Difference between revisions of "Template:MedianRanks"

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==== Median Ranks ====
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==== Median Ranks ====
  
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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Median ranks are used to obtain an estimate of the unreliability&lt;span class="texhtml" /&gt; 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:  
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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}

Revision as of 21:20, 6 March 2012

Median Ranks

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, Q(Tj), should have at the jt'hfailure 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'hfailure 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.