Exact conditional distribution of a three-phase invariant in the space group P1. II. Calculations and comparison with the Cochran approximation

The conditional probability density function of a three-phase invariant is computed from exact expressions derived and discussed in the first paper of this series [Shmueli, Rabinovich & Weiss (1989). Acta Cryst. A45, 361-367] and comparisons are presented of these computations with the approximate conditional density due to Cochran [Acta Cryst. (1955), 8, 473-478]. Conditional variances computed from the exact and approximate expressions are also compared. The computations are carried out for the space group P1. This is the first numerical comparison of conditional phase-invariant statistics evaluated from exact and approximate expressions. The discrepancy between these two kinds of statistics appears to be negligible if the E values involved are small and the number of atoms in the cell is moderately large. Very significant discrepancies are observed when the number of the atoms in the unit cell is small, as well as in the case of the presence of an outstandingly heavy atom. For a given triplet of magnitudes of E values or a fixed atomic composition of the unit cell the exact conditional density functions tend to have sharper peaks than the approximate ones.