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Historically, R-factors were introduced by ...  
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Historically, R-factors were introduced by ... ???
    
== Definitions ==
 
== Definitions ==
 
=== Data quality indicators ===
 
=== Data quality indicators ===
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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
 
* R<sub>sym</sub> and R<sub>merge</sub> : the formula for both is
 
<math>
 
<math>
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<br>
 
<br>
 
<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.
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where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection
* Redundancy-independant version of the above: R<sub>meas</sub>
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* Redundancy-independant version of the above:  
* measuring quality of averaged intensities/amplitudes: R<sub>p.i.m.</sub> and R<sub>mrgd-F</sub>
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<math>
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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}}
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</math>
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<br>
 +
<br>
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* measuring quality of averaged intensities/amplitudes:
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for intensities use
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<math>
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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}}
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</math>
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<br>
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<br>
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and similarly for amplitudes:
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<math>
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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}}
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</math>
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<br>
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<br>
 +
 
    
=== Model quality indicators ===
 
=== Model quality indicators ===
 
* R and R<sub>free</sub> : the formula for both is  
 
* R and R<sub>free</sub> : the formula for both is  
 
<math>
 
<math>
R=\frac{\sum_{hkl_{unique}}\vert F_{hkl}^{(obs)}-F_{hkl}^{(calc)}\vert}{\sum_{hkl_{unique}} F_{hkl}^{(obs)}}
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R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}}
 
</math>
 
</math>
 
<br>
 
<br>
 
<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]].
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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? ==
 
== what do R-factors try to measure, and how to interpret their values? ==
 
* relative deviation of
 
* relative deviation of
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