Template:MedianRanks
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(Tj), should have at the jt'hfailure 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 jt'hfailure in the following equation for the cumulative binomial:
- [math]\displaystyle{ 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]\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]
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.