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Forum index » Science and Technology » Math » Recreational
Sampling 1-2-3-4
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Mon Jul 17, 2006 11:47 am    Post subject: Re: Sampling 1-2-3-4 Reply with quote

On Sun, 16 Jul 2006 13:28:58 -0400, Stig Holmquist
<stigfjorden@hotmail.com> wrote:

Quote:
When sampling with replacement four times from an urn with four balls
numbered1,2,3and4 one can get 35 distinct combinations if oder is
not counted but 4^4=256 sets of four digits due to permutation when
order is observed. It is not hard to to tabulate all 35 combinations
and then calculate how many permutations are possible for each
specific combination.

The sample std.dev. for each combination can easily be calculated with
a handhelled scientific calculator or a simple computer program.
After mutiplication by the permutation factor for each combination one
can obtain the sum of all s.d's and the mean, which can then be
recalculated as the std.dev. for the entire population of 256 sets.

The following table illustrates how the std.dev. for some permutations
based on two or three ones was obtained

1111 1 0 0 Second column shows

1112 4 0.5 2 permutations, third

1113 4 1 4 column shows s.d. and

1114 4 1.5 6 last shows product.

1123 12 0.96 11.48

1124 12 1.41 16.97

1134 12 1.5 18

The samples 1122, 1133 and 1144 were listed in the "2" ,"3" and "4"
tables and the 1234 sample is unique. This table has 7 entries, the
2-table has 8, the 3-table has 9 and the 4-table has 10 entries for a
total of 7+8+9+10+1=35. After all four tables were calculated there
were twelve specific s.d's with varying permutation frequencies:

s.d. 0 0.5 1 1.5 0.58 0.82
freq. 4 24 16 32 18 24

s.d. 0.96 1.16 1.28 1.41 1.73 1.29
freq. 48 12 24 24 6 24

The total sum of s.d's was 271.24 for a mean of 1.06 and
a std.dev. for the population =0.92. It is clear that no regular
curve can illustrate the results.
In case anybody wonders, let me mention that the mean std.dev.
for all 35 combinations was 0.88
I don't have the computer capability to sample 12345
and much less 123456 that would apply to dice.

If somebody feels inclined to try it ,please let us know.

Stig Holmquist





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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Mon Jul 17, 2006 11:44 am    Post subject: Re: Sampling 1-2-3-4 Reply with quote

On Sun, 16 Jul 2006 19:12:55 -0700, William Elliot
<marsh@hevanet.remove.com> wrote:

Quote:
On Sun, 16 Jul 2006, Stig Holmquist wrote:

When sampling with replacement four times from an urn with four balls
numbered1,2,3and4 one can get 35 distinct combinations if oder is

Why stop at no spaces in "numbered 1, 2, 3 and 4"
or "numbered 1,2,3 and 4" whenyoucangothewholeway
and become completely unreadable.


Thankyouforyourconstructive nit-picking. As you can tell
prooofreading is not my strong suit.

Have a good day.

Stig Holmquist
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Mon Jul 17, 2006 2:12 am    Post subject: Re: Sampling 1-2-3-4 Reply with quote

On Sun, 16 Jul 2006, Stig Holmquist wrote:

Quote:
When sampling with replacement four times from an urn with four balls
numbered1,2,3and4 one can get 35 distinct combinations if oder is

Why stop at no spaces in "numbered 1, 2, 3 and 4"
or "numbered 1,2,3 and 4" whenyoucangothewholeway
and become completely unreadable.
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Sun Jul 16, 2006 5:28 pm    Post subject: Sampling 1-2-3-4 Reply with quote

When sampling with replacement four times from an urn with four balls
numbered1,2,3and4 one can get 35 distinct combinations if oder is
not counted but 4^4=256 sets of four digits due to permutation when
order is observed. It is not hard to to tabulate all 35 combinations
and then calculate how many permutations are possible for each
specific combination.

The sample std.dev. for each combination can easily be calculated with
a handhelled scientific calculator or a simple computer program.
After mutiplication by the permutation factor for each combination one
can obtain the sum of all s.d's and the mean, which can then be
recalculated as the std.dev. for the entire population of 256 sets.

The following table illustrates how the std.dev. for some permutations
based on two or three ones was obtained

1111 1 0 0 Second column shows

1112 4 0.5 2 permutations, third

1113 4 1 4 column shows s.d. and

1114 4 1.5 6 last shows product.

1123 12 0.96 11.48

1124 12 1.41 16.97

1134 12 1.5 18

The samples 1122, 1133 and 1144 were listed in the "2" ,"3" and "4"
tables and the 1234 sample is unique. This table has 7 entries, the
2-table has 8, the 3-table has 9 and the 4-table has 10 entries for a
total of 7+8+9+10+1=35. After all four tables were calculated there
were twelve specific s.d's with varying permutation frequencies:

s.d. 0 0.5 1 1.5 0.58 0.82
freq. 4 24 16 32 18 24

s.d. 0.96 1.16 1.28 1.41 1.73 1.29
freq. 48 12 24 24 6 24

The total sum of s.d's was 271.24 for a mean of 1.06 and
a std.dev. for the population =0.92. It is clear that no regular
curve can illustrate the results.

I don't have the computer capability to sample 12345
and much less 123456 that would apply to dice.

If somebody feels inclined to try it ,please let us know.

Stig Holmquist





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