ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 137 
A' alue of L. 
a. 
li^ “h ci-X. 
Discrepancy. 
1 
1 
Probable 
discrepancy. 
Ratio of actual 
to probable 
discrepancy. 
32- 5 
33- 5 
34- 5 
35 - 5 
-i-ooooo 
-0-99895 
-0-99232 
-0-96406 
-2-09762 
35-5425 
--0425 
-044-2 
0-96 
36-5 
-0-89812 
-1-63579 
36-4906 
+ -0094 
•0263 
0-36 
37-5 
-0-75541 
-1-16359 
37-4600 
+ -0400 
•0177 
2-26 
38-5 
-0-49267 
-0-66301 
38-4877 
+ -0123 
•0145 
0-85 
39-5 
-0-12212 
-0-15366 
39-5334 
--0334 
•0138 
2-42 
40-5 
+ 0-25541 
+ 0-32573 
40-5177 
-•0177 
•0139 
1-27 
41-5 
+ 0-58165 
+ 0-80928 
41-5104 
-•0104 
-0150 
0-69 
42-5 
+ 0-80705 
+ 1-30190 
42-5217 
-•0217 
•0195 
1-11 
43-5 
+ 0-91626 
+ 1-72938 
43-3993 
+ •1007 
•0290 
3-47 
44-5 
+ 0-97488 
+ 2-23952 
44-4467 
+ •0533 
•0525 
1-02 
45- 5 
46- 0 
47- 5 
48- 5 
+ 0-9923-2 
+ 0-99860 
+ 0-99965 
+ 1-00000 
i 
■ 
The extremities of the range are not considered, as the values of in (1 + a) or 
i u (I — a) are small when a is nearly equal to dz h so that the law of normal distri¬ 
bution does not hold with regard to the errors in these values; and, moreover, 2 is 
changing rapidly, so that is not exactly proportional to 0. For the ten values 
considered, the actual discrepancy is less than the probable discrepancy in four cases, 
and greater in six ; and for nine of them the ratio of the two is within the probable 
limit (§ 17). The remaining ratio is rather large (3‘47) ; but otherwise the data 
appear to justify the hypothesis of normal distribution.t 
* TLe value.s of x sEown in this column correspond to tlie fractional values of a given by tbe data 
(— 2763/2866, — 2574/2866, &c.), not to the nearest decimal values as shown in the second column 
(- -96406, ~ -89812, &c.). 
The quantities shown in the final column are the ratios of the quantities given in the preceding 
columns. If these were taken to the fifth place of decimals, the last figure in some of the ratios might 
be altered; but it is not necessary to make such exact calculations (§ 17). 
t It should be remembered that when the probable discrepancy is small, the possibility of errors of 
scale must be considered; thus an inaccuracy of one-hundredth of an inch in a division of the scale 
near 40 inches would make an appreciable difference in the ratio of the actual to the probable discrepancy. 
Also it should be noted in the present case that the observed individuals came from different parts 
of Scotland, so that the “original community” was really heterogeneous; and it is likely that the 
measurements in different regiments were taken by different observers, with different personal 
equations, and were not taken with as great care as would be observed at the present day. On the other 
hand, as the exact measurements are not given, but only the measurements to the nearest inch, the 
values of Lj and of a are fitted more closely to the class-indices than they should be; and the probable 
disci-epaucy should thei-efore be slightly less than that given by the theoretical formula. 
VOL. CXCII.—A. 
T 
