134 MR. W. F. SHEPPARD OE" THE APPLICATION OF THE THEORY OF 
Value of a. 
Probable error in L 
by direct observation. 
Probable error by 
averajire-and-average- 
square method. 
Ratio of 
probable 
eiTors. 
•0 
'8153.5 all'll 
■674-49 (ij yn 
1-25 
± -5 
'91908 aj \/n 
• 747 '28 a' yn 
1-23 
± -2 
■85528 aj yu 
'68523 a\ yn 
1-25 
± -I 
oc 
00 
00 
s 
•71937 a.'yn 
1-24 
± -6 
•96369 aj v/w 
•78489 aj yn 
1 23 
± -8 
P15298 aj yn 
•91023 a/ yn 
1-27 
Value of L. 
]\IecHan 
Qnartiles 
I )eciles 
(2.) If we take Li as equal to the average for the n individuals, and find X and Y 
by observing the values of L whose class-indices are a and ^ respectively, the mean 
scpiare of the resulting' error in Lj — {xY — y X.) j [x — y) is 
a/jn — 2 [x a-/n — y o^ln) j (x — y) + E'Y^ = (E'^ —- u-)jn, 
where E“ has the value given in § 22 (2.); and similarly, if we take a as equal 
to the standard deviation of the n individuals, the mean square of the error in 
a — (Y — X)/ {x — y) is (H“ — ^o-)ln. Hence E'^ and are respectively gTeater 
than and ^ or ; in other words, the probable errors in the values of Lj and of a as 
determined by the formulEe (li.) of § 22 (2.), are greater than the probable errors in 
their values as determined by the average-and-average-square method of § 22 (1.), 
If, for instance, a = — ^ so that the observed values are the two quartiles, 
the probable error in Li as determined by (ii.) of § 22 (2.) is ’75043 aj^n, which is 
11 per cent, greater'" than the probable error ’67449 aj^yn due to the average-and- 
average-square method; and the probable error in a is ’78672 ajy/n, which is nearly 
65 per cent, greater than the probable error ’47094 ajyn due to the average-and- 
average-square method. 
If we are unable to calculate the average and the standard deviation, we should 
* When the quartiles are observed, it is also usual to observe the “ median,” for which a = 0. If 
we take the arithmetic mean of the median and the two quartiles, the probable eiTor due to taking 
this as the value of L, is reduced to •72736 aj-/n, which is less than 8 per cent, in excess of the 
probable error due to taking the average. If X and Y are the quartiles and M the median, it 
may be shown that the best result from these data is obtained by giving to 4 (X -t- Y) and M 
weights in the ratio of 2 (exp. — Qb — 1: (exp. 4 Qb ~ ^>^^4 the probable error in the mean is 
then [4 Q« v/’T / {1 — 2 (exp. — 4 Qb + 2 (exp. — Qblb/'v/’^- The first two convergents to the above 
ratio are 2 : 1 and 7 : 3, so that { 7 (X -f Y) -|- 6 M}/20 is a slightly better value than (X -I- Y M)/3. 
I have assumed that the quartiles, &c., are found by actual observation. But there is reason to 
believe that their values are sometimes obtained by faulty methods of interpolation. This does not 
afl'ect the magnitude of the probable error, but it affects the calculated values of Lj and of a. 
