Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 3.2, pp. 353-359   | 1 | 2 |

Chapter 3.2. The least-squares plane

R. E. Marsha* and V. Schomakerb

aThe Beckman Institute–139–74, California Institute of Technology, 1201 East California Blvd, Pasadena, California 91125, USA, and  bDepartment of Chemistry, University of Washington, Seattle, Washington 98195, USA
Correspondence e-mail:


Arley, N. & Buch, K. R. (1950). Introduction to the theory of probability and statistics. New York: John Wiley. London: Chapman & Hall.
Clews, C. J. B. & Cochran, W. (1949). The structures of pyrimidines and purines. III. An X-ray investigation of hydrogen bonding in aminopyrimidines. Acta Cryst. 2, 46–57.
Deming, W. E. (1943). Statistical adjustment of data. New York: John Wiley. [First edition 1938.]
Hamilton, W. C. (1961). On the least-squares plane through a set of points. Acta Cryst. 14, 185–189.
Hamilton, W. C. (1964). Statistics in physical science. New York: Ronald Press.
Ito, T. (1981a). Least-squares refinement of the best-plane parameters. Acta Cryst. A37, 621–624.
Ito, T. (1981b). On the least-squares plane through a group of atoms. Sci. Pap. Inst. Phys. Chem. Res. Saitama, 75, 55–58.
Ito, T. (1982). On the estimated standard deviation of the atom-to-plane distance. Acta Cryst. A38, 869–870.
Kalantar, A. H. (1987). Slopes of straight lines when neither axis is error-free. J. Chem. Educ. 64, 28–29.
Lybanon, M. (1984). A better least-squares method when both variables have uncertainties. Am. J. Phys. 52, 22–26.
Robertson, J. M. (1948). Bond-length variations in aromatic systems. Acta Cryst. 1, 101–109.
Schomaker, V. & Marsh, R. E. (1983). On evaluating the standard deviation of Ueq. Acta Cryst. A39, 819–820.
Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959). To fit a plane or a line to a set of points by least squares. Acta Cryst. 12, 600–604.
Shmueli, U. (1981). On the statistics of atomic deviations from the `best' molecular plane. Acta Cryst. A37, 249–251.
Waser, J. (1973). Dyadics and variances and covariances of molecular parameters, including those of best planes. Acta Cryst. A29, 621–631.
Waser, J., Marsh, R. E. & Cordes, A. W. (1973). Variances and covariances for best-plane parameters including dihedral angles. Acta Cryst. B29, 2703–2708.
Whittaker, E. T. & Robinson, G. (1929). The calculus of observations. London: Blackie.