136 
MR. W. F. SHEPPARD OH THE APPLICATTOH OF THE THEORY OF 
use of the average and the average square. Let ©i/n and 0;/?i be the mean squares 
of the errors in II as determined by the two methods. Then it may be shown that 
the mean square of the error involved in taking pRi -b as the value of ( 2 ^ + q) E, 
is {{]r 4- ^pq)®\ 4 q-&l}/n = [(p + 5)“©i 4- S'" (©2 — ©i)}/^- Since this must be 
positive, it follows, by taking p q = 0, that ©; must be greater than ©f; and 
therefore Ri gives a better value of E thaTi E,. By taking 25 =—5'=4;lwe see 
that the quartile of Ej E 2 is Q (©2 — @?)Vv^ 
§ 24. Test of Hypothesis as to Normal Distribution. —To test whether any par¬ 
ticular distribution is normal, we use the result obtained at the beginning of the last 
section. Having found the average and the standard deviation of the n individuals, 
we calculate Lj ax, the value whicli should correspond to class-index a. The 
difference between this and the observed value X is a discrepancy whose mean 
square is ar [(1 — a~)/Az~ — (1 4- so that the probable discrepancy is 
Qa [([ — a'A/E" — (i + ")] Vv^; ^od the actual discrepancy has to be compared, 
for as many values of x as possible, with this probable discrepancy. 
Suppose, for instance, that we take the chest-measurements of Scotch soldiers,* to 
which Quetelet refers in the work quoted above :— 
Chest-measurements, to the nearest inch, of 5,732 Scotch soldiers. 
Inches. 
Number. 
Inches. 
Number. 
33 
3 
41 
935 
34 
19 
42 
646 
35 
81 
43 
313 
36 
189 
44 
168 
37 
409 
45 
50 
38 
753 
46 
18 
39 
1062 
47 
3 
40 
1082 
48 
1 
The values of the average and of the standard deviation cannot, of course, be 
calculated exactly; as the most probable values we tindt Li = 39'8489 inches, 
a = 2'05301 inches. Thiis we get the following results :— 
* ‘Edinburgli Medical Journal,’ vol. 13, pp. 260-262. Quetelet made some mistakes, whicli I hare 
corrected, in transcribing the figures. 
t The formula for calculating tbe standard deviation lias been given by me in a paper “ On the 
Calculation of the most Probable Values o1 Frequency-Constants,” in vol. 29 of the ‘ Proceedings of the 
London Matliematical Society ’ (p. 353). 
The values given in the text are obtained by a first approximation. A second approximation might 
be made by assuming that the data represent the result of a random selection from the normal 
distribution given by the first apj^roximation; but this correction would not alter any discrepancy 
shown in the table by as much as 1 per cent., and it may therefore be omitted. 
