The sign of κ is determined by the sign of the third moment about the mean μ3, that is, by the direction of the skewness or asymmetry. For positive skewness the mean must lie to the right of the median and hence μ1´, the first moment about the mean, must be positive which necessitates a positive sign for κ. Therefore, the sign of κ is the same as that of the skewness.
To fit a curve to the given data, after the constants have been determined it is necessary to find, by solving a cubic equation for each value, the values of t corresponding to the x’s of the respective classes. The cubic is
aλt3 + aκt2 + at - x = 0.
Any of the various methods of approximating to the solution of a cubic may be used in solving these equations.
The area of each class can now be obtained by computing the corresponding areas under the standard normal curve from a table of the probability integral.
The Method of Interpolation. The actual fitting of the curve can now be readily accomplished.[12] The distinctively geometrical operation is the interpolation for the values of λ and κ for a given pair of values of β and ϵ.
Within the limits of the table[13] the curves resulting from the assignment of a constant value to β are practically straight lines, β = 0 is the λ-axis; β = 1 is a line parallel to the λ-axis. Hence we may safely assume that the variation from one column to the next and from one line to the next is linear for values of β. That is, ordinary first difference interpolation methods are applicable.
As regards the system of ϵ curves we have for instance ϵ = .128 at (λ = .050, κ = 0); again, at approximately (.045, .060) and (.40, .085). We are therefore warranted in assuming the applicability of first difference methods to interpolation between the ϵ curves.
As an illustration let us find the values of λ and κ for ϵ = 0.112 and β = 0.044. On inspection of the table it is seen that λ lies between 0.30 and .035 and κ between .090 and .095. When κ = .090, λ = .033 for ϵ = .112. When κ = .095, λ = .031 for ϵ = .112. For β = .042 and κ = .090, λ = .033 and for β = .046 and κ = .095, λ = .031, ϵ = .112 in each case. Hence, to first differences, λ = .032 and κ = .093 for ϵ = .112 and β = .044. For interpolation in parts of the table showing more rapid variations appropriate methods will suggest themselves.
Taken geometrically the table represents two distinct systems of curves, with each curve of one system intersecting all the curves of the other system. Therefore, a pair of values for λ and κ can always be found for values of ϵ and β within the range of the table.