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 stepbystep 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withdrawn from the test at 2,500 miles, it could have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:
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:
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:
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.
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 2parameter Weibull distribution proves our calculations
of mean order numbers and median ranks, as shown next.
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]
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 i^{th} failure item.
MON is given as:
Let’s calculate the previous example using this
method.
For F1:
For F2:
For F3:
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 stepbystep
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
item’s 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.
