Difference between revisions of "Template:MedianRanks"

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(Created page with '====Median Ranks==== Median ranks are used to obtain an estimate of the unreliability, <math>Q({T_j})</math> for each failure. It is the value that the true probability of failur…')
 
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====Median Ranks====
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==== Median Ranks ====
Median ranks are used to obtain an estimate of the unreliability, <math>Q({T_j})</math> for each failure. It is the value that the true probability of failure, <math>Q({{T}_{j}}),</math> should have at the <math>{{j}^{th}}</math> failure out of a sample of <math>N</math> units at a ''50%'' confidence level. This essentially means that this is our best estimate for the unreliability. Half of the time the true value will be greater than the 50% confidence estimate, the other half of the time the true value will be less than the estimate. This estimate is based on a solution of the binomial equation.
 
  
The rank can be found for any percentage point, <math>P</math>, greater than zero and less than one, by solving the cumulative binomial equation for <math>Z</math> . This represents the rank, or unreliability estimate, for the <math>{{j}^{th}}</math> failurein the following equation for the cumulative binomial:
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Median ranks are used to obtain an estimate of the unreliability<span class="texhtml" /> for each failure. It is the value that the true probability of failure, <span class="texhtml">''Q''(''T''<sub>''j''</sub>),</span> should have at the <span class="texhtml">''j''<sup>''t''''h''</sup></span> failure out of a sample of <span class="texhtml">''N''</span> units at a ''50%'' confidence level. This essentially means that this is our best estimate for the unreliability. Half of the time, the true value will be greater than the 50% confidence estimate; on the other half, the true value will be less than the estimate. This estimate is based on a solution of the binomial equation.
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The rank can be found for any percentage point, <span class="texhtml">''P''</span>, greater than zero and less than one, by solving the cumulative binomial equation for <span class="texhtml">''Z''</span> . This represents the rank, or unreliability estimate, for the <span class="texhtml">''j''<sup>''t''''h''</sup></span> failure in the following equation for the cumulative binomial:  
  
 
::<math>P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
 
::<math>P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
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\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
 
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
  
where <math>N</math> is the sample size and <math>j</math> the order number.
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where <span class="texhtml">''N''</span> is the sample size and <span class="texhtml">''j''</span> the order number.  
  
The median rank is obtained by solving this equation for <math>Z</math> at <math>P=0.50,</math>
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The median rank is obtained by solving this equation for <span class="texhtml">''Z''</span> at <span class="texhtml">''P'' = 0.50,</span>  
  
 
::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
 
::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
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\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
 
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
  
 
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<br> For example, if ''N=4'' and we have four failures, we would solve the median rank equation four times; once for each failure with ''j=1, 2, 3'' and ''4'', for the value of ''Z''. This result can then be used as the unreliability estimate for each failure or the ''y'' plotting position. (See also the The Weibull distribution chapter for a step-by-step example of this method.) The solution of cumuative binomial equation for ''Z'' requires the use of numerical methods.
For example, if ''N=4'' and we have four failures, we would solve the median rank equation four times; once for each failure with ''j=1, 2, 3'' and ''4'', for the value of ''Z''. This result can then be used as the unreliability estimate for each failure or the ''y'' plotting position. (See also the The Weibull distribution chapter for a step-by-step example of this method.) The solution of cumuative binomial equation for ''Z'' requires the use of numerical methods.
 

Revision as of 21:12, 6 March 2012

Median Ranks

Median ranks are used to obtain an estimate of the unreliability for each failure. It is the value that the true probability of failure, Q(Tj), should have at the jt'h failure out of a sample of N units at a 50% confidence level. This essentially means that this is our best estimate for the unreliability. Half of the time, the true value will be greater than the 50% confidence estimate; on the other half, the true value will be less than the estimate. This estimate is based on a solution of the binomial equation.

The rank can be found for any percentage point, P, greater than zero and less than one, by solving the cumulative binomial equation for Z . This represents the rank, or unreliability estimate, for the jt'h failure in the following equation for the cumulative binomial:

[math]P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}[/math]

where N is the sample size and j the order number.

The median rank is obtained by solving this equation for Z at P = 0.50,

[math]0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}[/math]


For example, if N=4 and we have four failures, we would solve the median rank equation four times; once for each failure with j=1, 2, 3 and 4, for the value of Z. This result can then be used as the unreliability estimate for each failure or the y plotting position. (See also the The Weibull distribution chapter for a step-by-step example of this method.) The solution of cumuative binomial equation for Z requires the use of numerical methods.