K. Pearsson 
177 
for all truly normal distributions ought not to dilfer l)y more than two or 
three times the above expression from zero. 
We can form other expressions involving aud y8., and ask what their value is 
for the Gnussian curve. We can calculate their probable errors, and determine 
whether the given distribution satisfies the Gaussian value within the limits of 
random sampling. 
Thus I take the expression : 
^1 V;8.(/3, + 3) ^1 V^,(G+/3,-3) 
^ 2 5/8,-6^,-9 2 6 + 5(/3,-3)-6/8, 
Clearly this expression vanishes for the normal curve, and nearly when 
VyS, and — S are not very large, i.e. when we have not a very wide deviation 
from normality. The probable error of this expression, if the distribution be really 
normal, is ■07449 
Again, consider the expression : 
^ = ^"5/5,-0/3,^9 
This is a length which vanishes, if the distribution be truly normal. Its probable 
error is '67449 -y/s^ ^ ^'^^ case of the Gaussian curve, and accordingly d 
should not differ from a by more than two or three times the above probable 
error. 
Now let us write tj = /3.^ - 3. Then it is absolutely impossible for any 
distribution to be looked upon as Gaussian unless ^, d and tj are zero within the 
limits of random .sampling. These limits being deduced from theii' known probable 
errors. 
Now it will be asked why choose such an expression as instead of the simpler 
V/S, ? The answer is ipiite simple. We want to determine whether the mode 
coincides with the mean or not, and we cannot do this on the basis of the 
Gaussian curve where no distinction is made between the two. We must take 
some curve which is not Gaussian to determine this important quantity from. Now 
the equation to the Gaussian curve is 
(X - 111)'- 
y = y,e , 
where m is the mean value of x and cr„ the standard deviation, and we have for its 
differential equation : 
\ dy _ X — m 
y dx ~ c7„^ 
Biometiika xv 23 
