- 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

Jaap gave the theoretical distribution:

> 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

> With perfect random scrambles, the results should in theory have the

The theoretical average numbers for 20000 cubes then are

>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

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

--- In speedsolvingrubikscube@yahoogroups.com, h_kociemba

<no_reply@...> wrote:>

the

> 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

> >following distribution:

66.

> > 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

>

this

> With 6 degrees of freedom the tables show, that the chance that

> value is >18.55 is less then 0.005. So if I did not do anything

the

> false, the 25 move scramble is a very bad procedure. Especially

> high occurrence of 0 and 4 flipped edges contribute to the high

value

> of 66.

>

> Herbert

>