Template:Eyring-log sd: Difference between revisions

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====The Standard Deviation====
====The Standard Deviation====
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The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure),  <math>{{\sigma }_{T}}</math> , is given by:
The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure),  <math>{{\sigma }_{T}}</math> , is given by:


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The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:


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Revision as of 18:01, 9 March 2012

The Standard Deviation


The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure), [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:


[math]\displaystyle{ \begin{align} & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} =\ \sqrt{\left( {{e}^{2\left( -\ln (V)-A+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \end{align} }[/math]


The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:


[math]\displaystyle{ {{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)} }[/math]