Template:MedianRanks

Median Ranks

Median ranks are used to obtain an estimate of the unreliability, [math]\displaystyle{ Q({T_j}) }[/math] for each failure. It is the value that the true probability of failure, [math]\displaystyle{ Q({{T}_{j}}), }[/math] should have at the [math]\displaystyle{ {{j}^{th}} }[/math] failure out of a sample of [math]\displaystyle{ N }[/math] 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, the other half of the time 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, [math]\displaystyle{ P }[/math], greater than zero and less than one, by solving the cumulative binomial equation for [math]\displaystyle{ Z }[/math] . This represents the rank, or unreliability estimate, for the [math]\displaystyle{ {{j}^{th}} }[/math] failurein 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 [math]\displaystyle{ N }[/math] is the sample size and [math]\displaystyle{ j }[/math] the order number.

The median rank is obtained by solving this equation for [math]\displaystyle{ Z }[/math] at [math]\displaystyle{ P=0.50, }[/math]

[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.