Re: Measuring Turquoise Underwear

John Griffin wrote:

Evil Nigel <useweb@xxxxxxxxxx> wrote:


John Griffin wrote:


Stig Holmquist <stigfjorden@xxxxxxxxxxx> wrote:



On Thu, 10 May 2007 12:45:07 +0100, Evil Nigel
<useweb@xxxxxxxxxx> wrote:



Evil Nigel wrote:



Stig Holmquist wrote:



On Sun, 06 May 2007 16:49:12 -0400, Stig Holmquist
<stigfjorden@xxxxxxxxxxx> wrote:




On Sun, 06 May 2007 11:44:12 +0100, Evil Nigel
<useweb@xxxxxxxxxx> wrote:




Stig Holmquist wrote:



On Sat, 05 May 2007 18:42:08 +0100, Evil Nigel
<useweb@xxxxxxxxxx> wrote:





Stig Holmquist wrote:





Please explain the formula used for std.dev. and what
book you used The std.dev. for sums in the 6/49 game
is 32.8, and the std.dev. for 49 integers is 14.14.
Where does 12 come from?

Stig Holmquist>

Do you have Excel?
A B C D E F G

1 2 3 4 5 6 =STDEVPA(A1:F1)
1 2 3 47 48 49 =STDEVPA(A2:F2)
=AVERAGE(G1:G2)

The value in G3 = 12.36, a little higher than I
calculated the average population standard deviation
of the 14M combinations.


Make that 'a little lower' - my calculated average is
approx. 12.72.




Your calculation of std.dev.for F1 and F2 is based on
treating the data as a population, but they are just samples from a
14 million large population. Thus if you treat each set
as a sample you'll get 1.871 for F1 and 25.211 for F2
with a mean of 13.541.


Since I didn't really know what I was doing and I needed
a measure of diverseness, I assumed I could use either
population standard deviation or sample standard
deviation provided I was consistent throughout. I leaned
towards population rather than sample because the combo
1-2-3-4-5-6 is a complete population - there are no more
members, the values of which are unknown.




But there are 43 sets of samples with std.dev. of 1.871
and only one with 25.211. Also, there is a
distribution curve for the std.dev. of all possible
combinations. The shape of that curve is not known.


That's what I said. However the stats book didn't
specify that the distribution had to be normal. Apart

from the end bits, which are rather small in comparison

to 14 million, it probably is very bell-curve-ish.




Thus it seems to me that any calculation based on 12 is
poinless.


I'm open to suggestions for better methods of analysis.

BTW, I owe you substantial thanks. You're the first
person to have a good read of what I'm trying to
calculate and make intelligent feedback.

Evil Nigel



My memory is not what it used to be so I just recalled
that I corresponded some time ago with Harry Schneider,
who wrote the book "Lottery Numbers". It is about the UK
6/49 game and has stats for about 52 draws. He was the
first to calculate the std.dev. for the numbers and came
up with 14.1. I then modified a formula by J.E Freund

from his book "Mathematical statistics" (1962) on p. 184

where he claims the variance for the mean is
(N+1)(N-n)/12n. I suspect this is a misprint and should
not include the n in the denominator. The revised formula
would yield 50x43/12=179.17 ,which equals 13.4 as
std.dev. Keep in mind that the formula for the variance
of 49 integers taken one at a time is (N^2-1)/12 and thus
close to the above formula.

His book is now available with a co-author.
Stig Holmquist



I've just carried out a std.dev. for the old PB /53 with
302 draws. It yields numbers from 3.21 to 23.70.This range
can then be divided up into units 20 units of 1 and the
frequencies within each class can then be graphed. The
result is a broad tripple peak from 10 to 20 with a small
shoulder peak on each side. I suspect there are too few
data to yield a smooth curve. Can you generate a graph
based on your simulation?

Stig Holmquist


Cruncher running at the moment. I've arbitrarily chosen
integer ranges for the bins. I don't know about a graph -
I've only ever tried the graphing facility a couple of
times and each time they've looked like sh*t.

I'll post the bincounts here if/when it finishes.

Evil Nigel


Wow. I admit my spread*** is slow - 2 hours just to
generate and cycle through 14 million combos, but adding
processing to calculate the population SDev's and putting
them into bins, albeit using crap quality programming, the
macro got just over 5% of the way through in 21 hours.

1 3
2 68
3 417
4 1333
5 3052
6 5744
7 9743
8 15116
9 22278
10 31202
11 42496
12 55908
13 72210
14 90973
15 112116
16 111481
17 90214
18 62232
19 29725
20 8053
21 1208
22 98
23 1

The combo being processed when the macro was terminated was:
1 7 9 18 36 48

The distribution is somewhat skewed. The average SDev so far
is 14.84. Since that drops to 12.72 over the full range of
combos, that would suggest the 5% sample is not
representative of the whole distribution.

Evil Nigel


If you tested the last 500 draws of UK649 I'm sure you would
get a cumulative mean that approaches the theoretical limit.
But you must use the sample s.d not the population form.

Stig


s.d. frequency
2: 385
3: 4008
4: 17658
5: 51401
6: 112400
7: 208995
8: 351353
9: 539383
10: 764322
11:1015862
12:1284690
13:1530424
14:1701592
15:1755043
16:1621714
17:1289115
18: 865629
19: 499579
20: 245578
21: 93947
22: 25502
23: 4649
24: 555
25: 32

Maximum variance at: 1 2 3 43 48 49 Average s.d.: 13.9376251501268

Since you say the average should be 12.72, I wonder if I got
this right. Where did that number come from?

Are you using sample standard deviation? As that has more
degrees of freedom, it will be larger.

Evil Nigel


Yes.

This oughta do it:
Sample standard deviation ,frequency, cumulative proportion,
Population s.d.,...

1: 0 . 0 .
2: 385 .00003 719 .00005
3: 4008 .00031 7431 .00058
4: 17658 .00158 30557 .00277
5: 51401 .00525 83875 .00877
6: 112400 .01329 179586 .02161
7: 208995 .02824 328996 .04514
8: 351353 .05336 536645 .08351
9: 539383 .09193 800448 .14075
10: 764322 .14659 1104192 .21971
11:1015862 .21924 1420396 .32129
12:1284690 .31111 1705899 .44328
13:1530424 .42055 1894023 .57872
14:1701592 .54223 1899713 .71457
15:1755043 .66774 1637695 .83169
16:1621714 .78371 1166599 .91511
17:1289115 .87589 686199 .96418
18: 865629 .9378 336256 .98823
19: 499579 .97352 126475 .99727
20: 245578 .99108 32436 .99959
21: 93947 .9978 5168 .99996
22: 25502 .99963 490 1.
23: 4649 .99996 18 1.
24: 555 1. 0 1.
25: 32 1. 0 1.

Maximum variance at: 1 2 3 43 48 49 Average s.d.: 13.9376251501268 Average s.d.: 12.7232528212385

Thanks. Did you write a prog specially, and if so, in what language? How long did it take to run.

Your average Population SD is the same as mine, so if Stig accepts that we're not sock puppets fronting for the same person, that ought to convince him that your distribution is genuine. I notice he's been in some of the math newsgroups asking if there's a theoretical way to calculate the average, and the typical reply is (to paraphrase) "I don't know, that looks difficult".

Below are the Turquoise version 3 rankings of the drawn UK lottery balls since I began recording the stats. None of the data is retro generated.

By my calculation, the average Population Standard Deviation of the sets of Turquoise Rankings is 13.4474058610064 and the probability that this comes randomly from a population with mean 12.7232528212385 is 0.000703827297979179

Evil Nigel





Draw B1 B2 B3 B4 B5 B6 PSDev

1188 47 32 23 44 8 35 13.1244047484067
1187 13 32 26 27 47 48 12.2667119564381
1186 38 41 15 13 48 47 14.3255792979629
1185 14 22 7 44 42 20 13.7163730223733
1184 26 15 22 13 12 47 12.0381338531629
1183 2 1 43 44 11 24 17.7145577296063
1182 26 10 5 15 36 41 13.2591687354659
1181 32 7 3 4 39 8 14.3845982448822
1180 34 39 11 36 45 46 11.6535640709422
1179 22 35 27 43 11 49 12.7856256093404
1178 6 20 14 31 42 15 11.9117122568038
1177 4 2 36 16 44 9 15.9973956213712
1176 49 16 31 46 23 11 14.2672897060218
1175 33 31 1 7 48 2 17.9226734116934
1174 33 42 4 39 8 36 15.1437555888007
1173 9 2 20 17 25 4 8.43438728592
1172 21 27 4 37 26 14 10.4363148029688
1171 36 13 33 37 43 9 12.7769323391806
1170 36 8 41 16 20 7 13.0085442007252
1169 48 29 10 6 44 20 15.8578757159407
1168 38 1 27 3 2 43 17.6540835691538
1167 44 20 12 23 24 38 10.8691714904536
1166 19 5 7 14 44 48 17.0432456481218
1165 44 24 5 37 46 4 17.1820319584798
1164 23 33 17 18 46 42 11.3345587572795
1163 46 2 41 5 1 30 18.8097197096489
1162 25 13 21 12 42 6 11.6821326059167
1161 7 42 39 29 9 3 15.7665257217097
1160 9 8 44 35 48 11 16.9648329067974
1159 41 9 32 38 13 22 12.1025708930881
1158 43 33 9 22 15 10 12.4096736459909
1157 6 5 15 49 44 28 17.3469497799085
1156 4 23 38 15 39 25 12.2746350930146
1155 29 25 39 28 49 11 11.7815769553806
1154 18 13 7 44 40 29 13.6554832291729
1153 45 26 46 20 1 2 18.0523928854013
1152 8 42 28 4 39 31 14.4875425414005
1151 13 12 1 21 23 31 9.52919490594854
1150 47 22 35 17 21 12 11.8274633328068
1149 47 15 21 23 5 11 13.3499895963339
1148 27 17 32 26 16 18 5.99073358520381
1147 5 6 40 37 47 1 18.9619502044372
1146 41 12 10 37 43 7 15.502687938978
1145 38 11 8 45 5 33 15.8184140236062
1144 9 27 1 29 42 16 13.5973853695808
1143 40 3 42 21 16 20 13.5973853695808
1142 19 3 7 49 46 38 18.2847842025366
1141 39 5 49 35 44 16 15.6702123647242
1140 21 5 48 14 19 20 13.1582251420505
1139 21 6 30 42 43 28 12.6315302143309
1138 48 30 32 2 40 27 14.2643689738531
1137 12 19 15 36 32 10 9.89388138643722
1136 16 34 15 42 18 39 11.2792828771258
1135 30 8 25 22 46 42 12.6809130410848
1134 35 44 32 19 21 2 13.4752860204648
1133 44 42 41 7 6 5 18.1972220102105
1132 23 8 46 14 40 30 13.4463956343533
1131 14 39 9 27 49 36 14.0118997046558
1130 6 10 19 28 14 18 7.03364928200306
1129 41 15 44 46 43 18 12.8419884233972
1128 1 38 32 21 29 34 12.2667119564381
1127 28 49 5 6 34 25 15.4137384606504
1126 19 42 6 8 32 44 15.192286054296
1125 12 6 49 18 9 40 16.2958754154404
1124 3 15 49 21 34 9 15.5608126037456
1123 2 21 44 24 8 25 13.4370962471643
1122 17 39 12 34 20 37 10.5316981853197
1121 33 1 13 26 19 36 11.9814671704076
1120 29 41 44 42 38 45 5.33593686452738
1119 39 8 22 24 49 30 13.0085442007252
1118 2 48 9 32 49 27 17.7709188157381
1117 17 42 7 21 20 35 11.5998084275369
1116 35 23 29 11 24 17 7.75492674942123
1115 35 18 14 39 37 2 13.7527774972508
1114 35 39 19 13 11 12 11.310025051549
1113 40 18 46 38 19 36 10.5895650944167
1112 11 35 18 20 19 4 9.47657931721967
1111 2 22 16 44 5 38 15.6035964515307
1110 7 10 42 49 5 40 18.42778698958
1109 6 17 7 13 31 8 8.63455589799241
1108 42 23 44 39 37 22 8.77021474461525
1107 32 23 46 15 43 8 13.8974418109553
1106 19 18 12 6 28 33 9.08600878029268
1105 46 39 19 18 45 48 12.5620681241408
1104 44 12 26 29 4 8 13.8774397254441
1103 27 31 16 36 32 25 6.36177822799744
1102 15 41 7 47 40 44 15.4236470683457
1101 39 49 27 35 3 33 14.1891977691952
1100 14 38 5 20 15 1 11.8988794990677
1099 46 28 31 36 35 13 9.97914491994847
1098 33 4 36 32 1 35 14.9303940559741
1097 29 41 6 13 15 3 13.2465928533424
1096 49 11 35 24 17 5 14.8520481191428
1095 20 2 33 41 14 35 13.4835290467873
1094 3 2 24 29 4 41 15.0489940601431
1093 24 47 38 22 23 27 9.22707369044427
1092 11 14 16 20 43 9 11.3639292891539
1091 3 49 14 25 33 1 16.9156403629567
1090 8 15 45 49 21 25 15.0157324903656
1089 8 5 38 12 7 6 11.5421931287872
1088 17 10 36 29 16 9 9.84462628374824
1087 11 29 1 20 47 38 15.61694236683
1086 9 5 22 46 1 30 15.6994338185242
1085 11 26 35 41 14 10 12.0473602456675
1084 8 27 28 45 21 10 12.4018367815238
1083 2 11 31 12 34 43 14.6448701674758
1082 37 47 4 20 19 17 14.0712472794703
1081 49 38 33 27 5 37 13.5615879109589
1080 23 28 26 21 40 3 10.9962114688045
1079 4 39 25 2 21 19 12.5918845116827
1078 49 5 47 28 3 4 19.8382346884887
1077 46 49 1 25 37 26 16.0485374896143
1076 48 27 36 16 30 41 10.2632028788938
1075 46 21 28 45 3 19 15.0665191733194
1074 22 37 46 18 16 9 12.7758452644912
1073 15 27 41 43 7 16 13.4711626158332
1072 49 30 44 15 29 40 11.2657297440808
1071 25 42 46 35 18 6 13.9004396413287
1070 10 11 27 23 34 48 13.1497781983829
1069 17 39 43 11 9 31 13.3666251038423
1068 47 26 1 12 5 42 17.6580167503476
1067 35 2 33 48 41 28 14.4846662217517
1066 38 47 4 35 12 27 14.9378341431711
1065 13 24 16 37 1 18 10.915076220022
1064 13 46 6 32 20 11 13.7558068546422
1063 31 46 17 42 14 3 15.4569294061488
1062 47 8 6 15 14 30 14.3178210632764
1061 32 48 30 7 42 15 14.2594997574716
1060 46 40 41 4 6 11 17.8854379003951
1059 44 27 22 48 34 21 10.3869576339219
1058 22 45 15 31 49 1 16.6774964814534
1057 11 16 44 37 3 20 14.3226704524292
1056 42 26 31 12 41 20 10.7651701746368
1055 45 8 30 14 24 25 11.7851130197758
1054 39 42 15 26 49 10 14.2993783858678
1053 17 7 23 45 43 20 13.7648909266373
1052 17 26 19 32 23 14 5.98377435700542
1051 7 43 48 1 36 46 18.9509601046725
1050 26 28 48 25 44 14 11.6678570821248
1049 46 18 10 24 7 41 14.6818103636968
1048 5 15 31 46 18 4 14.7582368715086
1047 47 11 28 44 5 14 16.1804065324563
1046 23 44 35 8 33 13 12.6227308191743
1045 4 12 14 21 49 18 14.1499901845274
1044 39 10 26 22 48 15 13.1866936299017
1043 46 2 6 29 26 8 15.5750869446476
1042 47 34 32 5 33 12 14.2526800598655
1041 45 44 20 4 1 8 18.0800688297608
1040 35 12 25 48 38 2 15.7020876177518
1039 10 35 27 8 41 12 12.8765506078125
1038 17 33 4 47 21 22 13.3666251038423
1037 44 12 17 39 9 27 13.2245562832516
1036 8 4 47 9 38 43 18.0869811988869
1035 37 24 27 12 33 44 10.1775897605147
1034 21 44 39 18 11 41 12.7671453348037
1033 16 43 20 2 49 6 17.6225865171817
1032 22 34 31 44 14 7 12.4588210606872
1031 20 27 7 44 40 3 15.326991442115
1030 7 19 13 24 23 17 5.84284938098603
1029 36 30 37 35 2 27 12.0749971244533
1028 22 41 23 43 37 9 12.1712320201732
1027 43 21 5 49 2 17 17.7051467721175
1026 21 39 18 42 26 10 11.3284303119776
1025 41 45 18 24 9 34 12.7115433104456
1024 40 46 6 32 10 45 16.1288216832132
1023 44 43 34 2 35 22 14.456832294801
1022 19 27 1 29 10 3 10.9607886983049
1021 34 14 5 3 41 30 14.599276998841
1020 29 14 7 24 44 16 11.9814671704076
1019 32 3 37 26 49 46 15.2032525759749
1018 37 22 4 20 34 15 11.1504857891185
1017 29 3 32 5 14 23 11.1902735543963
1016 44 24 14 40 11 47 14.3643076176102
1015 14 29 18 3 24 33 9.95685135416257
1014 30 33 8 20 14 47 12.9571944837179
1013 42 48 41 38 49 14 11.6856987619721
1012 22 1 2 38 27 35 14.6220913536866

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