286 
PROFESSOR K. PEARSOX OX THE >L\THEMATICAL THEORY 
Table XX. — Bisection of Lines. 
1. 
2^ 
3. 
3-2. 
1-.3. 
2-1. 
n i 
16 
21 
16 
17 
21 
23 
X 5 
X- : 
p i 
11-36 
•7271 
17-27 
•6353 
14-90 
• 4590 
8-84 
•9199 
20-28 
• 4405 
23-99 
•3478 
2 o 
n 
17 
20 
17 
17 
22 
23 
F 2 
o 
X' 
13-30 
22-04 
20-31 
9-34 
21-38 
25-65 
P 
■6506 
i 
•2817 
•2066 
•8976 
■4364 
•2670 
Now looked at from this standpoint, we see that not one of the distributions are 
improbable on the basis of either the skew or of the normal curve. The longest odds 
against a random samj^ling on the basis of a normal distribution are only 4 to 1, 
and on the basis of a skew distribution only 2 to 1. There is a clear and marked 
advantage in favour of the skew distribution, but it is nothing like so enormous as in 
the case of the bright-line serie.s. If we take the distributions (1), (2), and (3) as giving- 
independent probabilities of random sanpDling,'^ then the odds against these distribu¬ 
tions as a result of random sampling from uormal material are 24 to 1. Thus it 
seems that even in this case the normal law is somewhat improbable as a general law 
of distribution. 
On the other hand, the combined odds against the system ot distributions repre¬ 
sented by the skew cui'ves, are only 3'7 to 1 ; or looking at tlie problem rather 
difterently : In the case of the normal distribution random sampling would give curves 
better than the observed in 20 per cent, of the trials, but in the case of skew distri¬ 
bution in only 5 per cent, of the trials. In other words, there is very great 
improvement in the closeness of fit produced by using skew distributions. 
I place here the equations to the skew and normal distributions (a) and (b) respec¬ 
tively ; remarking that the unit of y in either case is an observation, but the unit 
of X in the bright-line experiments is half a centimetre of the recording strip, and 
in the bisection experiments of the length of the line bisected. 
As we have found correlation in jndgraents, there is, of course, some assumption in this hypothesis 
hut it will serve as a rough comparative test. 
