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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(Tj), should have at the jt'h failure 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'h failure in the following equation for the cumulative binomial:

[math]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]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.