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 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>'''''f'''''ailure out of a sample of ''N ''units at the 50% confidence level.
 
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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