454 
PROF. KARL PEARSON ON SKEW VARIATION. 
and the right-hand side is clearly less than unity. This limited area covered by the 
real binomial explains its relative infrequency as a descriptive series in practical 
statistics. If, however, we take the negative binomial as admissible, i.e., allow forms 
of the type 
(p — q)~ n , where p — q—l 
we extend the possible area .of a binomial down to the line 2/3. — 3j3 1 — 6 = 0. 
Such a type of binomial is by no means of infrequent occurrence and can be more or 
less justified on a priori grounds.* Below Type III. line, the values of p and q become 
in the mathematical sense unreal, i.e., imaginary. It is by no means certain, however, 
that such imaginary binomials with real moment coefficients may not, like imaginary 
hypergeometricals, give statistically good fits and be ultimately provided with 
physical interpretations. 
(10) Concluding Remarks. —It is very difficult to assert finality for any scientific 
investigation, but I trust this second supplement to my original memoir on skew 
variation of 1894 has garnered the last harvest of possible types within the limits 
proposed in that investigation. The object was the discovery of a system of frequency 
curves providing for every possible variation of the first four moment coefficients of 
a distribution and provision for their rapid treatment and calculation. Since 1894 
much has been done by the provision of tables of the new functions and improved 
tables of old functions necessary to carry this out.t Diagrams like that accom¬ 
panying this memoir, enable the statistician who has calculated the characteristic /3 L 
and /3 2 , to select at once the appropriate type, from the position of the point fi 2 in 
the /3i, /3 2 plane. The first diagram, prepared by Mr. A. J. Bhind at my suggestion, 
has been long in use.J For the present very carefully prepared and much extended 
diagram I have to thank my colleague, Miss Adelaide G. Davin, whose labours cannot 
fail to be appreciated by those having to handle practically statistical data. 
Since the publication of my original memoir on skew variation, many attempts have 
been made to express the nature of skew distributions by other systems of curves or 
by expansions in series. I have given careful attention to these competing systems 
and have discussed some of them elsewhere (‘ Biometrika,’ vol. IV., pp. 169 to 212). 
My chief objections to them arise from the fact that they either (i.) cover far less 
than the necessary area ; or (ii.) involve constants the probable errors of which can be 
indefinitely great; or (iii.) involve constants the probable errors of which have not 
been or possibly cannot be calculated. In no case that I know of have they syste¬ 
matically been applied to extensive ranges of data, and the goodness of fit compared 
with that of other systems. The existence of such competing systems is at any rate 
* See ‘Biometrika,’ vol. IV., p. 209, and vol. XI., p. 139. 
t Now collected in “ Tables for Statisticians and Biometricians,” issued by the Cambridge University 
Press. 
1 ‘ Biometrika,’ vol. VII., p. 131, 
