124 
MR. W. F. SHEPPARD OFT THE APPLICATION OF THE THEORY OF 
If m values of X were observed, and if the discrepancies were independent, it 
would be an even chance that in one case at least the ratio of the discrepancy to the 
probable discrepancy would exceed the value given by the above table. As a 
matter of fact, the discrepancies are usually correlated; but, if we bear this in mind, 
the table may be used to decide whether the greatest value of the ratio is such as to 
negative the hypothesis under consideration. 
For calculating Qcr/\//?, in either class of cases, it will not always be necessarv to 
express cr“ in terms of a, /d, y, . . . If the value of X depends solely on the values of 
certain frequency-constants, and if 9, p, d, . . . are the errors in these frequency- 
constants, then /"(«', , y, . ■ •) —./'(«> y, • ■ ■) may be written in the form 
/r? Ir) mO + • • • The errors 6, . . . being of the form a (cc' — a) 4- b ifi' - /3) 
+ c (y' —- y) d- . . their mean squares and mean products can be found ; and thence 
the mean square of I’? -f can be obtained by the general formula 
given in § 12 . The expressions for the mean squares and mean products of the 
errors in frequency-constants of certain j)articular forms will be found in §§18 
and 19. 
The true values of a, y, . . ., or of the frequency-constants on which X depends, 
are not known; and therefore, in calculating Qajs/n, we can only use the observed 
values a', (i\ y, . . , But, n being great, the mistake so introduced in Qcr/\/n is 
small in comparison with Qo-/\/a itself. In general, it is sufficient to determine 
Qo-Zv^n within about 1 per cent, of its true value. It will therefore be found 
simplest to calculate and then to take out the corresponding value of Qo-/\/n 
from Table V. (p. 159). This table gives Qv'^N, for any given value of N, within 
from '8 to "08 per cent, of its true value. 
§ 18. Error in Mean, Mean Square, &c .—Let the mean value of a measure L (in 
an indefinitely great community), and the ;/?th power of the deviation from the 
mean, be denoted by Li and respectively. Also let the actual values of L be 
Li-f .Ti, Li-h.T 2 , L,- 1 -.T 3 , . . . ; and let the relative frequencies of these values be z^, Zr^, 
2 : 5 , • • • Thus we have %z — 1 , St'.r = 0 , Xzx^' = Now" let a random selection of 
n individuals be made, and let the numbers for which Ij has the values Lj + x^, 
Li + X., L, + a-;;, . . ., be respectively n (z, + e,), 71 (z. -b e..,), n (23 + € 3 ), . . . Then 
(§§ 15, 16) the mean value of A^ei Aoe., + A 363 + . . . = SAe is zero ; its mean square 
is {^A ‘2 — {SAzY]/n ; the mean product of !SAe and Bie, -f- Boe., -f B 3 e 3 + . . . = 2 !Be 
is (SAB 2 ~ SA 2 .SB 2 )/a; and, 11 being supposed to be great, the values of SAe or of 
SBe are normally distributed. 
Hence we obtain the following results :— 
(i.) The calculated value of Lj is Lj -j- -T x.x, + X 3 e 3 -f- . . .). Thus the error 
in L, is Xy^i + Xos, -f .^363 -f- . . •, and therefore this error is distributed normally 
with mean square [%zx' — {%zx)~}jn = X-ijn. 
