Weibull Probability Paper For Hand Plots

Sometimes you need a quick Weibull plot when you don’t have a computer nearby.  Download the attached PowerPoint file for both worked out problems and for Weibull probability plot paper.

The most difficult portion of making a Weibull probability plot is acquiring the data.  You need age-to-failure data, easy to say but difficult to obtain, which is the same type of information which will appear on your tombstone:  (death date) – (birth data) = (age-to-failure).  Some times this age-to-failure is years, months, days, etc.  Sometimes the age-to-failure data is cycle, number of hot cycles, number of cold cycles, etc.  Sometimes the age-to-failure data is storage time, etc.  Use the measure that motivates failure and the data must be consistent. 

Here is the age-to-failure data (weeks) on the PowerPoint file:
140
  37
  69
  73
108
  63
Yes, it is another “massive” data set of 6 data points. 

Most people think this data is too small to be meaningful.  My next question is:  “How many more data points would you like to purchase using money from your pocket to cover the expenses?”  The usual answer is none! 

Small data sets mean we need to use the data we have to get our nose pointed in the general direction.  We cannot wait for the dead bodies to pile up alongside the road while we delay corrective action waiting more data.  It’s important to take action to prevent the failures!!!  Usually you’ll get a reasonable indication of corrective action from a small set of data.  By the way, in most cases you’ll never live long enough to get a precise data set.  By the time you’ve got all the data you need for a risk free decision it doesn’t matter because you are dead and gone!

Step 1 for Weibull analysis-
Rank the data in ascending (rank) order:
  37
  63
  69
  73
108
140

This is X-axis data you will need for the Weibull plot

Step 2 for Weibull analysis-
Compute a Y-axis plotting position to be used with Step 1’s X-axis data.  Use Benard’s median rank plotting position. (You can read Benard’s 1953 paper as translated by Ron Schop or many other plotting positions are summarized by Mische’s unpublished 1979 ASME paper.) 

Benard’s median rank plot position equation is (i-0.3)/(N+0.4) where i is the rank number and N is number of data points.  We have N = 6 for the dataset in Step 1 and the Y-axis plot position for i = 1 though 6 is:
i = 1     10.9%
i = 2     26.6%
i = 3     42.2%
i = 4     57.8%
i = 5     73.4%
i = 6     89.1%
This is the Y-axis data you will need for the Weibull plot.

Step 3 for Weibull analysis-
Plot the data on 1:1 Weibull probability paper.  The 1:1 scale is important because of measuring X and Y dimensions on the graph to calculate the slope of the line, b.

Figure 1: Raw Data Plot On 1:1 Weibull Probability Paper

Step 4 for Weibull analysis-
Draw a trend line by estimating the best fit with the data.

Figure 2:  Draw Best Fit Trendline

Step 5 for Weibull analysis-
Where the trend line crosses the 63.2% horizontal line, drop a vertical line to define the characteristic value, h.  The h value is a mathematical property of the Weibull plot based on this premise: if you know the value of h, all trendline will pass through this point without regard to the line slope, b.

Figure 3:  Find The Characteristic Value, h at CDF=63.2%

Step 6 for Weibull analysis-
Measure the rise over the run for the line slope, b.

Figure 4:  Calculate the line slope b

Please note, not all printers will maintain the aspect ratios for 1:1 plots.  This means the b values may be in error by ±10% so don’t expect high accuracy unless you have precision graph paper.

Can you do the Weibull trendline regression in Excel?  The answer is yes, but----. 
            Problem #1: 
At the university you learned to do Y-onto-X regressions based on the pattern of error (your largest errors were in the Y-axis):  For your experiments, you carefully controlled error in the X-axis and took your variability in the Y-axis.  For reliability issues, you’ll find your smallest errors are in the Y-axis for the plot positions and larger errors are in the X-axis concerning age-to-failure.  Properly handle data errors for a Weibull plot requires regressing the X-axis onto the Y-axis.  Else you have not handled your data collection errors properly. 
            Problem #2: 
The Excel regression details are shown in the sample Excel spreadsheet which you can download and examine.
            Problem #3: 
By the time you’ve constructed your Excel spreadsheet you could have for less cost purchased SuperSMITH software consisting of SuperSMITH Weibull (for probability plots) and SuperSMITH Visual (for reliability growth plots) software.  Furthermore, it’s unlikely you will know and remember how to handle all of the Weibull rules learned during the past 80 years on how to analyze the different constraints such as when you have suspensions (censored data), etc.

You can download a PDF copy of this Problem Of The Month by clicking here.

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Last revised July 23, 2013
© Barringer & Associates, Inc. 2009