Yoh. 8, 1922 
STATISTICS: W. A. SHBWHART 
249 
In order to study the nature of the complex of causes controlUng a single 
quantity such as Xi, one of the first problems is to determine whether or 
not the causes represented by the U's, and F's satisfy the following con- 
ditions: (1) That all of the causes, n in number, are effective at the time 
of each observation, (2) that the probability, p, that a cause will produce 
a positive effect is the same for all of the causes. (3) That the probabil- 
ity, p, remains the same for all of the observations, (4) that the effect. Ax, 
of a single cause is the same for all of the causes. 
If these conditions are fulfilled, the distribution in Xi can be represented 
by the successive terms of the expansion {p -f q)^ where the ordinates 
are separated at intervals of 2 Ax. For most of the problems it has been 
found convenient to compare the observed distribution with the theoreti- 
cal distribution consistent with the above random conditions. In general, 
the following procedure has been followed : For each observed distribution 
two factors k = and ^2 = — have been calculated and compared 
with similar factors consistent with the above mentioned binomial ex- 
pansion, where the first four corrected moments of the observed distri- 
bution about the mean are represented by the symbols /Xb M2, Ms, and fXi. 
It will be noticed that k and ^2 are independent of the units used in making 
the measurements. 
For a symmetrical distribution k is always 0 and (32 may vary between 
1 corresponding to n = 1 and 3 corresponding to n = 00 . Irrespective 
of the values of p and q, it should be noted that ^2 does not increase be- 
yond 4 for 10,000 causes and even for 100 causes the value of 4 is not 
exceeded except for conditions in which the skewness k is very large. In 
general large values of /32 indicate either high skewness and few causes or 
that the complex of causes does not follow the random conditions con- 
sistent with the expansion of {p + g)". 
As a result of variations due to sampling the standard deviations, and 
cr^2 of k and ^2 are given in terms of the number of observations, s, by the 
following expressions (Tk = '\ — and o-^^ = \ — it being assumed that the 
distribution is practically normal. If, in practice, the values of k and 
02 are found to differ from 0 and 3 respectively by more than three times 
the standard deviations of k and 02, it is practically certain that the dis- 
tribution is either unsymmetrical or that the causes do not act at random 
as defined above. The method of application of these criteria depends 
in general upon whether or not the observed distribution is one involving 
attributes or variables. 
In general for the case of attributes the procedure is as follows: The 
factors k and 02 are calculated to determine whether or not they are signif- 
