Median Ranks

Median ranks are used to obtain an estimate of the unreliability, $$Q({{T}_{j}}),$$ for each failure at a $$50%$$ confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure. For example, when using a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours, Weibull++ estimates the median ranks ($$Z$$ values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units. For 10 failures at 100 hours, the median rank, $$Z,$$ is estimated by using:


 * $$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}}$$

with:


 * $$N=30,\text{ }J=10$$

where one $$Z$$ is obtained for the group, to represent the probability of 10 failures occurring out of 30. For 10 failures at 200 hours, $$Z$$ is estimated by using:


 * $$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}}$$

where:


 * $$N=30,\text{ }J=20$$

to represent the probability of 20 failures out of 30. For 10 failures at 300 hours, $$Z$$ is estimated by using:


 * $$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}}$$

where:


 * $$N=30,\text{ }J=30$$

to represent the probability of 30 failures out of 30.

=Alternative Computation= Any rank can be computed by:


 * $$M{{R}_{i}}=\frac{\frac{i}{N-i+1}}{{{F}_{1-\alpha ,2(N-i+1),2i}}+\frac{i}{N-i+1}}$$


 * where F is the F-distribution and
 * $$1-\alpha$$ is the confidence limit.

The Median Rank is obained by setting:


 * $$1-\alpha=0.50$$

or
 * $$M{{R}_{i}}=\frac{\frac{i}{N-i+1}}{{{F}_{0.50 ,2(N-i+1),2i}}+\frac{i}{N-i+1}}$$