282 
PEOFESSOR K. PEARSON OX THE MATHE^L4TICAL THEORY 
form improbably a normal distributior. Lastly, turning to Dr. Lee's {i.e., (1)), 'we 
see tliat the greatest divergence from normality is about 17 times the probable 
error in the case of ySo. The divergence from .symmetry is measured by about 
Do times the probable error. Thus the odds against .such a random selection are 
on the two counts 4 to 1 and 3 to 1, about, or since the two results, given 
normality, are independent, the combination gives about 12 to 1. These are 
certainly only moderately long odds, and we must conclude that though a skew 
curve describes Dr. Lee’s judgment considerably better than a normal curve, yet a 
normal curve might, without any great improbability, be adopted. 
The results for the absolute judgments indicate what we may expect to find for 
the relative judgments. The relative judgment of Dr. Lee and myself can well 
be described by a normal curve (see 2-1); the constants differing by less or by very 
little more from their normal values than the probable errors. On the other hand, 
i\Ir. Yule’s and Dr. Lee’s (see 1-3) relative judgments differ from normality on the 
score both of asymmetry and of flat-toppedness, by odds of more than 10 to 1 in 
each case, or of 100 (oral least 50) to 1 in the combination. Lastly, the odds against 
Mr. Yule’s and my relative judgments (see 3—2) on the basis of a random dis¬ 
tribution are only about 4 to 1 on the more unfavourable way of considering them, 
so that 3-2 might pass as a normal distribution. 
We thus conclude that while two out of the six distributions in the bisection 
series are very improbably random selections from normal material, two others are 
capitally represented by normal curves, while the remaining two are not very 
favourable cases. 
Taking these results in connection with those for the bright-line distributions 
we must conclude : That the distribution of errors of judgment can diverge in a 
very sensible way, both on account of asymmetry and of flat-toppedness, from the 
Gaussian curve of errors ; but that cases can be found which approximate with all 
probability to random sampling from normal material. Consequently it is necessarv 
to select a type or types of frequency curve which, while allowing for these points 
of sensible divergence, will yet pass into the normal distribution in certain special 
cases where within the limits j^i'escribed by the probable error the skewness and 
^2 — 3 are sensibly zero. 
Since it is incontestible that, if axioms (a), (/3), and (y) are adopted, our distribu¬ 
tion of errors must be normal, we must conclude that one or other or all of these 
axioms are not universally true. When therefore we get material for which the 
skewness is sensibly not zero, or is sensibly not three, we are quite at liberty to 
assert that the sources producing these errors do not fulfil axiom (a) or axioms (/3) 
or (y) resjDectively.’"' 
* It is very necessary to insist upon this. A recent critic has asserted that I have argued in the second 
memoir of my evolution series (‘ Phil. Trans.,’A, vol. 186, p. 34.3) in an illegitimate manner on the nature of 
the sources which lead to particular types of distril)Ution. He denies that it is possible to state anything 
