124 
On the Probable Errors of Frequency Constants 
a lower grade of inaccuracy, but there is little advantage to be gained. Thus for 
a,„ . = 1-0466 ^ , \s = -31660, X,,- = -68340 (33), 
ViY 
only a very slightly less inaccuracy with a considerable change in the values of 
the X's. 
It is desirable in such a case to take round numbers easily remembered, and 
thus we may propose 
?w 1 3 =hmi +im;, (34). 
In this case a,,, = 1-0469 (35). 
T7T > 
The practical difference of 1-0466 and 1-0469 is not worth considering and the 
coefficients are most easy to remember. 
If we wish to better this at all substantially we must take three pairs of decile- 
determined medians. For example, suppose we take 
Hi 1 fi 5 = X 1 m 1 + A- 3 m 3 + X 5 m 6 . 
Tij! Tu' TTf TU T7> To TTJ TF IT 
We find cr„, = 1-0321 -^, 
w 
here Xj =-33426 
X_3 = -44795 \ (36). 
TIT 
X_^i^ = -21779 
Or very approximately, 
which give also to four decimals 
<7„, = 1-0321-^ (38). 
Whether this 2 gain on using three instead of two median values is worth 
the extra labour, which is not, however, very great, may be discussable. Probably 
with the use of four to five medians we could make the accuracy substantially that 
of an arithmetic mean. The difficulties lie not so much in the reduction of the 
inaccuracy as in two other matters : (i) the whole superstructure of these probable 
errors of decile medians is based on the frequency distribution approaching that of 
the normal curve. We shall get in many cases larger errors than those of random 
sampling, due to definite deviations from such normality in our frequency ; and 
(ii) the decile values, if there has been grouping in classifying the frequency, can 
only be determined by use of the probability integral tables. We cannot pick out 
the character of the individual who stands individuals from the top. He will lie 
somewhere in a group of p individuals, and unless these individuals be arranged in 
order of character, we cannot find the character of the n^th and + l)th individual. 
Their characters would have to be found from those of the individuals who bound 
