CARL J. WEST, Ph. D.
=Introduction.= In curve tracing the graphic representation is
constructed from the equation. Due largely to the requirements of
statistics the converse, namely, to find the equation of the curve
when the distribution of points is given, has become of interest. This
problem is very different from the exercises of analytical geometry
in which a given law of distribution of points is to be translated
into algebraic language. For the presence in the statistical data of
accidental irregularities makes it undesirable as well as practically
impossible to obtain a curve passing _through_ the points. Instead, a
curve is “fitted” to the points, that is, a curve is passed _among_ the
points in accordance with some generally accepted principal such as
that of least squares or the agreement of moments.
Aside from the straight line and the parabolas, the curves proposed
by Pearson[6] have found acceptance. In order to derive curves which
can be fitted to widely varying distributions of points, Professor F.
Y. Edgeworth[7] has proposed to modify, to _translate_, the normal
probability curve with unit standard deviation.
1
y = —————— e^(—(t^2/2))
√(2π)
In this article we shall discuss the geometry of the curves which
Edgeworth obtains by this transformation and derive a method for an
approximate solution of the two equations, one of the fourth and the
other of the sixth degree, which arise in the fitting of a curve of
this class.
In order that the final curve may be written in terms of the
co-ordinates x and y the equation of the base or generating normal
probability curve is written:
1
z = —————— e^(—(t^2/2))
√(2π)
where t denotes abscissas and z ordinates.
Let the abscissas of the transformed curve be functions of the
corresponding abscissas of the base curve. Then it may be assumed that
x can be developed in powers of t, and hence we may write on omitting
fourth and higher powers,
x = a(t + κt^2 + λt^3),
where a, κ and λ are constants to be determined in “fitting” the curve.
Since x denotes the value of a measurement and y the frequency of x,
that is, the number of individuals possessing that value of x, the
magnitude of an element of area denotes the number of individuals
between two values of x. Obviously, therefore, if the transformation is
to be of concrete value the magnitude of an element of area must not be
altered, though of course the shape will be changed. Hence
y dx = z dt,
and y = z dt/dx
1 1
= —————— e^(—(t^2/2)) · ——————————————————
√(2π) a(1 + 2κt + 3λt^2)
The formulas of transformation are thus:
x = a(t + κt^2 + λt^3),
1 1
y = —————— e^(—(t^2/2)) · ——————————————————
√(2π) a(1 + 2κt + 3λt^2)
=Maximum and Minimum Points.= Since only curves with one maximum
point or mode are practically useful it is desirable to determine what
values of the constants a, κ and λ give unimodal curves.