ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 123 
small. Thus X is of the form ■) +fa{^'— +—/^) +fyiy — y) + . • • ; 
and therefore, by § 16, its mean value is . . .), and the different possible 
values are distributed normally about this mean value with mean square 
{(“/a" -r /d// + yfy +•••)“ (“/a + /^fp + y4 + • ■ 
If we denote the expression in curled brackets by cr“, the quartile deviation from the 
mean is Qcr/\/n, where Q is the deviation of the quartile ordinate from the central 
ordinate in the standard normal curve (= '67449 approximately"^). 
The applications are of two kinds. In one class of cases X is a “ frequency- 
constant ” whose value is required. Its observed value f (a', y', . . .) differs from 
its true value /"(a, /3, y, . . .) by an error due to the paucity of observations, and 
Qcr/\/ n is then the prohahle error. In the other class of cases the theory is applied 
to the testing of any hypothesis with regard to numerical statistics. The difference 
between the observed and the calculated values of X is a discrepancy, and we test 
the hypothesis that this discrepancy is due to paucity of observations by comparing 
it with the prohahle discrepancy Q.(Tj\/n. If the comparison is made for several 
different values of X, we ought to find that for about half of them the discrepancy 
(z= d) is less than the probable discrepancy (= q), and that, amongst the remaining 
values, d is in no case a very large multiple of q. The following considerations will 
enable us to determine whether, in any particular case, the values of djq are or are 
not greater than we might reasonably expect. 
Let the different values of a magnitude § be distributed normally, with quartile 
deviation q, about a mean value zero ; and let m values be taken at random. Then, 
if the area of the standard normal figure lying between the ordinates at the points 
X = — pjq and as = + pjq is (f), the probability of one at least of the values of 8 
being numerically greater than p is 1 — If we choose cf) so that this probability 
may be equal to the corresponding value of p may, by analogy with the “ probable 
error,” be called the prohahle limit of 8. The following table gives the values of pjq 
determined by this condition, for values of m from 1 to 20t 
m 
P' 1'1 
'1 m 
pIi 
VI 
pIi 
VI 
pIi 
1 
1-000 
6 
2-37b 
11 
2-777 
16 
3-009 
2 
1'559 
7 
2-481 
12 
2-832 
17 
3 046 
3 
1-874 
8 
2-570 
13 
2-882 
18 
3-080 
4 
2-088 
9 
2-648 
14 
2-928 
19 
3-112 
i ^ 
2-248 
10 
2-716 
15 
2-970 
20 
1 
3-142 
* The value of Q to 20 places of decimals is '67448 97501 96081 74320, and its logarithm to 
1 2 27 29 201 230 
13 places is 1'82897 53543 532. The successive convergents to Q are -v > “v> 777 > 7 :, > xtv > 7777 1 • • • 
t For larger values of m, the value of pjq may be taken as equal to that given by Chauvexet’s 
criterion for the rejection of one out of 7n/log« 4 + f observations. 
n 2 
