© Barringer & Associates, Inc Weibull Probability Paper For Quick Hand Plots Plus Two Worked Out Examples Paul Barringer, P.E Barringer & Associates, Inc. P.O. Box 3985 Humble, TX Phone: FAX: Web:

© Barringer & Associates, Inc You’ve got 6 data points (N=6) representing age-to-failure which is [(death date) – (birth date)]: Step 1, Sort  69 X-data Calculate the Y-plot position using Benard’s median rank = (i-0.3)/(N+0.4) for N=6: i=1 10.9% i=2 26.6% i=3 --Step 2  42.2% Y-Data i=4 57.8% i=5 73.4% i=6=N 89.1% Step 3  Plot the data on 1:1 Weibull paper (see attached Weibull paper) Step 4  Draw a trend line through the data Step 5  Where the trend line crosses 63.2%, drop down and read eta which is the characteristic life Step 6  On the 1:1 paper literally measure the rise over run of the trend line to calculate beta which is the Weibull slope or shape factor See the worked out example on the next page for the case of no censored (suspended) data. For plotting position details see plot position papers at All Data Are Failures

© Barringer & Associates, Inc mm = rise 57mm = run  = rise/run = 134/57 = 2.35  = 92 W/1:1 Age-To-Failure weeks Weibull Example-All Failures 3 © Barringer & Associates, Inc. 2005

4 By Computer Calculations  =  = Goodness of fit criteria: R 2 = 0.95 (CCC 2 =  OK) PVE=70.8% (PVE>10%  OK Weibull Example-All Failures 4 © Barringer & Associates, Inc Age-T-Failure weeks

© Barringer & Associates, Inc Computer Regression The New Weibull Handbook, 5 th edition by Dr. Robert B. Abernethy, ISBN , 2006 describes best practices for small datasets: –Regress Y-on-to-X using median rank regression –Use PVE% for goodness of fit For medium data sets (say 10 to 100 data) –If maximum likelihood estimation is required, use the reduced bias adjustment to correct for small number of failures For large size data sets (say over 500 data) –Use maximum likelihood estimation For grouped or interval data use the inspection option –Regress data through top most data in a stack

© Barringer & Associates, Inc You’ve got 7 data points (N=7) representing cycles-to-failure for the wire (W) failure mode (bond (B) failure mode is suspended) >Signifies suspension 28W >12B  No plot >12B 18W >50B --Step 1, Sort  >22B  No plot 36W 28W X-data >22B 36W 42W 42W 18W >50B  No plot Calculate the Y-plot position using Benard’s median rank = (i-0.3)/(N+0.4) for N=7: Inverse Rank Rank Wire Mode i=1 7 No plot i= % i=3 5 --Step 2  No plot i= % Y-Data i= % i= % i=7=N 1 No plot Step 3  Use Drew Auth’s correction for the failure data following the suspension by correcting the “i” value in Bernard’s median rank. (i-th adjustment) = {(Inverse Rank)*(Previous Adjusted Rank)+(N+1)}/{(Inverse Rank)+1} = {6*0+(7+1)}/{6+1} = 8/7 =  ( )/(7+0.4) = 0.843/7.4 = 11.4% {4*1.143+(7+1)}/{4+1} =  ( )/(7+0.4) = 29.9% {3*2.514+(7+1)}/{3+1} =  ( )/(7+0.4) = 48.5% {2*3.886+(7+1)}/{2+1} =  ( )/(7+0.4) = 67.0% Step 4  Follow steps 3-6 from uncensored data. Then see next page for plot Failure Data Plus Censored Data (Suspensions)

© Barringer & Associates, Inc mm = rise 50mm = run  = 41  = 128/50 = 2.56  = 41 Weibull Example With Suspensions Cycles-To-Failure cycles W/1:1 6 © Barringer & Associates, Inc. 2005

8 By Computer Calculations  =  = 41.3 Goodness of fit criteria: R 2 = (CCC 2 =0.783  OK) PVE=96.07% (PVE>10%  OK Cycles-To-Failure Cycles Weibull Example With Suspensions 7 © Barringer & Associates, Inc. 2005

9 8 4 Cycle Weibull Paper © Barringer & Associates, Inc

10 3 Cycle Weibull Paper © Barringer & Associates, Inc. 2005

11 2 Cycle Weibull Paper © Barringer & Associates, Inc. 2005

12 1 Cycle Weibull Paper © Barringer & Associates, Inc. 2005

13 Automate! Life is too short to make these plots by hand! Automate (the automation remembers all the rules for making Weibull plots)! Use SuperSMITH Weibull software. See for details and for costs. SuperSMITH Weibull follows all the requirements of The New Weibull Handbook. See for details.