•003 
•000009 
-•004 
•000016 
-•006 
•000036 
-•010 
•000100 
-•014 
•000196 
Sc 2 =*001050 
Hence the “ weight” of a is 
(number of observations ) 2 
w = 22 ? 
100 
— -0021 
= 47619. 
The largeness of this figure indicates the high degree of probability that a is very near 
to the true value sought. But the question, What is the true value ? does not admit 
of absolute answer; for one of the observations was as low as 203 '62 8, and one as high 
as 203-666. The number of possible values between 203-628 and 203-666 is infinite, 
the arithmetic mean a being but one of these values; and although more likely 
than any other that could be named , it is not more likely than one or other of all the 
possible values. The odds are many to one that a is not the truth ; but they are also 
many to one that a is very near the truth. The question “How near'?” cannot be 
answered ; alter the question to “ What is the probability that the truth is comprised 
within the limits a±4?” and the answer may be easily given, however small k may be. 
Thus if 7r=- 01, — in other words, if the question be “ What is the degree of likelihood 
that the truth lies between 203-632 and 203-652 ? ’ — the answer is given by the formula? 
x=H, | 
t —k w j 
having recourse to tables calculated from the celebrated Definite Integral, 
H l =4=’ (e-‘.dx. 
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