International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 198
Section 2.1.6.3. Intensities scaled to the local average^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
When the 's are a subset of the 's, the beta distributions of the second kind are replaced by beta distributions of the first kind, with means and variances readily found from Table 2.1.5.1. The distribution of such a ratio is chiefly of interest when Y relates to a single reflection and Z relates to a group of m intensities including Y. This corresponds to normalizing intensities to the local average. Its distribution is in the acentric case, with an expected value of of unity; there is no bias, as is obvious a priori. The variance of is which is less than the variance of the intensities normalized to an `infinite' population by a fraction of the order of . Unlike the variance of the scaling factor, the variance of the normalized intensity approaches unity as n becomes large. For intensities having a centric distribution, the distribution normalized to the local average is given by with an expected value of of unity and with variance less than that for an `infinite' population by a fraction of about .
Similar considerations apply to intensities normalized to in the usual way, since they are equal to those normalized to multiplied by .