Median Rank for Multiple Censored Data: Difference between revisions

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{{UConstruction}}{{Banner Weibull Reference Examples}}__NOTOC__
{{Reference Example}}
This example validates the median rank calculation for multiple censored data.
 
=Data Source=
This example validates the median rank calculation for multiple censored data in Weibull++ standard folios.
Table 3.1 on page 78 in book “''Reliability & Life Testing Handbook Vol 2''” by Dr. Kececioglu, Prentice-Hall, 1994.
 
===Data===
 
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
{{Reference Example Heading1}}
|-
Table 3.1 on page 78 in the book ''Reliability & Life Testing Handbook Vol 2'' by Dr. Kececioglu, Prentice-Hall, 1994.
|- align="center"
 
 
{{Reference Example Heading2}}
{| {{table}}
!Num. In Stage
!Num. In Stage
!State F or S
!State F or S
!Time to Failure
!Time to Failure
|- align="center"
|-
|1 ||F |||5100
|1 ||F |||5100
|- align="center"
|-
|1 ||S |||9500
|1 ||S |||9500
|- align="center"
|-
|1 ||F |||15000
|1 ||F |||15000
|- align="center"
|-
|1 ||S |||22000
|1 ||S |||22000
|- align="center"
|-
|1 ||F |||40000
|1 ||F |||40000
|}  
|}  
===Result===
 
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
 
|-
{{Reference Example Heading3}}
|- align="center"
{| {{table}}
!Num. In Stage
!Num. In Stage
!State F or S
!State F or S
!Time to Failure
!Time to Failure
!Median Rank (%)
!Median Rank (%)
|- align="center"
|-
|1 ||F |||5100 |||12.94
|1 ||F |||5100 |||12.94
|- align="center"
|-  
|1 ||S |||9500 |||
|1 ||S |||9500 |||
|- align="center"
|-  
|1 ||F |||15000 |||36.1
|1 ||F |||15000 |||36.1
|- align="center"
|-  
|1 ||S |||22000 |||
|1 ||S |||22000 |||
|- align="center"
|-  
|1 ||F |||40000 |||70.84
|1 ||F |||40000 |||70.84
|}  
|}  


=Results from Weibull++=
 
 
{{Reference Example Heading4}}
The coordinates of each point in the following plot shows the failure time and the corresponding median rank.  
The coordinates of each point in the following plot shows the failure time and the corresponding median rank.  


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The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used.
The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used:


::<math>MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4}</math>
::<math>MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4}</math>
Line 56: Line 61:




In Weibull++, the following exact method is used.
In Weibull++, the following exact method is used:


::<math>MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}}</math>
::<math>MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}}</math>


   
   
where <math>m=2(N-MON_{i}+1), n=2xMON_{i}\cdot F_{0.5,m,n}\,\!</math> is the 50 percentile of a F distribution with degree of freedom of ''m'' and ''n''.
where <math>m=2(N-MON_{i}+1), n=2\times MON_{i}\cdot F_{0.5,m,n}\,\!</math> is the 50 percentile of a F distribution with degree of freedom of ''m'' and ''n''.

Latest revision as of 16:20, 28 September 2015

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Median Rank for Multiple Censored Data

This example validates the median rank calculation for multiple censored data in Weibull++ standard folios.


Reference Case

Table 3.1 on page 78 in the book Reliability & Life Testing Handbook Vol 2 by Dr. Kececioglu, Prentice-Hall, 1994.


Data

Num. In Stage State F or S Time to Failure
1 F 5100
1 S 9500
1 F 15000
1 S 22000
1 F 40000


Result

Num. In Stage State F or S Time to Failure Median Rank (%)
1 F 5100 12.94
1 S 9500
1 F 15000 36.1
1 S 22000
1 F 40000 70.84


Results in Weibull++

The coordinates of each point in the following plot shows the failure time and the corresponding median rank.


Median Ranks Multiple Censored Data.png


The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used:

[math]\displaystyle{ MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4} }[/math]


where [math]\displaystyle{ MR_{i}\,\! }[/math] is the median rank at the [math]\displaystyle{ ith\,\! }[/math] failure time; [math]\displaystyle{ MON_{i}\,\! }[/math] is the mean order number; [math]\displaystyle{ N\,\! }[/math] is the total samples. For the step by step calculation of mean order number (MON), please refer to the book “Reliability & Life Testing Handbook Vol 2” by Dr. Kececioglu, Prentice-Hall, 1994.


In Weibull++, the following exact method is used:

[math]\displaystyle{ MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}} }[/math]


where [math]\displaystyle{ m=2(N-MON_{i}+1), n=2\times MON_{i}\cdot F_{0.5,m,n}\,\! }[/math] is the 50 percentile of a F distribution with degree of freedom of m and n.

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