Weibull Analysis - Median Rank

Discussion in 'Probability' started by Robert Scott, Jan 13, 2009.

  1. Robert Scott

    Robert Scott Guest

    Weibull statistical analysis is used to project failure rates on a population of
    identical parts based on a sample of failure times. Part of the calculation
    involves something called the median rank. For a complete explanation see this
    reference:

    http://www.mathpages.com/home/kmath122/kmath122.htm

    So I decided to develop my own implementation of the analysis, including the
    calculation of median rank. I wanted to calculate the value from the
    definition, which involves numerical iteration.

    The only on-line reference I could find for median rank tables was this page:

    http://www.weibull.com/GPaper/hread.htm

    (look only at the 50% column - that's the median rank). My calculations are
    pretty close to those in the tables, In fact we agree up to sample size = 14.
    But starting with sample size = 15 and higher, we start to diverge a little.

    My question is, who is right? The paper at www.weibull.com, or me? I took
    great care to use 64-bit double precision calculations to make everything as
    precise as I could. What I am looking for is a tie-breaker. Does anyone have a
    source for median rank tables that they could check to see who is right?

    I won't bother posting my tables for sample sizes 2-14, since they agree with
    the cited on-line table, but here are my findings for sample sizes 15-25:

    Sample Size 15

    1 0.0451584
    2 0.1093961
    3 0.1743207
    4 0.2393934
    5 0.3045198
    6 0.3696705
    7 0.4348328
    8 0.5000000
    9 0.5651672
    10 0.6303295
    11 0.6954802
    12 0.7606066
    13 0.8256793
    14 0.8906039
    15 0.9548416

    Sample Size 16

    1 0.0423967
    2 0.1027030
    3 0.1636542
    4 0.2247447
    5 0.2858859
    6 0.3470504
    7 0.4082265
    8 0.4694083
    9 0.5305917
    10 0.5917735
    11 0.6529496
    12 0.7141141
    13 0.7752553
    14 0.8363458
    15 0.8972970
    16 0.9576033

    Sample Size 17

    1 0.0399533
    2 0.0967816
    3 0.1542176
    4 0.2117850
    5 0.2694005
    6 0.3270381
    7 0.3846872
    8 0.4423423
    9 0.5000000
    10 0.5576577
    11 0.6153128
    12 0.6729619
    13 0.7305995
    14 0.7882150
    15 0.8457824
    16 0.9032184
    17 0.9600467

    Sample Size 18

    1 0.0377762
    2 0.0915057
    3 0.1458097
    4 0.2002382
    5 0.2547122
    6 0.3092075
    7 0.3637138
    8 0.4182263
    9 0.4727418
    10 0.5272582
    11 0.5817737
    12 0.6362862
    13 0.6907925
    14 0.7452878
    15 0.7997618
    16 0.8541903
    17 0.9084943
    18 0.9622238

    Sample Size 19

    1 0.0358240
    2 0.0867752
    3 0.1382712
    4 0.1898852
    5 0.2415426
    6 0.2932202
    7 0.3449086
    8 0.3966030
    9 0.4483008
    10 0.5000000
    11 0.5516992
    12 0.6033970
    13 0.6550914
    14 0.7067798
    15 0.7584574
    16 0.8101148
    17 0.8617288
    18 0.9132248
    19 0.9641760

    Sample Size 20

    1 0.0340637
    2 0.0825097
    3 0.1314737
    4 0.1805500
    5 0.2296676
    6 0.2788046
    7 0.3279519
    8 0.3771052
    9 0.4262619
    10 0.4754205
    11 0.5245795
    12 0.5737381
    13 0.6228948
    14 0.6720481
    15 0.7211954
    16 0.7703324
    17 0.8194500
    18 0.8685263
    19 0.9174903
    20 0.9659363

    Sample Size 21

    1 0.0324682
    2 0.0786438
    3 0.1253131
    4 0.1720895
    5 0.2189054
    6 0.2657397
    7 0.3125841
    8 0.3594343
    9 0.4062879
    10 0.4531435
    11 0.5000000
    12 0.5468565
    13 0.5937121
    14 0.6405657
    15 0.6874159
    16 0.7342603
    17 0.7810946
    18 0.8279105
    19 0.8746869
    20 0.9213562
    21 0.9675318

    Sample Size 22

    1 0.0310155
    2 0.0751240
    3 0.1197041
    4 0.1643864
    5 0.2091065
    6 0.2538444
    7 0.2985919
    8 0.3433450
    9 0.3881016
    10 0.4328602
    11 0.4776200
    12 0.5223800
    13 0.5671398
    14 0.6118984
    15 0.6566550
    16 0.7014081
    17 0.7461556
    18 0.7908935
    19 0.8356136
    20 0.8802959
    21 0.9248760
    22 0.9689845

    Sample Size 23

    1 0.0296872
    2 0.0719057
    3 0.1145755
    4 0.1573433
    5 0.2001472
    6 0.2429682
    7 0.2857985
    8 0.3286342
    9 0.3714733
    10 0.4143145
    11 0.4571570
    12 0.5000000
    13 0.5428430
    14 0.5856855
    15 0.6285267
    16 0.6713658
    17 0.7142015
    18 0.7570318
    19 0.7998528
    20 0.8426567
    21 0.8854245
    22 0.9280943
    23 0.9703128

    Sample Size 24

    1 0.0284681
    2 0.0689518
    3 0.1098684
    4 0.1508789
    5 0.1919241
    6 0.2329856
    7 0.2740562
    8 0.3151320
    9 0.3562111
    10 0.3972924
    11 0.4383750
    12 0.4794582
    13 0.5205418
    14 0.5616250
    15 0.6027076
    16 0.6437889
    17 0.6848680
    18 0.7259438
    19 0.7670144
    20 0.8080759
    21 0.8491211
    22 0.8901316
    23 0.9310482
    24 0.9715319

    Sample Size 25

    1 0.0273451
    2 0.0662310
    3 0.1055327
    4 0.1449246
    5 0.1843499
    6 0.2237909
    7 0.2632406
    8 0.3026954
    9 0.3421534
    10 0.3816136
    11 0.4210750
    12 0.4605373
    13 0.5000000
    14 0.5394627
    15 0.5789250
    16 0.6183864
    17 0.6578466
    18 0.6973046
    19 0.7367594
    20 0.7762091
    21 0.8156501
    22 0.8550754
    23 0.8944673
    24 0.9337690
    25 0.9726549



    Robert Scott
    Ypsilanti, Michigan
     
    Robert Scott, Jan 13, 2009
    #1
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