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| ==== Median Ranks ====
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| 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>failure out of a sample of ''N ''units at the 50% confidence level.
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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:
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| ::<math>P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
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| N \\
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| k \\
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| \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>
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| where <span class="texhtml">''N''</span> is the sample size and <span class="texhtml">''j''</span> the order number.
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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>
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| ::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
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| N \\
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| k \\
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| \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 for the value of ''Z'' four times; once for each failure with ''j= 1, 2, 3'' and ''4''. This result can then be used as the unreliability estimate for each failure or the ''y'' plotting position. (See also the chapter [[The Weibull Distribution|The Weibull distribution]] for a step-by-step example of this method.) The solution of cumuative binomial equation for ''Z'' requires the use of numerical methods.
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