28',) 
OF EKKUFS OF JUOGMENT AND UN 
THE rEi;,S(jNAL EQUATION. 
( 3 - 1 .) 
(«.) 
y = 56'5924 ( 1 + 
16 28385 
S".’16,093 
\6'074.832 
12-03998 
Origin at mode + 1‘8495 hundredth of line. 
ij = 59-083 expt. (= a;722-797,492). 
Origin at mean + 1-5890 hundredths of line. 
( 1 - 2 .) 
X 
\ 35'390655 
1 
A 
384605 
(«.) y + 28-15012J 37-69023/ 
Origin at mode — 1-8245 hundredths of line. 
(h.) y = 56-688 expt, (- .^724-763,446). 
Origin at mean — 1-7120 hundredths of line. 
Summing ipj the results of tlie above investigation as to random sampling, ve 
conclude :— 
(i.) That outlying observations render the skew curves a bad fit in one, and a very 
bad fit in a second case, and that in ten cases the observed results are very probable 
as random samplings from skew distributions. 
(ii.) That the normal distribution is bad in one case and preposterously l^ad in 
four others ; it is probalde in seven other cases, but in all cases less probable, and in 
five very much less ^Ji-obable, than the skew distribution. 
We are thus led to much the same result as in our previous investigation of 
typical physical constants of the distribution, namely : that the axioms on whicli 
normality depends are not universally true, but that we require to use curves which 
^vill allow of a distinction between mode and mean, that will not assume an arbitrary 
relation between the fourth and second moments, yet which will pass gradually into 
the normal curve as we deal with material more and more nearly satisfying the 
fundamental axioms (a) {B) and (y) (see pp. 274-275). 
Such curves are supplied by the skew curves. If it be argued that these curves 
themselves involve relations between the first four and the higher moments, the 
answer is simply that we need only take such a number of independent moments 
that the bulk of frequency distriljutions can be represented as random samplings 
from our theoretical curves. It is idle to assert witli Lifts that if we have 
n frecpiency groups we must take n independent moments to describe the distribution, 
tlie suLC qud non of tlie problem is to describe with the fewest possible constants the 
distribution of a very great number of groups. Nor will arbitrary curves with six (jr 
seven constants do as Avell as a well-cliosen curve with three or four.'^ Tlie ii(.>rm;d 
* Tested in a variety of ■\vay3 in a memoir on the general theory of curve fitting, -which will shortly 
appear in ‘ Biomctriha.’ 
2 F 
VOL. cxcvm. 
A. 
