Editorial 115 
Accordingly 
Mean(5.,M = i^^|(l-|)-iv|(^ 
or Mean (SnM,) = ^' ('^)- 
Thus Mean(S./',S.r,) = ^^ ^[l -^j (8). 
Equations (5) and (8) are the fundamental equations of our present subject. They 
give the standard deviations and correlations of the errors in any lengths measured 
along the a;-axis as determined by the frequencies of the corresponding ranges. 
If we have 
z =y (a'l, X.2, . . . Xp), 
then to our degree of approximation, 
a. 
where S denotes summation of s from 1 to and 8' summation of every pair of 
different s and s from 1 to jj. 
With our degree of approximation, i.e. to first order terms in I/VN, the above 
results are true whatever be the law of frequency. 
For the special case z = x.,- x^ we find 
'y.y.^NV n)] 
.(10). 
For the special case that Xo — = ^'i„'Il' i^traquartile range, = \' 
1 [3 N'\ 2 
^,fl.. NXlQKy^^^-^ y\) IQy.j,} ^^^^^ 
For a symmetrical distribution y^^ — U ,~ ll<i '^'^^^ ^'^^^ 
For the special case that x., — Xj - 1,,^,^^ , the median to right quartile range, 
, ^1/3^ li\^^ 1 \ 
i\^U6 3/^,/4y^„, 4.y,jjJ ^ 
and for x^ — Xi = 1^^,^, the left quartile to median range, 
1 /I N"- 3 N'' 1 \ ..o, • X 
'"^•-=N[if:.'-Wy^-iy;j;J 
(12) and (12 6is-) will only give the same result provided y,^^ = y,^^, which is of 
course satisfied by symmetrical curves but might be satisfied by other curves as well, 
8—2 
