The values of the μ’s are computed from the data[8] and equated to the corresponding integrals which of course involve the four constants. In this way four equations are obtained from which the values of the constants may be determined. Since it is our present object to discuss the solution only of these equations, merely the principal results will be given.
The general form for the moments about the median of the area under the translated curve is
On applying the two well known formulas:
the determination of μ1´, μ2´, μ3´ and μ4´ is reduced to a matter of algebraic detail. Then on transferring to the arithmetic mean as origin the values of μ2, μ3, and μ4 can be determined in terms of a, κ and λ. It is most convenient however, to make use of the quantities β1 = μ32 ⁄ μ23 and β2 = μ4 ⁄ μ22 or rather β = β1 ⁄ 8 and ϵ = (β2 - 3) ⁄ 12 and express the constants in terms of these quantities. It is to be noted that both ϵ and β are zero for a normal distribution, that is, for λ = κ = 0.
Omitting the detailed reduction[9] which is straightforward and direct, we have
Obviously no algebraic solution can be obtained from equations (3) and (4) for κ and λ in terms of the computed values β and ϵ, and hence a resort to tables is necessary. The values of β and ϵ for values of κ from 0 to 0.0335 and of λ from -0.040 to +0.100 have been computed.[10] The process of determining the constants of the translated normal curve consists first in computing β and ϵ from the given data, and then in entering the table and interpolating for the corresponding values of κ and λ.[11] On substituting these values in (2) the value of a can be found and thence on multiplying a by κ the position of the median of the distribution is obtained.