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Historically, R-factors were introduced by ... ???

== Definitions ==

=== Data quality indicators ===

In the following, all sums over hkl extend only over unique reflections with more than one observation!

* R<sub>sym</sub> and R<sub>merge</sub> : the formula for both is

<math>

<br>

<br>

where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection~~, and the first summation is over all unique reflections with more than one observation.~~* Redundancy-independant version of the above: ~~R~~<~~sub~~math>R_{meas} = \frac{\sum_{hkl} \sqrt \frac{n}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}</~~sub~~math><br><br>* measuring quality of averaged intensities/amplitudes: ~~R~~ for intensities use <~~sub~~math>R_{p.i.m~~.~~} (or R_{mrgd-I}) = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}</~~sub~~math> <br><br> and ~~R~~similarly for amplitudes: <~~sub~~math>R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert F_{hkl,j}-\langle F_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}F_{hkl,j}}</~~sub~~math><br><br>

=== Model quality indicators ===

* R and R<sub>free</sub> : the formula for both is

<math>

R=\frac{\sum_{~~hkl_{unique}~~hkl}\vert F_{hkl}^{~~(~~obs~~)~~}-F_{hkl}^{~~(~~calc~~)~~}\vert}{\sum_{~~hkl_{unique}~~hkl} F_{hkl}^{~~(~~obs~~)~~}}

</math>

<br>

<br>

where <math>F_{hkl}^{~~(~~obs~~)~~}</math> and <math>F_{hkl}^{~~(~~calc~~)~~}</math> have to be scaled w.r.t. each other. R and R<sub>free</sub> differ in the set of reflections they are calculated from: R is calculated for the [[working set]], whereas R<sub>free</sub> is calculated for the [[test set]].

== what do R-factors try to measure, and how to interpret their values? ==

* relative deviation of