Statistics

Allen, F. H.; Watson, D. G.; Brammer, L.; Orpen, A. G.; Taylor, R.

doi:10.1107/97809553602060000621

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.5, p. 791

Section 9.5.2.4. Statistics

F. H. Allen,^a D. G. Watson,^a L. Brammer,^b A. G. Orpen^c and R. Taylor^a

^a Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, England,^bDepartment of Chemistry, University of Missouri–St Louis, 8001 Natural Bridge Road, St Louis, MO 63121-4499, USA, and ^cSchool of Chemistry, University of Bristol, Bristol BS8 1TS, England

9.5.2.4. Statistics

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Where there are less than four independent observations of a given bond length, then each individual observation is given explicitly in the table. In all other cases, the following statistics were generated by the program STATS.

(i) The unweighted sample mean, d, where $[d=\textstyle\sum\limits^n_{i=1}d_i/n]$ and is the ith observation of the bond length in a total sample of n observations. Recent work (Taylor & Kennard, 1983, 1985, 1986) has shown that the unweighted mean is an acceptable (even preferable) alternative to the weighted mean, where the ith observation is assigned a weight equal to $[1/\sigma^2(d_i)]$ . This is especially true (Taylor & Kennard, 1985) where structures have been pre-screened on the basis of precision.
(ii) The sample median, m. This has the property that half of the observations in the sample exceed m, and half fall short of it.
(iii) The sample standard deviation, denoted here as σ, where: $[\sigma=\textstyle\sum\limits^n_{i=1}\, [(d_i-d)^2/(n-1)]^{1/2}.]$
(iv) The lower quartile for the sample, . This has the property that 25% of the observations are less than and 75% exceed it.
(v) The upper quartile for the sample, . This has the property that 25% of the observations exceed , and 75% fall short of it.
(vi) The number (n) of observations in the sample.

The statistics given in the final table correspond to distributions for which the automatic 4σ cut-off (see above) had been applied, and any manual removal of additional outliers (an infrequent operation) has been performed. In practice, a very small percentage of observations was excluded by these methods. The major effect of removing outliers is to improve the sample standard deviation, as shown in Fig. 9.5.2.1 in which a single observation is deleted.

Figure 9.5.2.1| top | pdf |

Effect of the removal of outliers (contributors that are > 4σ from the mean) for the C—C bond in C_ar—C≡N fragments. Relevant statistics (see text) are: $[\matrix{ & d & m & \sigma &q_l & q_u & n\cr(a)\hbox{ before}\hfill & 1.445 & 1.444 & 0.012 & 1.436 & 1.448 & 32\cr (b)\hbox{ after} \hfill & 1.455 & 1.444 & 0.008 & 1.436 & 1.448 & 31.}]$

The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical, normal distribution: the mean (d) will be approximately equal to the median (m); the lower and upper quartiles [(q_l,q_u)] will be approximately symmetric about the median: $[m-q_l\simeq q_u-m]$ , and 95% of the observations may be expected to lie within ±2σ of the mean value. For a skewed distribution, d and m may differ appreciably and [q_l] and [q_u] will be asymmetric with respect to m. When a bond-length distribution is negatively skewed as in Fig. 9.5.2.2,i.e. very short values are more common than very long values, then it may be due to thermal-motion effects; the distances used to prepare the table were not corrected for thermal libration.

Figure 9.5.2.2| top | pdf |

Skewed distribution of B—F bond lengths in $[{\rm BF}_{4}^{-}]$ ions: d = 1.365, m = 1.372, σ = 0.029, q_l = 1.352, q_u = 1.390 for 84 observations. Note that d ≠ m and that q_l, q_u are asymmetrically disposed about the mean d.

In a number of cases, the initial bond-length distribution was clearly bimodal, as in Fig. 9.5.2.3(a). All cases of bimodality were resolved on chemical grounds before inclusion in the table, on the basis of hybridization, conformation-dependent conjugation interactions, etc. For example, the histogram of Fig. 9.5.2.3(a) was resolved into the two discrete unimodal distributions of Figs. 9.5.2.3(b), (c), which correspond to planar N(sp²), pyramidal N(sp³), respectively. The mean valence angle at N was used as the discriminator, with a range of 108–114° for Nsp³ and $[\ge]$ 117.5° for Nsp².

Figure 9.5.2.3| top | pdf |

Resolution of the bimodal distribution of C—N bond lengths in C_ar —N(Csp³)₂ fragments: (a) complete distribution; (b) distribution for planar N, mean valence angle at N > 117.6°; (c) distribution for pyramidal N, mean valence angle at N in the range 108–114°.

References

Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517–525.Google Scholar

Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85–89.Google Scholar

Taylor, R. & Kennard, O. (1986). Cambridge Crystallographic Data Centre. 7. Estimating average molecular dimensions from the Cambridge Structural Database. J. Chem. Inf. Comput. Sci. 26, 28–32.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 9.5, p. 791