Template:MedianRanks

Median Ranks

Median ranks are used to obtain an estimate of the unreliability for each failure. It is the value that the true probability of failure, Q(T_j), should have at the j^t'h failure out of a sample of N units at the 50% confidence level.

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 j^t'h failure 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 chapter The Weibull distribution for a step-by-step example of this method.) The solution of cumuative binomial equation for Z requires the use of numerical methods.