Median Ranks: Difference between revisions

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Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),</math> for each failure at a <math>50%</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.
Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),</math> for each failure at a <math>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>Z</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.
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>Z</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.

Revision as of 14:24, 12 July 2011

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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]