The area between the two ordinates corresponding to t = ±3 is 0.99998 of the total area under the curve, so that when neither of the minimum points corresponds to points closer than three units to the origin of the base curve the curve may be practically valuable. A moment’s consideration will show that the abscissas of the two minimum points must be practically the same as that of the corresponding infinite ordinates. The roots of the quadratic
3λt2 + 2κt + 1 = 0
are numerically greater than 3 for all pairs of values of (λ, κ) lying above the line
27λ - 6κ + 1 = 0
As statistically promising within the discriminant of the cubic we then have the shaded area of the (λ, κ) plane.
The Origin. The generating curve is the symmetrical normal probability curve with origin at its center. Since x = 0 when t = 0, the origin of the translated curve coincides with that of the base or generating curve. The translated curve may not be symmetrical so that the mean ordinate may not coincide with the modal ordinate. Because of the relation between corresponding areas the ordinate at the origin must continue to divide the area under the curve into equal parts, that is, the origin and median always coincide.
Determination of the Constants. Since the exact position of the median can not ordinarily be determined by inspection or direct computation there are in reality four constants to be determined: the distance between the median and the mean, a, κ and λ.
In determining the constants it is usual to compute the value of the first four moments. The third and fourth moments are extensions of the idea of the well known formulas for the first and second moments. Denoting the moments about the median by μ, we have
where N is the total area under the curve.