Template:MedianRanks: Difference between revisions

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==== Median Ranks ====
[[Category: For Deletion]]
 
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.
 
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}
  N  \\
  k  \\
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
 
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 <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}
  N  \\
  k  \\
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
 
<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.

Latest revision as of 12:24, 20 July 2012

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