Chapter 8.2 Other refinement methods

Mathematical, physical and chemical tables

First Online Edition (2006)

Part 8. Refinement of structural parameters

  1. E. Prince1,
  2. D. M. Collins2

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000610

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Prince, E. and Collins, D. M. 2006. Other refinement methods. International Tables for Crystallography. C:8:8.2:689–692.

Author Information

  1. 1

    NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

  2. 2

    Laboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 20375-5341, USA

Publication History

  1. Published Online: 1 JAN 2006


Least squares is a powerful data fitting method when the distribution of statistical fluctuation in the data is approximately normal, or Gaussian, but it can perform poorly if the distribution function has longer tails than a Gaussian distribution. Chapter 8.2 discusses several procedures that work better than least squares if the normality condition is not satisfied. Maximum likelihood methods, which are identical to least squares for a normal distribution, can be designed to be optimum for any distribution. Other methods are robust, because they work well over a broad range of distributions, and resistant, because they are insensitive to the presence in the data of points that disagree with the model. Maximum entropy methods are particularly useful when there are insufficient data.


  • entropy maximization;
  • maximum-likelihood methods;
  • robust/resistant methods