# Difference between revisions of "Template:MedianRanks"

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− | ==== Median Ranks ==== | + | ==== Median Ranks ==== |

− | Median ranks are used to obtain an estimate of the unreliability | + | 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: | + | 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*(*T*_{j}), should have at the *j*^{t'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 *j*^{t'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.