PROF. KARL PEARSON ON SKEW VARIATION. 
455 
noteworthy evidence that to attempt to describe frequency by the Gaussian curve is 
hopelessly inadequate. It is strange how long it takes to uproot a prejudice of that 
character ! If the reader will turn again to the present diagram, he will see that the 
Gaussian frequency occupies a single point in an indefinitely extended area. Those 
who support the Gaussian theory have to prove that no distribution occurs at a 
distance from the point G of our diagram greater than could be accounted for by 
the probable errors of sampling of and /3 2 . These errors are known and have been 
tabled'" and that position is'quite untenable. Frequency distributions occur every day 
which by no manner of means can be described by Gaussian systems. 
It has been said that my skew curves suddenly change their algebraic type and 
that the statistician is puzzled by a slight change in the constants (3 X and (3 2 involving 
such radical changes in the equation to the type. But if the reader examines the 
present diagram, he will see that the main Types 1^, Ij, I L , IV., VI. and VIj occur in 
areas , while the remaining types occur in the critical curved or straight lines which 
bound these areas. Special cases like the Gaussian, the exponential or the rectan¬ 
gular distributions occur where critical lines intersect. Now all these critical lines 
are really critical in the sense that a change of important physical significance occurs 
in this neighbourhood, and it is very unlikely that physical changes will be 
unaccompanied by sharp algebraical changes of form, such as are directly obvious 
in my curves, but are disguised by discontinuities in some of the proposed alternative 
expressions in series, t 
Any one illustration that the frequencies which occur in actual statistical data can 
practically cover the whole possible area of the /3 X , /3 2 planes, and can present 
frequency distributions which change abruptly in type, will suffice to confute both 
the argument that frequency is concentrated in or near the Gaussian point, and the 
argument that it is undesirable that skew-frequency curves should be so manifold in 
form, although how they are to change from U to J, to “ cocked hat,” to rectangle 
and to exponential forms without this abrupt change will be a puzzling problem to 
solve for the professed mathematician. An illustration of this character has been 
several times referred to in the course of this paper. Let us suppose there exists 
,an indefinitely large population, each individual of which carries any number of 
characteristics which are correlated together, for simplicity we will say according to the 
normal law. We may suppose that there are enough pairs of characters to give all 
values of the correlation p from +1 to —I. 
* ‘ Tables for Statisticians and Biometricians,’ pp. 68-71. 
t An analogy might be given in the case of the expression of a “ cocked-hat ” shape of finite range 
and a U-shaped distribution by a single Fourier’s series. Here the trigonometrical expression by the 
Fourier’s series would be superficially the same if kept in symbolic form, while the algebraic form of the 
U-curve would require two vertical asymptotes*and its equation would be wholly different from that of 
the “ cocked-hat’ form. The Fourier expression would only disguise the real discontinuity. In the same 
manner real discontinuity of form is disguised in the series which express skew frequency in terms 
of a long series of moment coefficients. 
3 Q 2 
