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:
χ² = Σ [( ƒi
- Ni ) ² / ( Ni )]
where
ƒi is the "Observed amount"
for each digit and
Ni is the "Expected amount"
for each digit.
Σ means that we must sum
together all the results for each digit.
To compute Chi-Square for 100 decimal places of π, we:
Chi-Square ( χ ² ) =
4.2 [ For the first 100 decimal places of π . ] [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
[0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
4 4 4 1 0 4 1 4 4 16
[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
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 ...
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)?"