284 
PROFESSOR K. PEARSON OX THE MATHEMATICAL THEORY 
theoretical curves, partly hy reading ordinates where the contoiir of the curves was 
nearly straight, and partly hy quadrature near the tails. Till tables have been 
calculated for the skew curves, such processes are all that are available ; but thev 
are quite sufficient, for we do not want great exactness in the determination of 
We merely desire to know whether the observations are with reasonable probability 
the residt of a random sampling from the proposed theoretical distribution. Xow 
let us examine the results. We see that for (2), (3-2), and (2-1) the normal curve is 
for practical purposes “ impossible.” As a matter of fact, we might have gone to the 
tenth figure without the probability being sensible in these cases. Further, (l) is 
highly improbable, the odds being about 1666 to 1 against its occurrence as a 
random sample. In the case of (1-3) the odds are 22 to 1 against a deviation as 
bad or worse than this, so that this is an improbable result. Lastly in case 3, and 
this only, we find the odds short, only 2'4 to 1, about, against it; a case such 
as this would occur on the average about twice in five trials. It is really the only 
case in which, under our present test, we could admit the normal curve. 
Turning to tlie skew curve, we see that in three out of the six cases the odds are in 
its favour, namely, in (1), (3), and (1-3). It is not improbable in (2), the odds being 
onl}^ aljout 5 to 1 against it. It is improbable in (2-1) and very improbable in (3-2) ; in 
both of these cases, however, it is at least a million times as probable as the normal 
curve. Thus the skew curve is always markedly and often immensely superior as 
a method of describing the frequency to the normal curve. 
Nor is it hard to discover grounds for its failure in cases (3-2) and (2-1), or for its 
lesser success in (2). The skew curve depending, as its constants do, on the fourth 
moment, takes much more account of outlying observations than the normal curve 
does. Let us consider how the y- of these distributions is made up. 
Absolute Efjuation (2). If the reader will look at the diagram (p. 294) of this distribu¬ 
tion, he will observe the outlying observation on the left. There is a less marked one on 
the right. The skew curve endeavours, and fairly successfully endeavours, to account 
for these outlying observations by thinning its peak and stretching its tails—it thus 
becomes a much worse fit for tlie body of the observations than the normal curve. 
Beyond 3A on the left the skew curve leads us to expect '3, about, of an observation, 
the normal curve only ‘013 of an ol)servation. Thus the outlying observation 
increases the y' of the skew curve by about 3, but the y' for the normal curve by 
about 73 1 In other words the outlying observation is not very probable from the 
standard of the skew curve ; it is improbable enough to be considered practically 
impossible from the standpoint of the normal curve. If we reject this outlying 
observation as due to a momentary eccentricity of the observer, then with the same 
values of the constants of tlie curves the y~’s are as 17 to 10, about, or the normal 
curve fits the body of the observations better than the ske^^■ curve. But this 
position is, of course, again entirely reversed if we fit the two curves afresh, re- 
calculatiup' the constants without includino- the outlviiip' observation in our data. 
O O o 
