Median Ranks

Median ranks are used to obtain an estimate of the unreliability, [math]\displaystyle{ Q({{T}_{j}}), }[/math] for each failure at a [math]\displaystyle{ 50% }[/math] 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 ([math]\displaystyle{ Z }[/math] 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, [math]\displaystyle{ Z, }[/math] is estimated by using:

[math]\displaystyle{ 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]

with:

[math]\displaystyle{ N=30,\text{ }J=10 }[/math]

where one [math]\displaystyle{ Z }[/math] is obtained for the group, to represent the probability of 10 failures occurring out of 30. For 10 failures at 200 hours, [math]\displaystyle{ Z }[/math] is estimated by using:

[math]\displaystyle{ 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]

where:

[math]\displaystyle{ N=30,\text{ }J=20 }[/math]

to represent the probability of 20 failures out of 30. For 10 failures at 300 hours, [math]\displaystyle{ Z }[/math] is estimated by using:

[math]\displaystyle{ 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]

where:

[math]\displaystyle{ N=30,\text{ }J=30 }[/math]

to represent the probability of 30 failures out of 30.

Alternative Computation

Any rank can be computed by:

[math]\displaystyle{ M{{R}_{i}}=\frac{\frac{i}{N-i+1}}{{{F}_{1-\alpha ,2(N-i+1),2i}}+\frac{i}{N-i+1}} }[/math]

• where F is the F-distribution and
• [math]\displaystyle{ 1-\alpha }[/math] is the confidence limit.

The Median Rank is obained by setting:

[math]\displaystyle{ 1-\alpha=0.50 }[/math]

or

[math]\displaystyle{ M{{R}_{i}}=\frac{\frac{i}{N-i+1}}{{{F}_{0.50 ,2(N-i+1),2i}}+\frac{i}{N-i+1}} }[/math]