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Frequency Distributions of π (Pi) from
100 to 3-Billion Decimal Digits

and

How To Compute CHI-SQUARE ( χ ² )

(Note: Chi-Square is already Given for Each of
the 55 Sample Distributions of π Listed below)

For a definition of Chi-Square and instructions on how to compute it for
each sample distribution of π, see the discussion following the Table.

Here's a simplified meaning for Chi-Square (as applied to the frequency distributions of its digits; see Table 1 below):

The smaller the number, the closer π (Pi) is to being evenly distributed among all its digits.

So, for the first seven cases listed below, we attain the most even distribution of digits in case#1 (where 100 digits gives us a Chi-Square of only 4.20). The worst of these seven cases being for 5000 digits (where Chi-Square = 10.77). If you can find a Chi-Square value for one of Pi's distributions that's smaller than 2.785 or larger than 10.77, please let me know; I may add it to the list.


Table 1. Various Frequency Distributions for the Decimal Digits of π (Pi).


    CHI- |
  SQUARE |          DIGITS :         0          1          2          3          4          5          6          7          8          9
  ------------------------------------------------------------------------------------------------------------------------------------------
   4.20  |  to         100 |         8          8         12         11         10          8          9          8         12         14
   6.80  |  to         200 |        19         20         24         19         22         20         16         12         25         23
   6.88  |  to         500 |        45         59         54         50         53         50         48         36         53         52
   5.12  |  to         800 |        74         92         83         79         80         73         77         75         76         91
   4.74  |  to        1000 |        93        116        103        102         93         97         94         95        101        106
  ------------------------------------------------------------------------------------------------------------------------------------------
   4.34  |  to        2000 |       182        212        207        188        195        205        200        197        202        212
  10.77  |  to        5000 |       466        532        496        459        508        525        513        488        492        521
   8.52  |  to        8000 |       754        833        811        781        809        834        816        786        764        812
   9.318 |  to       10000 |       968       1026       1021        974       1012       1046       1021        970        948       1014
  ------------------------------------------------------------------------------------------------------------------------------------------
   7.72  |  to       20000 |      1954       1997       1986       1986       2043       2082       2017       1953       1962       2020
   5.86  |  to       50000 |      5033       5055       4867       4947       5011       5052       5018       4977       5030       5010
   4.46  |  to       80000 |      7972       8141       7920       7975       7957       8044       8026       8031       7953       7981
   4.09  |  to      100000 |      9999      10137       9908      10025       9971      10026      10029      10025       9978       9902
  ------------------------------------------------------------------------------------------------------------------------------------------
   7.31  |  to      200000 |     20104      20063      19892      20010      19874      20199      19898      20163      19956      19841
   7.73  |  to      500000 |     49915      49984      49753      50000      50357      50235      49824      50230      49911      49791
   6.27  |  to      800000 |     79949      79851      79872      79962      80447      80298      79650      79884      80167      79920
   5.51  |  to     1000000 |     99959      99758     100026     100229     100230     100359      99548      99800      99985     100106
  ------------------------------------------------------------------------------------------------------------------------------------------
   9.00  |  to     2000000 |    199792     199535     200077     200141     200083     200521     199403     200310     199447     200691
   7.88  |  to     5000000 |    499620     499898     499508     499933     500544     500025     498758     500880     499880     500954
   3.79  |  to     8000000 |    799111     800110     799788     800234     800202     800154     798885     800560     800638     800318
   2.785 |  to    10000000 |    999440     999333    1000306     999964    1001093    1000466     999337    1000207     999814    1000040
  ------------------------------------------------------------------------------------------------------------------------------------------
   4.17  |  to    20000000 |   2001162    1999832    2001409    1999343    2001106    2000125    1999269    1998404    1999720    1999630
   6.17  |  to    50000000 |   4999632    5002220    5000573    4998630    5004009    4999797    4998017    4998895    4998494    4999733
   5.95  |  to    80000000 |   7998807    8002788    8001828    7997656    8003525    7996500    7998165    7999389    8000308    8001034
   7.27  |  to   100000000 |   9999922   10002475   10001092    9998442   10003863    9993478    9999417    9999610   10002180    9999521
  ------------------------------------------------------------------------------------------------------------------------------------------
   4.90  |  to   150000000 |  14998689   15001880   15001586   14999130   15003829   14993562   14998434   14999462   15001416   15002012
   4.13  |  to   200000000 |  19997437   20003774   20002185   20001410   19999846   19993031   19999161   20000287   20002307   20000562
   3.55  |  to   300000000 |  29998356   30000582   30006337   29999867   29999810   29993099   29998913   29999071   30003683   30000282
   7.19  |  to   400000000 |  39996048   39997375   40011791   39995030   40001014   39992123   40001899   40000314   40005735   39998671
   7.42  |  to   500000000 |  49995279   50000437   50011436   49992409   50005121   49990678   49998820   50000320   50006632   49998868
   8.42  |  to   600000000 |  59991725   59997597   60008591   59992558   60007991   59990211   60003895   59998772   60010958   59997702
   5.14  |  to   700000000 |  69989891   69997755   70006497   69994028   70009581   69994537   70003795   69997014   70005161   70001741
   6.62  |  to   800000000 |  79991897   79997003   80003316   79989651   80016073   79996120   80004148   79995109   80002933   80003750
   5.20  |  to   900000000 |  89991208   89998381   90000968   89990083   90013132   89996086   90006412   89995658   90001979   90006093
   4.92  |  to  1000000000 |  99993942   99997334  100002410   99986911  100011958   99998885  100010387   99996061  100001839  100000273
  ------------------------------------------------------------------------------------------------------------------------------------------
   5.46  |  to  1100000000 | 109995255  109995734  109998117  109989540  110014752  109995714  110010983  109992446  110002111  110005348
   3.76  |  to  1200000000 | 119994545  119998376  119995764  119996027  120011938  119997838  120006708  119988389  120003777  120006638
   6.57  |  to  1300000000 | 129992349  129999635  129994947  129998712  130015452  129995659  130006321  129981472  130005694  130009759
   6.63  |  to  1400000000 | 139993771  139997681  139993896  139998838  140017106  139994769  140007554  139981894  140002996  140011495
   8.33  |  to  1500000000 | 149996271  149997564  149991500  149996961  150021213  149996095  150008791  149977629  150004997  150008979
   8.59  |  to  1600000000 | 159999135  159992488  159992888  160000067  160022723  159996471  160004613  159975707  160008114  160007794
   7.74  |  to  1700000000 | 169998554  169990269  169994151  170001888  170020303  169991375  170007272  169978128  170009119  170008941
   7.37  |  to  1800000000 | 180000150  179993762  179992712  179998857  180017546  179996039  180008910  179974121  180009406  180008497
   8.07  |  to  1900000000 | 189998446  189991837  189989637  189998135  190016583  189993115  190010418  189975948  190015171  190010710
   6.69  |  to  2000000000 | 199994317  199995284  199992575  199999470  200014368  199989852  200004785  199979293  200017844  200012212
   8.10  |  to  2100000000 | 209994273  209996242  209989787  209994216  210014550  209983731  210009320  209982350  210020047  210015484
   8.56  |  to  2200000000 | 219996755  219995714  219991229  219996795  220015059  219976424  220012310  219982905  220017469  220015340
   9.33  |  to  2300000000 | 230003038  229990821  229990406  229996953  230011961  229978319  230014126  229978125  230019314  230016937
   9.72  |  to  2400000000 | 240003142  239989919  239992104  239999733  240006438  239976466  240015731  239976896  240019683  240019888
  10.62  |  to  2500000000 | 250000846  249990712  249991477  249996031  250006163  249976863  250015411  249975895  250024241  250022361
  10.77  |  to  2600000000 | 260000393  259992864  259991867  259993731  260002469  259976903  260016872  259975520  260025651  260023730
  10.19  |  to  2700000000 | 270000980  269993416  269988862  269991028  270005078  269975157  270016526  269980153  270025420  270023380
   9.00  |  to  2800000000 | 279995033  279993109  279991190  279991268  280006730  279978152  280019097  279980331  280023843  280021247
   9.63  |  to  2900000000 | 289999708  289992362  289989206  289991027  290003323  289977780  290024444  289978313  290020595  290023242
   9.24  |  to  3000000000 | 299999143  299995932  299989126  299992290  300002257  299979016  300025447  299975510  300016550  300024729

 

CHI-SQUARE ( χ²) is essentially a measurement of the difference between an expected distribution and an observed distribution.

    For the number π, mathematicians have decided that the expected distribution should be the same amount for each of the digits within π. This is due to the assumption that π is a random number and what they believe about randomness itself.
    For example, this means that the 100 decimal-digit expansion of π would be expected to have a distribution 10 for each of its 10 decimal digits (0-9).

    The formula for Chi-Square (χ²) which we will be using here is:


To compute Chi-Square for 100 decimal places of π, we:
  1. Subtract the expected frequency for each digit (0-9) from the observed frequency.
  2.      [0]          [1]           [2]          [3]          [4]
      8-10 = -2    8-10 = -2    12-10 = 2    11-10 = 1    10-10 = 0
     
         [5]          [6]          [7]           [8]          [9]
      8-10 = -2    9-10 = -1    8-10 = -2    12-10 = 2    14-10 = 4
    
  3. Square each of the values from above:
  4.  [0]    [1]    [2]    [3]    [4]    [5]    [6]    [7]    [8]    [9]
      4      4      4      1      0      4      1      4      4     16    
    
  5. Divide each of these values by the expected frequency:
  6.  [0]    [1]    [2]    [3]    [4]   [5]    [6]    [7]    [8]     [9]
     4/10   4/10   4/10   1/10    0    4/10   1/10   4/10   4/10   16/10
    
  7. Sum (add up) the results from each of these terms:
  8. SUM =
     .4  +  .4  +  .4  +  .1  +   0  +  .4  +  .1  +  .4  +  .4  +  1.6
    
    SUM = 6 x (.4) +  2 x (.1) +  1.6
        =    2.4   +    .2     +  1.6   =  4.2      so ...
    
    
  Chi-Square ( χ ² ) = 4.2    [ For the first 100 decimal places of π . ]


For only 10,000 decimal places, the calculations are:


        0      1      2      3      4      5      6      7      8      9

  1.   968   1026   1021    974   1012   1046   1021    970    948   1014
       -32    +26    +21    -26    +12    +46    +21    -30    -52    +14

  2. Square the differences...
      1024    676    441    676    144   2116    441    900   2704    196

  3. Divide each value in line #2 by 1,000.

  4. Sum-up all of the terms:

    Sum =

   = 1.024 + .676 + .441 + .676 + .144 + 2.116 + .441 + .9 + 2.704 + .196
   = 1.024 + 2 x (.676) + 2 x (.441) + .144  + 2.116  + .9 + 2.704 + .196
   = 1.024 +   1.352    +  .882    +   .144  + 2.116  + .9 + 2.704 + .196
   =      2.376         +       1.026        +     3.016   +      2.9
   =                  3.402                  +           5.916

 Chi-Square  =  9.318  [ For 10,000 decimal places of Pi.]
                -----

At 10,000,000 decimal places, the calculations are:

        0         1        2          3         4         5          6        7         8        9

  1. 999440    999333   1000306    999964    1001093   1000466    999337   1000207    999814  1000040
       -560      -667      +306       -36      +1093      +466      -663      +207      -186      +40

  2. Square the differences...

     313600    444889     93636      1296    1194649    217156    439569     42849     34596     1600

  3. Divide each value in line #2 by 1,000,000 (1 million).

  4. Sum-up all of the terms:

    Sum =
      .3136 + .444889 + .093636 + .001296 + 1.194649 + .217156 + .439569 + .042849 + .034596 +  .0016

 Chi-Square  =  2.785136  [ For 10,000,000 decimal places of Pi.]
                --------
NOTE: This is the smallest value you'll encounter for all the Chi-Square values listed in Table 1.




Back to: "The Randomness of π (Pi)?"