2i8 
Oil Corrections for Moment-Coefficients 
the abruptness coefficients are found from such small groupings the remaining sub- 
ranges can safely be made fairly coarse, as in the above examples, where five 
divisions of the total range are clearly adequate. 
(9) Illitstvatiuti III. Mean Age and Variahilitij of Infants at Deatli. It is 
very important in practical statistics to obtain the mean and standard deviation of 
J-shaped curves. A good illusti'ation of such curves may be found in infantile 
mortality statistics. These have the advantage that in the early part of the year 
of infancy the frequencies are in certain cases given by much smaller intervals. 
Thus in the Prussian official statistics they are given for the first fortnight by days. 
Professor Raymond Pearl in a paper of 190G {Biometrika, Vol. iv, p. 510) has 
endeavoured to ascertain the mean age at death of infants in the first year of life 
from the Prussian data. It will be of interest to determine what changes are likely 
to be made in his results by the use of our present abruptness corrections. He 
writes (p. 512) : 
It is evident that the grouping here [i.e. in the Prussian data] is sufficiently fine to make 
possible a very accurate determination of the mean age of death A standard month of 
30 days was assumed : then with a unit of 30 days the first and second moment-coefficients about 
an arbitrary axis were determined. From these the position of the mean and the value of the 
second moment about it were easily found. Only the " rough " second moment was calculated, as 
it was deemed sufficiently accurate for present purposes, and furthermore it was difficult to deter- 
mine the proper corrective terms to apply in this case. In the calculations each frequency 
element was for practical convenience centred at the midpoint of its range. The error made by 
so doing is negligible. 
With our present corrections we can test how far the errors made by concentra- 
tion at the midpoints of the subranges are really negligible. It is certainly right 
to concentrate at those points provided we allow for terminal abruptness which is 
very marked in this case. If we make the proper terminal corrections theory shows 
that quite considerable subranges, say in this case one month, may be used to 
determine the raw moments. It will be sufficient to illustrate the method on the 
Prussian male infant deaths. 
We have deaths per 1000 inflxnts born : For the birth terminal we have* : 
Months 
Deaths 
Days 
Deaths 
0—1 
63'99 
0—3 
18-25 
1—2 
22-59 
3—6 
6-58 
2—3 
18-58 
6—9 
7-89 
3—4 
15-96 
9—12 
5-65 
4—5 
13-30 
12—15 
5-82 
5—6 
11-51 
6—7 
10-61 
7-8 
9-30 
8—9 
8-74 
9—10 
8-29 
10—11 
7-51 
11—12 
6-94 
Total 197-32 
i;/= 3-759,224) . 
= 25-809,801 j ^"™«"ths. 
Three day intervals taken with a view to smoothing auomalous values. 
