116 
Miscellanea 
We see here just as before that the approximate value is largely in defect of the true value. 
(e) Considering Table II in connection with Table I it is evident that there is no simple 
relation between what I have called the " Relative Error " and any of the constants of the curves. 
That is to say, even though we know the ordinary constants of a given curve it will not be 
possible to say beforehand whether the approximate formula for the p.e. of cr will give a reason- 
ably good value in that case. 
Taking into account all these facts I think there can be no doubt regarding the answer 
to our first question. The answer definitely is that when a distribution is not closely normal, 
there is a considerable likelihood that the assumption of normality made in calculating the 
probable error of the standard deviation by the approximate formula may introduce an error 
great enough to be of practical importance in drawing conclusions. Whenever a piece of bio- 
metric work involves the comparison of standard deviations to determine whether they are 
significantly different among themselves great caution should be exercised about making con- 
clusions depend on probable eri'ors calculated by the approximate formula. 
We may now turn to the consideration of our second problem (cf p. 113 supra). In attempting 
to get at an answer to it the plan was to calculate for each of the curves given in Table I (with 
the exception of the Type III curve) the probable error of o- first according to equation (iv) and 
then according to equation (iii), using, of course, the proper values of /n and n. The same 
calculations were also made for the curve obtained from the Italian marriage statistics (Type VI) 
cited above. The results of these computations are set forth in Table III. The " Relative 
Errors" in this case are obtained as before by taking the percentage which the approximate 
is of the absolutely true value and subtracting this percentage from 100. 
TABLE III. 
Probable Errors of Standard Deviations. Influence of Sheppard's Correction. 
Curve 
of 
Type I 
Curve 
of 
Type 11 
Curve 
of 
Type IV 
Curve 
of 
Type V 
Curve 
of 
Type VI 
Curve of 
Type VI 
(Italian Marriages) 
•0118 
1 ^0301 
•0312 
•0234 
•0355 
•01775 
•0108 
roi49 
•0310 
•0226 
•0340 
•01771 
Difierence 
Relative Error 
-1- •ooog 
V-8% 
4- -0152 
1-5 7„ 
-f- ^0002 
0-6 7„ 
-1- -0009 
3 "8 7„ 
-^-•0015 
4-2 °U 
-f -00004 
0"2 7„ 
From this table we note : 
(a) That the absolute differences resulting from calculating the probable errors in the two 
different ways are in every case very small. In only one case (the Type II curve) is the absolute 
difference greater than 1 in the third decimal place. Now it ia, of course, obvious that in 
ordinary statistical work the probable errors will very rarely be tabled to more than the third 
place of figures. It would appear then, from the absolute differences here, that the error made 
by using the approximate formula (iii) instead of (iv) is for practical purposes insignificant. 
(6) Relative to the magnitude of the probable errors themselves the differences are very 
small. It is seen from the last line of the table that approximately 8°/„ is the maximum 
relative error made and in the majority of cases the relative error is much smaller than this. 
