OF ERRORS OF JUDGMENT AND ON THE PERSONAL EQUATION. 
271 
when we have to consider to what degree the testimony of a number of apparently 
independent witnesses of the same event is strengthened by the concurrence of their 
judgments as to what actually took place. Without some estimate of tlie correlation 
of judgments we cannot assert what weight is to be given to combined testimony. 
(9.) On the Nature of the Frequeiicij Distribution in the case of Errors of Judgment. 
Having completed our investigation of the nature of fluctuations in personal 
equation and of the correlation between judgments—an investigation which demands 
no hypothesis as to the form of their law of distribution—we now turn to a considera¬ 
tion of the manner in which errors of judgment are distributed. 
In Tables XV. and XVI. will be found the frequencies for the two series of experi¬ 
ments, the results being grouped (see pp. 272-273).* 
The question to be answered is this ; Is the general nature of these distributions 
capable of being described by the “ normal ” curve of errors, on the assumption that 
they are random samplings of the whole “ populations ” of err<)rs that the observers 
respectively would produce if they continued to experiment indefinitely under the 
same conditions ? So far as I am aware no thorough investigation has yet been 
made as to how far actually observed errors are capable of being described by the 
normal curve of errors. In most text-books on the tlieory of errors certain axioms 
are laid down as ruling the distribution of err()rs of judgment, and on tbe basis of 
these axioms the normal curve of errors is deduced. One or two limited series of 
errors of observation are then cited, and the axi(.)ms declared to Ije satisfactory by com¬ 
paring a graph of the theoretical with the observed distribution, or Ijy a table com¬ 
paring the observed and theoretical frequencies of errors occurring witliin each small 
range. As a rule a vague inspection of the amount of agreement is the only thing 
appealed to to test the accordance of theory and experiment. So far as 1 am aware 
writers on the theory of errors have quite overlooked the point that that theory 
itself provides a perfectly general test of whether the accordance between theory and 
experiment is a reasonable or an unreasonalde one. It is not a question of whether 
there is a “practical accordance” l)etween the two, wliatever that may mean, l)ut of 
the degree of probability that a given system of errors or deviations is a random 
sampling from an indefinitely large distribution of errors obeying the axioms from 
which the normal curve of errors has been deduced. To talk of “practical accord¬ 
ance ” between theory and observation is simply to shuffle out of an examination of 
the truth, when the odds are 3000 to 1, or even 70 to 1, against the observed results 
being a random sample of errors obeying certain fundamental axioms.t Now in the 
In Table XV. a group such as 4'75.5 embraces all the frequency between 4'50.5 and -S'OOS; and in 
Table XVI. a group such as "04 embraces all the frequency between '035 and ’04.5. 
t A recent writer on statistics seems to find that an agreement measured ])y the odds of 3000 to 1 is 
very satisfactory, and one against which the odds are 70 to 1 represents with all ‘practicuhle accuracy the 
observ ed frequency. Comment is needless. 
