Miscellanea 
115 
approximate one was and then subtracting the percentage from 100. The figures so obtained 
are given in the last line of the table opposite the entry " Relative Error." 
TABLE II. 
Probable Errors of Standard Deviations. 
P. E. of a- 
Curve of 
Type I 
Curve of 
T.ype II 
Curve of 
Type III 
Curve of 
Type IV 
Curve of 
Type V 
Curve of 
Type VI 
Calculated by (i) 
Calculated by (ii) 
•0114 
•0134 
1-0149 
1 -0893 
•0028 
■0027 
•0310 
•0229 
•0200 
•0084 
•0337 
■0231 
Absolute Difference . . . 
Relative Error 
- '0019 
17% 
- ^0644 
7% 
+ •OOOl 
4 7„ 
+ •oosi 
26 7„ 
+ •0116 
58 7, 
+ •0106 
31 7o 
From this table we note at once the following facts : 
(a) As was to be expected the approximate formula for p.e. of a in the case of these skew 
curves never gives the true value. 
(b) The probable error obtained by the approximate formula may be either in excess or 
defect of the true value. 
(c) The amount of the deviation of the approximate from the true value varies in the different 
curves but may be very considerable. Thus in the ease of Type V curve the probable error from 
the approximate formula is less than half as large as it should be. In the Type VI curve the 
approximate value is but 69 7„ of the true, and the Type IV curve 74 7o- It will be noted that 
not only are these deviations of the approximate from the true values large in amount, but 
further they are in the direction most likely to cause serious error in practical biometrical work. 
It is not so serious a matter when an approximate formula makes a 2>robable error too large as 
when it makes it too small. 
(d) In the case of the curves of Types I, II and III the deviations of approximate from 
true values are all small, and in the first two cases the approximate value is in excess of the 
true. So far as we may judge from the curves here discussed, it would appear that the most 
serious difficulty from the use of the approximate probable error formula is to be expected in 
curves of Types IV, V and VI. Thinking that possibly the great deviations observed in the 
case of Type V and VI curves might be exceptional and due to the particular examples chosen, 
I took another Type VI curve and calculated the p.e. of a- as before. This curve is one discussed 
by Pearson (Pkil. Trans. Vol. 197, A, p. 451). It gives the variation in age of the brides 
in 28,454 Italian marriages. The values of the moments indicate a curve of Type VI but the 
criterion k2 is so nearly 1 that a Type V curve gives a good graduation. The results of the 
probable error calculations (using corrected values of the moments) are shown in the following 
table : 
P. E. of a- 
Curve of Type VI 
Calculated by (i) 
Calculated by (ii) 
•0177 
•0103 
Absolute Difference ... 
Relative Error 
+ ^0074 
42 7, 
15—2 
