Median Rank

Techniques

Cumulative Distribution Function

Cumulative Hazard Function

Weibull Distribution

Normal Distribution

Lognormal Distribution

Exponential Distribution

Exam

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The cumulative distribution function, F(x), is usually estimated from the median rank, but other estimates such as the mean rank and the Kaplan-Meier product limit estimator are also used. The median rank estimate for F(x) is

where Oi is the modified order of failure of the ith data point.

A modified order of failure is only needed if censored data is involved; if not the original order of failure, i, is equivalent to the modified order of failure. The logic for a modified order of failure is as follows. Consider three items; the first was tested for 3 hours and the test was stopped without failure, the second item was tested and failed after 4 hours, and the third item was tested and failed after 4.5 hours. For this data set the failure order is unclear. The first item could have been either the first failure, the second failure or the third failure; thus it is not certain that the first item to fail, the second item, is the first ordered failure The modified order of failure is computed from the expression

where I_i is the increment for the ith failure, and is computed from the expression

Where n is the total number of points in the data set, both censored and uncensored,
c is the number of points remaining in the data set including the current point, and
Op is the order of the previous failure.

Example
Construct a probability plot for the failure data below given that an additional 7 items were tested for 149 cycles without failure.

43, 67, 92, 94, 149

Solution
The table below contains the calculations necessary for plotting.