1 28 Ou the Prohahle Errors of Freqneiieri Constants 
The \alues of A, are so nearly one-third that without much greater inaccuracy 
we can assume them J, , we have then 
o- , , , = \{a. -Va a,) (43), 
with = 1-0822 ^^ (44). 
It is quite conceivable that another choice of the three pairs of grades might 
lead to an even smaller increase of inaccuracy over the mean and moment method 
of finding tr. But it suffices here to have shown that the excess inaccuracy by 
discarding the use of quartiles for quatuordecimals and then these for the. and 
j-jj pairs of grades and lastly the latter for three pairs of grades can be reduced 
from 65 7, to 24 7^, from 24 7,, to 12 7„ and ultiuiately to 8 7„. 
(7) In the course of this paper we have seen that it is quite possible to better 
vastly the determination of median and quartile deviation by not determining the 
median and quartile directly but from pairs of appropriately chosen grades. We 
have further seen, however, that either these chosen gi'ades will not be those of 
classification, or on the other hand if the individual ranking has been obtained as 
in small series, the measurement of the ranked individuals is hardly likely to be 
fine enough to give the mean and standard deviation to an adequate number of 
decimal points. In short while the disadvantages in aecui'acy are great, the 
advantages in brevity of treatment will not be compensatory in the majority of 
cases, and mean and moment methods should be always given the preference. 
Why then is it needful to discuss at length the probable errors of grade methods ? 
The answer lies in the fact that in all cases of classification by broad qualitative 
categories no other method of solution is feasible. Further it is certain that hitherto 
too little attention has been paid to the best means of deducing the constants of 
such distributions. In such cases it is futile to attempt to obtain directly median 
quartiles or deciles. Such attempts throw us back on interpolations of an unsatis- 
factory character. We are compelled to accept the grades given by the data 
themselves, and these grades will usually not be symmetrical and are not capable 
of satisfactory I'eadjustment. From every pair of gi'ades we can find the standard 
deviation in terms of a chosen variate interval, and from every grade we can find 
the position of the mean in terms of this standard deviation. The standard 
deviations as found from each pair of grades will not, however, be independent ; if 
there be p broad categories only |> — 2 values of the standard deviation in terms 
of the variate sub-ranges are a priori open to choice and only p — 1 determinations 
of the median. Let us consider how this, the really practical problem of grading, 
is associated with the methods discussed in this paper. 
We start with the sampled population, the mean of which may be treated as 
a fixed origin for measurements in the samples. This population is divided up by 
dichotomic ordinates into broad categories; the distances between these dichotomic 
ordinates will be the same in the sample as for the sampled population, and 
although their values have to be found from the sample they are really constant 
