418 
Miscellanea 
II. On the Representation of Statistical Data. 
By L. ISSERLIS, D.Sc. 
§ 1. In a paper entitled "On the Mathematical Representation of Statistical Data*" Professor 
Edgeworth gives several examples of the fitting of what he calls the generahzed curve of error 
and some of his recent modifications of it to frequency data. The examples to which he applies 
his methods are taken, with a single exception, from a paper by the present writer, in which 
double hypergeometrical series were fitted to frequency distributions in two variables!. Professor 
Edgeworth describes these data as suited to test and illustrate methods of representing frequeTicy 
distributions." Judged by the usual criterion for the testing of goodness of fit these examples 
indicate that Professor Edgeworth's curves are decidedly unsatisfactory, but he maintains that 
the fault lies in the criterion and not in the curves. Professor Edgeworth proposes therefore 
a new criterion for the testing of the goodness of fit of theory to observation, viz. the magnitude 
of the sum S (e^/U), where U is the ratio of the frequency in any category of the data to the 
total frequency (called by Professor Edgeworth the relative frequency) and e is the difference 
between this and the theoretical relative frequency. He adds erroneously that this is Professor 
Pearson's divided by the total frequency N. To quote from Professor Edgeworth's paper 
(p. 461) "His (i.e. Professor Pearson's) x~ being, in our notation, equal to S (N~e~)/S {NU), where 
N is the total number of observations, is N times as great as our criterion S (e^/U)." As a matter 
of fact Professor Pearson's )('^ is equal to S {(m' - m)^/w?'} where m' is the theoretical and m the 
observed frequency, and in Professor Edgeworth's notation this would be N8 (e^ju) where u is 
the theoretical relative frequency. The fact that the denominator U employed by Professor 
Edgeworth is the observed' relative frequency must make the test nugatory in practice, for 8 (e^/V) 
will be infinite whenever the data contain one category with zero frequency. Very little difference 
results in Professor Edgeworth's examples when the f/'s are replaced by m's in the criterion 
proposed by him, but the later part of his paper claims to give methods of curve fitting whichf 
"not only do or may give better values for the coefficients than the use of moments, but also 
mnst^ give better results than moments or any other process." Professor Edgeworth attempts in 
the section entitled "some new constructions" to achieve this highly desirable end by, as he says, 
minimising the Pearsonian criterion x^- Actually he minimises S (e^/U), a very much easier 
thing to do since the denominators are constant, but leading to equations which are altogether 
irrelevant. 
Professor Edgeworth's proposed criterion is an absolute one, independent of the number of 
categories. The value S [e-jlJ) is usually small and apparently we are to judge of the goodness 
of fit by the number of zeroes between the decimal point and the first significant figure. 
In what follows we fit Pearsonian frequency curves of the appropriate type to the examples 
used by Professor Edgeworth. In addition to calculating and the corresponding value of P 
for each, we give also the values of S (e-jV) and 8 [e-ju). All these are compared with the 
corresponding values for the curves fitted by Professor Edgeworth || . It will appear that Professor 
Edgeworth's curves are a poorer fit than the Pearsonian types not only when judged by the 
legitimate criterion but even according to the criterion proposed by Professor Edgeworth, and 
this in spite of the fact that he does not as a rule attempt to fit his curve to the whole of each 
distribution but confines himself to the central compartments. 
* Journal of the Royal Statistical Socidy, Vol. lxxix. Part iv. (July, 1916). 
t Phil. Mag. September, 1914. % hoc. cit. p. 476. 
§ Professor Edgeworth's italics. 
II So far as that was possible. Several of Professor Edgeworth's calculations in the later sections 
of the paper are not finished, and some of the arithmetic would have to be rectified before completing 
the calculations. 
