240 
rROFESSOE K. PEARSON ON THE MATHEMATICAL THEORY 
(3.) Current Theorii of Errors of Ohservation. 
The assumption usually made is that the error of an observation is due to the 
result of the combined action of a great number of independent sources of error ; 
each source follows a j^ermanent law and attributes equal probability of occurrence to 
numerically equal errors. From this statement, or some modified form of it,* is 
deduced the well-known normal curve of error frequency :— 
y = .(v.). 
An Important point to be considered is, therefore, whether actual errors of 
observation in any case are such that they may be su23posed to be a random 
sampling of errors obeying this law. I have in a recent j^aj^erf obtained a criterion 
» 
for the probability of any* system being the result of a random sampling from a 
series following any law of frequency, and I have shown that it is most highly 
improbable that the series cited by Airy and Merriman as evidence of the 
suitability of the normal curve can really have been random samples from material 
actually obeying such a distribution. 
Assuming the applicability of the normal curve, or, indeed, the indej^endence of 
judgments of independent observers,^ we have at once 
3 2 1 S 
— ^U1 + ^02 ? 
Similarly :—• 
2 2 1 2 
— ^02 + f^()3 ^ 
and, 
2 ‘ 2 1 2 
<^13 — <^03 1 ^01 
Hence we deduce :— 
o , o o \ 
2 _ "I ^13 ' 
^01 — 
2 t 2 2 
3 _ ^32 ■>" _ ^13 
O'os — 
rr 3 — 
•^03 — 
fy 2 _ _ 2 
^y'32_ 
9 
(vi.) 
(Vii.). 
It was this simple result which led to the whole of the j^i’esent investigation. I 
liad not seen it noticed before, and it seemed of wide-reaching importance. I mean 
in the following manner ; The astronomer, and often the jjhysicist, can, as a rule, onH 
determine relative and not absolute judgments. He cannot deduce the absolute 
* These are really additional assumptions. See pp. 274-275 later, 
t ‘ Phil. Mag.,’ July, 1900, p. 157 ef seq. 
I If z'l and z -2 be judgments of two observers and .^12 their relative judgment, Sn, Sn, S~i 2 , errors 
measured from the means of the respective systems, then = Sn - S,?o, whence the result follows at 
once, if the correlation = S(5..^ x 8^.,) ^ero. 
