Reliability HotWire

Issue 104, October 2009

Reliability Basics

Median Rank Based on Mean Order Number

In a probability plot such as the Weibull probability plot, the points represent the "observed unreliabilities," while the straight line represents the predicted values from a model. However, one cannot observe an unreliability value; only failures or suspensions can be observed. Reliability or unreliability values must be estimated from the data. Rank methods determine the way the estimated unreliabilities are associated with the failure times. The median rank method, which is used in Weibull++, estimates unreliability values based on the failure order number and the cumulative binomial distribution. To accommodate suspension data, it is necessary to adjust the order of each failure. [2] In this article, we will provide step-by-step methods to calculate the median rank using the adjusted failure order number (mean order number).

Suppose we have the testing failure data for a product, as shown in Table 1. We want to know the median rank for each failure.

Table 1: Failure and Suspension Data

Item Number State* Life of Item (hrs)
1 S1 2,500
2 F1 2,730
3 F2 3,900
4 S2 4,100
5 F3 5,000

*F = Failure, S = Suspension

Table 1 contains three failure data points and two suspension data points. To get the rank positions for each failure, we need to evaluate the locations for the suspension items.

Let's first decide the rank position of F1. The position of the first failure F1 is affected by S1. Had S1 not been withdrawn from the test at 2,500 miles, it could have either operated successfully past 2,730 miles or failed before 2,730 miles, which would place F1 (2,730) at either first place, n=1, or second place, n=2. Thus, the failure order number for F1 will be in the range of 1 to 2.

To determine this number, we need to determine the number of possible orders of the success and failure data points for n=1 and n=2. Table 2 and Table 3 list these possible orders, or combinations.

Table 2: Combinations for F1 in the First Place

F1 in Position 1 (n=1)
1 2 3 4 5 6 7 8
F1 F1 F1 F1 F1 F1 F1 F1
S1 S1 F2 F2 F2 F2 F2 F2
F2 F2 S1 S1 S2 S2 F3 F3
S2 F3 S2 F3 S1 F3 S1 S2
F3 S2 F3 S2 F3 S1 S2 S1

 

Table 3: Combinations for F1 in the Second Place

F1 in Position 2 (n=2)
1 2
S1 S1
F1 F1
F2 F2
S2 F3
F3 S2

 

There are eight possible combinations when F1 is in position 1, and there are two possible combinations when F1 is in position 2. The most likely rank position is the average of these possible combinations in Table 2 and 3. The expected order, or the mean order number (MON), for F1 is given by:

Equation

Using the same methodology, the second failure F2 could be located in position n=2 or n=3. These possibilities are illustrated in Tables 4 and 5.

Table 4: Combinations for F2 in the Second Place

F2 in Position 2 (n=2)
1 2 3 4 5 6
F1 F1 F1 F1 F1 F1
F2 F2 F2 F2 F2 F2
S1 S2 S1 S2 F3 F3
S2 S1 F3 F3 S1 S2
F3 F3 S2 S1 S2 S1

 

Table 5: Combinations for F2 in the Third Place

F2 in Position 3 (n=3)
1 2 3 4
S1 S1 F1 F1
F1 F1 S1 S1
F2 F2 F2 F2
S2 F3 S2 F3
F3 S2 F3 S2

 

The mean order number for F2 is given as:

Equation

The possible positions for F3 are illustrated in Tables 6, 7 and 8.

Table 6: Combinations for F3 at the Third Place

F3 in Position 3 (n=3)
1 2
F1 F1
F2 F2
F3 F3
S1 S2
S2 S1

 

Table 7: Combinations for F3 at the Fourth Place

F3 in Position 4 (n=4)
1 2 3
S1 F1 F1
F1 S1 F2
F2 F2 S1
F3 F3 F3
S2 S2 S2

 

Table 8: Combinations for F3 at the Fifth Place

F3 in Position 5 (n=5)
1 2 3 4
S1 F1 F1 F1
F1 S1 F2 F2
F2 F2 S1 S2
S2 S2 S2 S1
F3 F3 F3 F3

 

The mean order number for F3 would be calculated as follows:

Equation

With the established mean order number for each failure, we can calculate the median rank positions for the failure items using the Quick Statistical Reference tool in Weibull++, as shown next.

Quick Statistical Reference

Table 9 gives the calculated median ranks for this example.

Table 9: Median Rank Position

Rank Positions for the Failure (Sample Size=5)
Failure Number MON Median Rank Position 
F1 1.2 16.60%
F2 2.4 38.80%
F3 4.2 72.30%

The probability plotting analysis for the above data using the 2-parameter Weibull distribution proves our calculations of mean order numbers and median ranks, as shown next.

Weibull Probability Plot

The concept behind this method is simple, but the calculation can sometimes be rather laborious. Over the past years, other techniques have been developed. (For more details, see Kececioglu. [1])  Here, we will introduce one of these methods. This method calculates MON using an increment, I, which is defined by: [1]

Equation

where:

  • N = the sample size, or total number of items in the test.
  • PMON = previous mean order number.
  • NIBPSS = the number of items beyond the present suspended set.
  • i = the ith failure item.

 MON is given as:

Equation

Letís calculate the previous example using this method.

For F1:

Equation

For F2:

Equation

For F3:

Equation

The MON obtained for each failure item via this method is same as from the first method, so the median rank values will also be the same.

Conclusion
In this article, we provided step-by-step examples for calculating the median rank based on the mean order number, which accounts for suspension items by using the adjusted failure rank number. The first method considers the number of possible combinations for each failure item. The second method uses an increment number for the calculation by considering the previous MON and the failure itemís present position. While the resulting MON for each failure item is identical regardless of which of these methods we use, it is clear that the second method is much simpler to implement when the data set is large.

References
[1] Kececioglu, Dimitri, Reliability & Life Testing Handbook, Vol. 1 and 2, Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1993 and 1994.  
[2] ReliaSoft Corporation, Life Data Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2007.

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