Tables for
Volume B
Reciprocal space
Edited by U. Shumeli

International Tables for Crystallography (2006). Vol. B, ch. 3.1, pp. 351-352   | 1 | 2 |

Section 3.1.11. Mean values

D. E. Sandsa*

aDepartment of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail:

3.1.11. Mean values

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The weighted mean of a set of quantities [X_{i}] is [\langle X \rangle = {\textstyle\sum} w_{i} X_{i} / {\textstyle\sum} w_{i}, \eqno(] where the weights are typically chosen to minimize the variance of [\langle X \rangle]. The variance may be computed from the variance–covariance matrix M of the [X_{i}] by [\sigma^{2} (\langle X \rangle) = {\bf w}^{T} {\bi M} {\bf w} / ({\textstyle\sum} w_{i})^{2}. \eqno(] Minimization of [\sigma^{2} (\langle X \rangle)] leads to weights given by [{\bf w} = {\bi M}^{-1} {\bf v}, \eqno(] where the components of vector v are all equal ([v_{i} = v_{j}] for all i and j); since ([link] and ([link] require only relative weights, we can assign [v_{i} = 1] for all i. Placing these weights in ([link] yields [\sigma^{2} (\langle X \rangle) = 1 / {\textstyle\sum} w_{i}. \eqno(] For the case of uncorrelated [X_{i}], the weights are inversely proportional to the corresponding variances [w_{i} = 1/\sigma^{2} (X_{i}). \eqno(] For the case of two correlated variables, [w_{i} = 1 / [\sigma^{2} (X_{i}) - \hbox{cov} (X_{1}, X_{2})]. \eqno(] Derivation and discussion of these equations may be found in Sands (1966[link], 1982b[link]).

The presence of systematic errors in the experimental data often results in these formulae producing estimates of the standard uncertainties of molecular dimensions that are too small; it has been suggested that such error estimates should be multiplied by 1.5 to make them more realistic (Taylor & Kennard, 1983[link]). It is essential also that averages be computed only of similar quantities, and interatomic distances corresponding to different bond orders or different environments may not represent the same physical quantities; that is, there are reasons for the discrepancies, and averaging may obscure important information. Another source of error in molecular geometry parameters determined from crystallographic measurements is thermal motion, and distances should be corrected for such effects before making comparisons (Busing & Levy, 1964[link]; Johnson, 1970[link], 1980[link]).

A discussion of the appropriateness of weighted and unweighted means may be found in Taylor & Kennard (1985[link]), which suggests that the unweighted mean might even be preferable if environmental effects are large.


Busing, W. R. & Levy, H. A. (1964). Effect of thermal motion on the estimation of bond lengths. Acta Cryst. 17, 142–146.
Johnson, C. K. (1970). The effect of thermal motion on interatomic distances and angles. In Crystallographic computing, edited by F. R. Ahmed, pp. 220–226. Copenhagen: Munksgaard.
Johnson, C. K. (1980). Thermal motion analysis. In Computing in crystallography, edited by R. Diamond, S. Ramaseshan & K. Venkatesan, pp. 14.01–14.19. Bangalore: Indian Academy of Sciences.
Sands, D. E. (1966). Transformations of variance–covariance tensors. Acta Cryst. 21, 868–872.
Sands, D. E. (1982b). Molecular geometry. In Computational crystallography, edited by D. Sayre, pp. 421–429. Oxford: Clarendon Press.
Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517–525.
Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85–89.

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