## chi-square test

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• Ron posted the results of random scrambles a few days ago, looking ... The theoretical average numbers for 20000 cubes then are 0:9.77, 2:644.53, 4:4834, 6:
Message 1 of 2 , Apr 12 4:08 AM
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Ron posted the results of random scrambles a few days ago, looking
how many edges were flipped:

> Here are the results of 20,000 random scrambles sequences of 25
> moves (1 in 3 moves in a half turn).
> 0: 28 0,14%
> 2: 701 3,51%
> 4: 5095 25,48%
> 6: 8950 44,75%
> 8: 4590 22,95%
> 10: 627 3,14%
> 12: 9 0,05%
> Average: 5.929

Jaap gave the theoretical distribution:

> With perfect random scrambles, the results should in theory have the
>following distribution:
> 0: 1 0.05%
> 2: 66 3.22%
> 4: 495 24.17%
> 6: 924 45.12%
> 8: 495 24.17%
> 10: 66 3.22%
>12: 1 0.05%
> Total: 2048

The theoretical average numbers for 20000 cubes then are

0:9.77, 2:644.53, 4:4834, 6: 9023.44, 8:4834, 10:644.53, 12:9.77

The chi-square test
http://en.wikipedia.org/wiki/Pearson's_chi-square_test
is the appropriate tool to test if 25 move scrambles are "good":

(28-9.77)^2/9.77 + (701-644.53)^2/644.53 + .....gives a result of 66.

With 6 degrees of freedom the tables show, that the chance that this
value is >18.55 is less then 0.005. So if I did not do anything
false, the 25 move scramble is a very bad procedure. Especially the
high occurrence of 0 and 4 flipped edges contribute to the high value
of 66.

Herbert
• That seems to be the right test to perform from what I remember of AP Stats class. So I was right! 25-turn scrmables are biased in some way. -Doug ... the ...
Message 2 of 2 , Apr 12 7:14 AM
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That seems to be the right test to perform from what I remember of
AP Stats class. So I was right! 25-turn scrmables are biased in some
way.

-Doug

--- In speedsolvingrubikscube@yahoogroups.com, h_kociemba
>
> Ron posted the results of random scrambles a few days ago, looking
> how many edges were flipped:
>
> > Here are the results of 20,000 random scrambles sequences of 25
> > moves (1 in 3 moves in a half turn).
> > 0: 28 0,14%
> > 2: 701 3,51%
> > 4: 5095 25,48%
> > 6: 8950 44,75%
> > 8: 4590 22,95%
> > 10: 627 3,14%
> > 12: 9 0,05%
> > Average: 5.929
>
> Jaap gave the theoretical distribution:
>
> > With perfect random scrambles, the results should in theory have
the
> >following distribution:
> > 0: 1 0.05%
> > 2: 66 3.22%
> > 4: 495 24.17%
> > 6: 924 45.12%
> > 8: 495 24.17%
> > 10: 66 3.22%
> >12: 1 0.05%
> > Total: 2048
>
> The theoretical average numbers for 20000 cubes then are
>
> 0:9.77, 2:644.53, 4:4834, 6: 9023.44, 8:4834, 10:644.53, 12:9.77
>
>
> The chi-square test
> http://en.wikipedia.org/wiki/Pearson's_chi-square_test
> is the appropriate tool to test if 25 move scrambles are "good":
>
> (28-9.77)^2/9.77 + (701-644.53)^2/644.53 + .....gives a result of
66.
>
> With 6 degrees of freedom the tables show, that the chance that
this
> value is >18.55 is less then 0.005. So if I did not do anything
> false, the 25 move scramble is a very bad procedure. Especially
the
> high occurrence of 0 and 4 flipped edges contribute to the high
value
> of 66.
>
> Herbert
>
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