|
|
| Line 1: |
Line 1: |
| {{InProgress|}}
| | #REDIRECT [[Parameter_Estimation#Median_Ranks]] |
| | |
| | |
| 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 10 failures at 100 hours, the median rank, <math>Z,</math> is estimated by using:
| |
| | |
| ::<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>
| |
| | |
| | |
| with:
| |
| | |
| ::<math>N=30,\text{ }J=10</math>
| |
| | |
| | |
| where one <math>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>Z</math> is estimated by using:
| |
| | |
| ::<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>
| |
| | |
| | |
| where:
| |
| | |
| ::<math>N=30,\text{ }J=20</math>
| |
| | |
| | |
| to represent the probability of 20 failures out of 30.
| |
| For 10 failures at 300 hours, <math>Z</math> is estimated by using:
| |
| | |
| ::<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>
| |
| | |
| | |
| where:
| |
| | |
| ::<math>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>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>1-\alpha</math> is the confidence limit.
| |
| | |
| The Median Rank is obained by setting:
| |
| | |
| ::<math>1-\alpha=0.50</math>
| |
| or
| |
| ::<math>M{{R}_{i}}=\frac{\frac{i}{N-i+1}}{{{F}_{0.50 ,2(N-i+1),2i}}+\frac{i}{N-i+1}}</math>
| |