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Resources & Downloads => Mathematics => Topic started by: Bayes on Aug 30, 01:23 PM 2010

Title: Standard Deviation and z-score in plain English
Post by: Bayes on Aug 30, 01:23 PM 2010
Well, mostly, I can't avoid numbers, but I'll try to keep the explanation simple without too much technical stuff.

Basically, the standard deviation is a number which tells you how "spread out" a set of numbers is around an average. It's what's called a measure of dispersion in the statistical lingo. It's important to note that these numbers could represent anything - they could be the heights of a group of people, or their lifespans, a set of temperatures over a year in a particular location, the number of goals scored by a football team in each match over a season, or the set of stakes you place in a session of roulette. The list is endless.

To calculate the standard deviation of a set of numbers, you can follow a procedure (an algorithm) or use a formula. It isn't necessary to know the details, because there are calculators available or you can use a spreadsheet. The actually calculations aren't difficult, but it's tedious and time-consuming to do them. Most spreadsheets have a Standard Deviation function, I think the one in excel is called STDDEV().

You can find an online calculator here (link:://invsee.asu.edu/srinivas/stdev.html).

Standard deviation is related to the AVERAGE of a set of numbers. Specifically, it tells you how much variation there is from the average - a higher SD (Standard Deviation) means higher variation (or volativity) and a lower SD means less. So a higher SD suggest that the lowest or highest number in your set is further away from the average than that in another set of numbers which has a lower SD, even though both AVERAGES may be the same.

Example:

Results of betting on R/B for 20 spins. The numbers represent your profits and losses for each spin.

-1,1,1,-1,1,-1,1,-1,1,1,-1,-1,1,-1,-1,1,1,-1,-1,-1

The average is -2/20 = -0.1

The standard deviation = 0.9949

Now, here are results from betting on a double street over 20 spins.

-1,-1,-1,-1,-1,-1,5,-1,-1,5,-1,-1,-1,5,-1,-1,-1,-1,-1,-1

Once again, the average is -0.1, but the standard deviation is 2.1424, more than twice as much.

In general, the fewer numbers you're betting on, the higher the SD will be.

Ok, but what does it mean when you hear that some sample of spins is 3 SD? or that an SD is increasing (or decreasing)?

To answer this, I need to introduce the idea of a distribution. Basically, a distribution describes the range of values that a set of outcomes can take. For example, if you were to play 1000 sessions of roulette betting on R/B for 100 spins and recorded the number of reds in each of the sessions, you would end up with an AVERAGE of close to 50 reds in each session, but the numbers would be spread out (as measured by the SD). So in some sessions you might see only 40 reds, in others maybe 60.

It turns out (and anyone can verify this for themselves) that in roulette the way the numbers are distributed for most events we're interested in follows the so-called "Bell Curve" (other names are "Normal" curve or "Gaussian" curve). This curve tells you that in most of your sessions the numbers of reds will be centered around the average, and the further away from the average a result is, the rarer it will be. The horizontal axis on the graph tells you WHAT happened (e.g. you got 45 reds in that session) and vertical axis is how many times that result occurred.

Another fact about distributions which follow the bell curve tells you that in the vast majority of your sessions, there will be a "limit" to the number of reds which you will see in any 100 spin session.

The 68 - 95 - 99.7 rule

The standard deviation is best understood when the group of measurements has a particular shape (the Bell shape). If the distribution of measurements is symmetric and approximately bell shaped, that is, most of the measurements are clustered in the center of the distribution and the values trail
off to the left and to the right in about the same rate. If the dataset is bell shaped, then approximately,

ââ,¬Â¢ 68% of the observations will fall within one standard deviation from the average.
ââ,¬Â¢ 95% of the observations will fall within two standard deviations from the average.
ââ,¬Â¢ 99.7% of the observations will fall within three standard deviations from the average.

We call these statements collectively the 68 - 95 - 99.7 rule.

But what does this actually MEAN?

If you were to record the numbers of reds in each of your 1000 sessions, you would end up with a set of numbers:

45, 56, 41, 52, 39, 58, 47....

Use a computer to calculate the Standard Deviation and the average of this set of numbers. Ignoring the effect of the zero, your average will be 50 and the SD will be 5.

The 68 - 95 - 99.7 rules says that:

in 680 of your sessions, the number of reds fall within the average, plus or minus one standard deviation, ie: between 50 - 5 = 45 and 50 + 5 = 55.

in 950 of your sessions, the number of reds will fall within the average, plus or minus two standard deviations, ie: between 50 - 10 = 40 and 50 + 10 = 60.

in 997 of your sessions, the number of reds will fall within the average, plus or minus three standard deviations, ie: between 50 - 15 = 35 and 50 + 15 = 65.

More on what it means

In the previous example, I used the number of reds in a sample of 100 spins as the variable of interest, but I could have used the number of WINS in each of the sessions, and the rule would still be valid. As long as the distribution follows the bell-curve, the 68-95-99.7 rule applies.

What about if you wanted to measure results of a street or number? supposing you suspect that a particular wheel is biased, you want to find out which numbers are hitting more (or less) than they "should" be - do you still do the calculations the same way?

Yes. Although the bell curve always has the same BASIC shape. The differences will be in where it is "centred" (ie; where the average is) and how "flat" or "spread out" it is (which depends on the Standard Deviation).

Another important point: Note that the standard deviation depends on the number of "samples" you take, the SD will NOT be the same for a session of 100 spins and a session of 200 or 300 spins.

I think I'll stop here for now. If you have any questions or don't understand something, please tell me!

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Title: Re: Standard Deviation in plain English
Post by: VLS on Aug 30, 03:10 PM 2010
Thank you for this Bayes, the more math-oriented information to educate the fellows the better.

I'm glad our community counts with people like you and mr.ore  :thumbsup:
Title: Re: Standard Deviation in plain English
Post by: Bayes on Aug 30, 03:52 PM 2010
I'm not going to kid myself that this stuff will get much attention, but I don't really mind. I actually enjoy the challenge of trying to make these topics more accessible, and often in the process you discover things you might not have noticed otherwise.

It's all too easy if you're familiar with something to assume that others are too, but it's usually not the case, particularly for 'geeky' stuff like this. For example, looking at the text under your avatar, I wonder how many members actually know what an IDE is?  :)
Title: Re: Standard Deviation in plain English
Post by: Blood Angel on Aug 30, 04:17 PM 2010
I enjoy your posts bayes.
Title: Re: Standard Deviation in plain English
Post by: VLS on Aug 30, 05:07 PM 2010
Quote from: Blood Angel on Aug 30, 04:17 PM 2010
I enjoy your posts bayes.

Count 2 with me.

There you go dear Bayes; you've got audience :)

Quote from: Bayes on Aug 30, 03:52 PM 2010
It's all too easy if you're familiar with something to assume that others are too, but it's usually not the case, particularly for 'geeky' stuff like this. For example, looking at the text under your avatar, I wonder how many members actually know what an IDE is?

You are right! Over and out. I'm back to programming ;)
Title: Re: Standard Deviation in plain English
Post by: MrJ on Aug 30, 05:49 PM 2010
One of the best threads I have seen in years!  Ken
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Aug 30, 07:35 PM 2010
Really, really nice thread  :thumbsup:

Deviation is something players should know of, because it can tell, what is at least possible. You just can't win 100 units by flat betting one unit on EC, yet many beginner does not know that simple fact. Many systems fails without even using all possibilities the game allows because they do not consider standard deviation when size of unit is being decided. It is very important to know, when the hole is so big that system cannot recover, or what should the unit size to have good chance to recover.

One can have some fun in RX when computing standard deviations for possible profit, because it allows for progression in risk in turbo mode. Number is very good bet, nothing has such a good probability of recovering hole than betting straight, when really down. It at least ALLOWS for recover to happen, even if expectation is negative. And that is what should good systems aim for - to ALLOW to happen something good, to hammer a good trend to maximal possible extend, to make it at least possible to win.

Many people use positive progressions starting with one unit, and most of time they make small profit. But most of time they could have had much better profit, were they more aggressive. Best progressions I know of are PLAY ONCE IN LIFETIME things, and they just and only ALLOW for a big win to happen, the probability is small, yet maximal possible. Math CAN answer the question, how to play game with negative expectation, it can give you the optimal strategy, the best response, if you ask for just that. How many asked that simple question - how should I play to maximize my chance?
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Aug 30, 07:51 PM 2010
With the knowledge of standard deviation, a lot of things can be accomplished - I just made an experiment with double zero roulette in RX, and survived 10000 spins betting only one unit, progression in risk, sessions were 100 spins long, after each session if too down I switched to bet with higher deviation, and on win back. It does not mean that it will always work, but at least it is possible to play so long. It was just an experiment, and means nothing, of course.
Title: Re: Standard Deviation in plain English
Post by: Bayes on Aug 31, 02:24 AM 2010
Thanks for the positive feedback guys!

You make some very good points mr.ore, I'll try to expand on them later in this thread.

I found an interesting article on Standard Deviation in trading and how to exploit things like mean reversion, which sounds a bit like gambler's fallacy but isn't. Roulette may be even more suited to these techniques because the outcomes are closer to the "ideal" bell curve, which market prices only approximate.

Article here (link:://:.zealllc.com/2003/stddev.htm).
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Aug 31, 05:29 AM 2010
Well, I have programmed some graphs, it is for single number flat betting, uses simple moving average of size 37*3 (I'm not sure what window size is best...)  and shows all six standard deviations.
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Aug 31, 05:44 AM 2010
Simulation on some 65k spins from Wiesbaden, single number, window size 750, so it is correct from spin 750, only then is history big enough. Gnuplot script  included.
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Aug 31, 05:48 AM 2010
Hmm, I made a BIG mistake, I used moving average computed from bankroll instead of expected value, I will make another graphs later. The images above DOES NOT demonstrate the idea. Roulette is NOT Forex or comodities.
Title: Re: Standard Deviation in plain English
Post by: Blood Angel on Aug 31, 09:41 AM 2010
Hi Bayes

Is SD and Z score the same thing? And how long(many spins) should one prove an EC system until it is deemed to  work...in your opinion....just an opinion  :)
Title: Re: Standard Deviation in plain English
Post by: Blood Angel on Aug 31, 09:42 AM 2010
Hi Bayes

Is SD and Z score the same thing?
Title: Re: Standard Deviation in plain English
Post by: Bayes on Aug 31, 10:31 AM 2010
Ah, you must have read my mind, I was going to cover that next.  :thumbsup:

Not exactly, but they're very closely related. Details to follow...
Title: Re: Standard Deviation in plain English
Post by: esoito on Sep 01, 02:15 AM 2010
Congratulations, Bayes!

And a big THANK YOU for this EXCELLENT explanation.

I'm actually starting to understand SD at last.

You certainly have a gift for rendering difficult matter into clear, concise English.   :thumbsup:

And many thanks, too, to mr.ore for his valuable comments, examples and explanations.

We are so lucky to have such expert and generous members.

Title: Re: Standard Deviation in plain English
Post by: Bayes on Sep 17, 08:04 AM 2010
Thanks esoito. Ok, moving on to the z-score.

The z-score

In the first post of this thread, I gave the 68 - 95 - 99.7 rule, which said that in a sequence of fixed length (say 100 spins) the numbers of reds would conform to the rule, ie; that in 68% of these 100 spin sequences you would get between 45 and 55 reds (1 standard deviation), in 95% of them you would find between 40 and 60 reds (2 standard deviations), and in 99.7% of them you would find between 35 and 65 reds (3 standard deviations), but what if you have a sequence of a different length? or you're not interested in the number of reds, but maybe the number times a particular dozen or street hits? Also, suppose you want to know the "interval" of hits for a particular % (not just one of 68%, 95% or 99.7%)?

The z-score is a measure of the dispersion (how "spread out" the hits are) which is defined in terms of the standard deviation. So the z-score tells you how many standard deviations away from the average a particular measurement is. It turns out there's a simple formula for calculating the standard deviation for the sort of outcomes we're interested in, so for example, instead of having to record the numbers of reds in each of your 1000 sessions, you can use this:

s = √(np(1 − p))

where s is the standard deviation, n is the number of spins in the sequence, and p is the probability of the outcome you're interested in.

So to find the SD of a sequence of 100 spins, where red is the outcome of interest, we have:

n = 100
p = 0.5 (I've ignored the zero just to keep the arithmetic simple).
Also (1 − p) also equals 0.5 in this case.

So s = √(100 âÅ"• 0.5 âÅ"• 0.5)
      = √25
      = 5

The z-score is another simple formula, which uses the standard deviation and also the AVERAGE number of hits you would EXPECT to get in your sequence (you have to  calculate this) together with the measurement you're interested in (or what actually occurred):

z = (X − expected no. of hits)/s

In other words, z is the difference between what you ACTUALLY got (X) and the AVERAGE (what you would expect to get in the long term), divided by the standard deviation.

The 'expected number of hits' is just the probability of a hit, multiplied by the number of spins in the sequence.

Example, in 100 spins you would expect to get 50 reds, on average. ie:

Average = n âÅ"• p

Where n = 100 and p = 0.5

So, Average = 100 âÅ"• 0.5 = 50 hits.

So the formula for the z-score is:

z = (X − np)/s

To make sense of this formula, notice what happens when X = np. What does this mean? it means that the actual outcome you got is the same as the expected outcome. So in 100 spins, if the actual number of reds was 50, then this tells you that (X − np) is zero, so the z-score is also zero. Remember what the z-score tells you - it is a measure of the dispersion - how far the outcome is from what you would expect on average. In this case, the outcome is the SAME as the average, so the z-score reflects that. If the absolute value of the z-score is large, it means the dispersion is also large.

What do I mean by 'absolute value'?

To be continued....
Title: Re: Standard Deviation in plain English
Post by: Bayes on Sep 24, 11:56 AM 2010
If your actual measurement (X) is more than what's expected (np) the z-score will be positive, and if it's less it will be negative. If in 100 spins you get 35 reds, then:

z = (35 - 50)/5 = -3.00

And if you get 65 reds it will be:

z = (65 - 50)/5 = +3.00

The absolute value disregards the "sign" of the number (whether it's positive or negative) and is concerned only with the "distance" from zero - if you imagine a number line with zero in the middle with the positive numbers increasing to the right and the negative numbers increasing to the left, the absolute value only takes account of how far the number is from the zero.

If the z-score is negative, it means worse than average "performance" for the number or group of numbers in question. If it's positive, this indicates a better than average showing.

A couple of step-by-step examples.

Example 1

What is the z-score of a street which has hit twice in the last 75 spins?

First thing is to identify what numbers go into the formula.

z = (X - np)/s, where s = √(np(1 − p))

What is n? the number of spins over which you want to measure the z-score, which is 75
What is p? this is the probability of a street hitting. Since a street consists of 3 numbers, and there are 37 numbers in total (all of which are equally likely), then p = 3/37 = 0.081081
So (1 − p) = (1 − 0.081081) = 0.918919

Therefore the standard deviation, s = √(75 âÅ"• 0.081081 âÅ"• 0.918919) = 2.363898

What is X? this is the number of times the street has hit in the last 75 spins, which is 2.

So now we're ready to plug the numbers into the formula:

z = (2 − 75 âÅ"• 0.081081)/2.363898 = -1.726

The negative number tells us that this particular street has hit well below expectation over the last 75 spins. Remember the 68-95-99.7 rule which says that in a sequence of 75 spins, a z-score of between -1 and +1 will occur in 68% of the sessions, a z-score between -2 and +2 will occur in 95% of the sessions, and a z-score of between -3 and +3 will occur in 99.7% of the sessions. So -1.726 means that this result is outside the 68% interval, but inside the 95% interval (maybe about 80%), meaning that it only occurs in about 20% of sessions or less.

Example 2

You notice that a dozen has hit 13 times in the last 18 spins, what is its z-score?

First find s, the standard deviation.

s = √(np(1 − p)

What is n? the number of spins over which you want to measure the z-score, which is 18.
What is p? this is the probability of a dozen hitting. Since a dozen consists of 12 numbers, the probability, p is 12/37 = 0.324, so (1 − p) = (1 − 0.324) = 0.676.

Therefore, the standard deviation, s = √(18 âÅ"• 0.324 âÅ"• 0.676) = 1.986

What is X? this is the number of times the dozen has hit in the last 18 spins, which is 13.

Plug the numbers into the formula:

z = (13 - 18 âÅ"• 0.324)/1.986 = +3.609

Since the z-score is positive it means that the dozen is hitting above expectation in this sequence of 18 spins. In fact, because the score is above +3.00, it indicates a rare event which occurs in less than 0.3% of all 18 spin sequences (because +3.609 lies outside the +3.00, and +3.00 is the outside "limit" for which 99.7% of results will occur).

Now here's one for you to try  :) :

You have a system which bets on sleepers - the plan is to tick off all numbers until there are only 4 left unhit and then start betting on them.  Eventually, after 87 spins, you have your 4 numbers remaining. What is the z-score?
Title: Re: Standard Deviation in plain English
Post by: Bayes on Sep 27, 03:01 AM 2010
All members go to the back of the class for not doing their homework.  >:(

Seriously though, is there too much maths? I know it's not a popular subject, I tried to keep it to a minimum but sometimes you can't avoid formulas. I read somewhere that for every equation in a book, the sales drop by half.

If you don't understand something, let me know and I'll try to make it clearer, there are no st*pid questions!  :thumbsup:

Maybe you're just not interested, and are thinking, "what's the point of all this, just give me a winning system!". It's a fair point, but IMHO, an understanding of this stuff can generate ideas, and can save a lot of manual testing and experiment. It can also help you understand why something won't work.

I'll be covering more applications in future posts.
Title: Re: Standard Deviation in plain English
Post by: mr.ore on Oct 10, 05:02 AM 2010
z = (0-87*4/37)/sqrt(87*4/37*(1-4/37)) = -3.24737656354395485351
Title: Re: Standard Deviation in plain English
Post by: Bayes on Oct 10, 05:58 AM 2010
Z = (0-87*4/37)/sqrt(87*4/37*(1-4/37)) = -3.24737656354395485351 âÅ"“

Thanks mr.ore.

This brings me to a specific application of the formula which can be used to find the longest losing runs. Notice that in that last problem X was zero (corresponding to no wins). You can set X = 0 in the formula and do a little algebra to find another formula:

z = (X - np)/s, where s = √(np(1 − p))

ie; z = (X - np)/√(np(1 − p))

Now set X = 0

z = -np/√(np(1 − p))
z2 = n2p2/np(1 − p)
z2 = np/(1 − p)

n = z2(1 − p)/p

We now have a formula for the longest losing run in terms of the z-score and the probability, p.
But what is realistic number for z to use in the formula? If we take z = 3.00 that will cover 99.7% of cases.

Example 1

What is the longest losing run for a dozen, assuming z = 3 (3 standard deviations from the mean)?

Here, z = 3 so z 2 = 9, and p = 12/37 = 0.324, so (1 − p) = 0.676 (rounded up).

So n = (9âÅ"•0.676)/0.324 = 18.78 or 19 rounded up to the nearest whole number.

But hang on. That doesn't seem right. We know from published stats that a dozen can sleep for 30 spins, sometimes more (I think the record is something like 35).

Don't forget that Ã,± 3 standard deviations from the mean (a z-score of Ã,±3) still leaves 0.3% of sequences which will exceed it - it by no means represents a "cap" on how long a losing run can be. In fact, there's no way of knowing how long it could be, there is no "law" of probability which says when a losing run must end; in fact the maths says that theoretically, the losing run could be infinite. However, practically speaking, we fairly safely assume that a sequence (assuming a random wheel) is extremely unlikely to exceed 5 standard deviations (a z-score of Ã,±5).

More later...
Title: Re: Standard Deviation in plain English
Post by: Bayes on Nov 02, 08:50 AM 2010
Here is a sample of some "maximum sleeps":

460 - 540 spins with a straight up not showing. (maths expectancy is a hit once every 37 spins)
309 splits with a split not showing. (maths expectancy is a hit once every 18.5 spins or so)
178 spins with a street not showing. (maths expectancy is a hit once every 12.3 spins or so)
155 spins with a corner(square) not showing. (maths expectancy is a hit once every 4.6 spins or so)
93 spins with a line (double-street) not showing... (maths expectancy is a hit every 4.6 spins or so)
36-40 spins with a dozen/column not showing.(maths expectancy is a hit once every 3 spins or so)
20- 36 spins with any even money chance not showing. (expectancy is a hit once every two spins)
10 numbers may sleep for about 45-55 spins

We can re-arrange the formula n = z2(1 − p)/p to find z in terms of n, then use this together with the stats to find the highest z-score which we can reasonably expect. We can then  use z2(1 − p)/p  with other groups of numbers to find out what the maximum sleep is likely to be.

So, after a little algebra, we have:

z = −√(np/(1−p))

Take the first stat in the above list: 460 - 540 spins with a straight up not showing

Let's call it 500 spins. Then we can put the numbers into the formula:

z = −√(500 âÅ"• 0.027 / (1 − 0.027)) = -3.725

Now if we do the same for the other stats, and take the average, this should give us a good indication of which value of z to use to calculate the maximum sleep for any set of numbers.

To be continued...
Title: Re: Standard Deviation in plain English
Post by: Fripper on Nov 02, 10:00 AM 2010
Hi bayes and thanks for all this stats and effort.

I have a quesiton tho:
We know that a even chance can sleep up to 40 spins. Now I wonder how long half the wheel can sleep, do you have any statistics on this or is it the same as all the other even chance bets?

I mean like numbers 32,15,19.....and so on until and number 10.
How long can this sector sleep? Is it the same?

I'm a little curious :)
Title: Re: Standard Deviation in plain English
Post by: Bayes on Nov 03, 05:59 AM 2010
Quote from: Fripper on Nov 02, 10:00 AM 2010
I have a quesiton tho:
We know that a even chance can sleep up to 40 spins. Now I wonder how long half the wheel can sleep, do you have any statistics on this or is it the same as all the other even chance bets?

Hi Fripper,

I don't have any stats from live wheels on this, but assuming there is no bias or dealer influence, then any 18 numbers will have the same distribution, so the fact that they are adjacent on the wheel shouldn't make any difference - you should find the same deviations.  :thumbsup:
Title: Re: Standard Deviation in plain English
Post by: MrJ on Nov 03, 02:47 PM 2010
This is still one of my favorite threads on any board. Good job Bayes! 

Ken      :thumbsup:
Title: Re: Standard Deviation in plain English
Post by: Bayes on Nov 05, 06:47 AM 2010
Thanks ken. I was thinking that maybe I'd overdone the maths, because its supposed to be "in plain English". But you can still use the results even if you don't follow all the equations.  :thumbsup:

I still have quite a few posts to add in this thread, it's a big subject!
Title: Re: Standard Deviation in plain English
Post by: Bayes on Jan 11, 07:21 AM 2011
So, what is the longest losing run we can realistically expect? Based on the above stats and others, we can be pretty sure that it won't get any worse than a z-score of -5.0.

Remember, a z-score of -3.0 covers 99.7% of cases, this is 1 chance in 741 (see this site (link:://:.fourmilab.ch/rpkp/experiments/analysis/zCalc.html) to calculate the chance of a particular z-score occurring).

So we can plug z = -5.0 into the formula for the longest losing run, which gives:

n = 25(1−p)/p -  Formula for the longest losing run.

The only thing you need to supply is the value of the probability p.

Remember, p is the quantity of numbers you're betting on, divided by 36, 37, 38 (no-zero, single-single or double-zero wheels respectively).

For example, the longest losing run for a street (3 numbers):

p = 3/37 = 0.0811 (single-zero, to 4 decimal places)

n = 25âÅ"•(1− 0.0811) / 0.0811 = 283

This is nearly 100 spins more than the maximum of 178 spins given in the stats above, but remember this is the worst case scenario, and records are being broken all the time.

Actually, that number (178 spins) is very close to a z-score of -4.0 (181 spins).

This is the master formula:

n = z2(1−p)/p  -- Longest losing run (general case)

For those not sure, z2 means zâÅ"•z (z multiplied by itself once).

For a z of -3.0, z2 = 32 = 9 (1 chance in 741 of this)
For a z of -4.0  z2 = 42 = 16 (1 chance in 31,574 of this)
For a z of -5.0  z2 = 52 = 25 (1 chance in 3,486,914 of this)
Title: Re: Standard Deviation in plain English
Post by: Toby on Feb 28, 01:39 PM 2011
Hi Bayes, it happens that you beging to collect trials and when you have about 2000 you may have a 6-number-sector with +3sd. The same sector has 4 sd after 4000 trials keeping the same edge.

You could have a 6-number-sector after 20k with 5sd and an edge of 3% or less.

Is there a way to compare what is better during the wheel clocking?

Example, +4sd on 10k trials or +4sd on 5000 trials? what is better and why?
Title: SD
Post by: Toby on May 29, 07:01 PM 2011
There is a confussion when you measure 1 2 3 o r more SDs.

Some players believe that having past data hey can look for -3SD there and try to explote the advantage.

The way is to pick a sector/dozen or so randomly, such as neighbor numbers, ECs or double streets.

It easier to find -3SDs in any group in 200 trials than to decide beforehand what group to check for a drawdown.

I guess we would need more SDs to pick from any of a group from past spins.

Best Regards
Title: Re: Standard Deviation in plain English
Post by: iggiv on May 29, 07:20 PM 2011
thanx Bayes. long time no see. u OK?
Title: Re: Standard Deviation in plain English
Post by: Bayes on Jun 04, 11:54 AM 2011
Just noticed this, I'm fine iggiv, how's yourself.

I see that the unicode math symbols have been messed up in this thread for some reason. Don't know why that is, maybe Vic can shed some light on it when he returns.
Title: Re: Standard Deviation in plain English
Post by: albertojonas on Jun 05, 07:13 PM 2011
hy bayes!

Maybe this is a silly question...
Is there any instrument to "tell" or give a clue on when or if the trend is inverting?

like completing a cycle?

i am more or less familiar with bollinger trends analises...

Great posts by the way
Congrats.
AL
Title: Re: Standard Deviation in plain English
Post by: Bayes on Jun 08, 05:27 AM 2011
Hi AJ,

If you know about bollinger bands you probably also know about other technical indicators like moving averages, elliot waves, fibonacci retracements etc used by technical traders. I suppose you could try any of these on roulette but you'd have to define the number of spins over which you choose to identify a trend, and it's all quite subjective. Personally I like point and figure charts (link:://en.wikipedia.org/wiki/Point_and_figure_chart); they're quite simple to use and are in some ways more objective than other methods because there are clear rules for when to enter markets. Just keep in mind that they're based on supply & demand, a concept which doesn't apply to roulette outcomes.  ;)
Title: Re: Standard Deviation in plain English
Post by: albertojonas on Jun 08, 01:31 PM 2011
That is very true: supply and demand. And roulette is of a different nature. maybe one can also use

link:://en.wikipedia.org/wiki/Stochastic_oscillator (link:://en.wikipedia.org/wiki/Stochastic_oscillator)

Precisely what I struggle with is to focus any observation in a window.
we all have made the naive question once: where do we enter the randomness? ???

-I try to develop some kind of detection to center myself on random before I start play.
it could be either z-score=x, wait till all 37 numbers appear, or other elaborate events depending on what I am betting.
do you use any or know of others? This is kind of a fun exploration to me.
Very interesting stuff,
thanks bayes.


Title: Re: Standard Deviation in plain English
Post by: Gizmotron on Jun 08, 02:23 PM 2011
Quote from: Bayes on Nov 02, 08:50 AM 2010
Here is a sample of some "maximum sleeps":460 - 540 spins with a straight up not showing. (maths expectancy is a hit once every 37 spins)
309 splits with a split not showing. (maths expectancy is a hit once every 18.5 spins or so)
178 spins with a street not showing. (maths expectancy is a hit once every 12.3 spins or so)
155 spins with a corner(square) not showing. (maths expectancy is a hit once every 4.6 spins or so)
93 spins with a line (double-street) not showing... (maths expectancy is a hit every 4.6 spins or so)
36-40 spins with a dozen/column not showing.(maths expectancy is a hit once every 3 spins or so)
20- 36 spins with any even money chance not showing. (expectancy is a hit once every two spins)
10 numbers may sleep for about 45-55 spins

That's great information. It's mathematical data on the existence of the elegant pattern  concept. That's very interesting about the sleeping dozen. By experience I see runs up to 30 every once in a while. A sleeping dozen occurs for 16 spins all the time, meaning during almost every four hour session. But that's from playing experience.

Also you mentioned later about this:
Quote from: Bayes on Jun 08, 05:27 AM 2011
Personally I like point and figure charts (link:://en.wikipedia.org/wiki/Point_and_figure_chart); they're quite simple to use and are in some ways more objective than other methods because there are clear rules for when to enter markets. Just keep in mind that they're based on supply & demand, a concept which doesn't apply to roulette outcomes.  ;)

Here is the easy to read charting I use in real casinos:


| L M H | 1 2 3 |  | B _R | L  H | O  E | -- ## -- Line
| X     |   X   |  | X    | X    |    X | --  8 --  1
|     X |     X |  |    X |    X |    X | -- 36 --  2
|   X   |     X |  | X    |    X |    X | -- 24 --  3
| X     |     X |  |    X | X    | X    | --  9 --  4
| X     |     X |  |    X | X    | X    | --  3 --  5
|   X   |     X |  | X    | X    | X    | -- 15 --  6
| X     |     X |  |    X | X    |    X | -- 12 --  7
|---------------------------------------|  -  0 --  8
|   X   |   X   |  |    X |    X | X    | -- 23 --  9
| X     |     X |  |    X | X    | X    | --  3 -- 10
|   X   | X     |  | X    | X    | X    | -- 13 -- 11
|   X   |     X |  | X    | X    | X    | -- 15 -- 12
| X     | X     |  | X    | X    |    X | -- 10 -- 13
|   X   | X     |  |    X |    X | X    | -- 19 -- 14
|   X   |     X |  |    X | X    |    X | -- 18 -- 15
|     X | X     |  | X    |    X | X    | -- 31 -- 16
| X     |   X   |  | X    | X    |    X | --  2 -- 17
| X     |   X   |  | X    | X    |    X | --  8 -- 18
|     X |   X   |  | X    |    X |    X | -- 26 -- 19
|     X |     X |  |    X |    X | X    | -- 27 -- 20
|     X | X     |  |    X |    X |    X | -- 34 -- 21
|   X   |     X |  |    X | X    |    X | -- 18 -- 22
|   X   |   X   |  | X    | X    | X    | -- 17 -- 23
|   X   |   X   |  | X    |    X |    X | -- 20 -- 24
|     X | X     |  | X    |    X | X    | -- 31 -- 25
|   X   |   X   |  | X    | X    | X    | -- 17 -- 26
| X     |   X   |  | X    | X    |    X | --  2 -- 27
|     X | X     |  |    X |    X | X    | -- 25 -- 28
Title: Re: Standard Deviation in plain English
Post by: albertojonas on Jun 08, 03:29 PM 2011
gizmotron, nice to have you here.  ;D

i play even chances in pairs looking for inbalance as a constatant
from the window chart above it would have the following results:

+7 on BR
+6 on lH
+3 on OE

how do you apply it gizmotron?
Title: Re: Standard Deviation in plain English
Post by: Gizmotron on Jun 08, 04:22 PM 2011
Quote from: albertojonas on Jun 08, 03:29 PM 2011
Gizmotron, ... how do you apply it gizmotron?

I use a juxtaposed form of condition identification. I picked up a good tip from John Patrick, mainly his use of the Regression System combined with "Up & Pull." I had to invent my own MM for betting on two dozens or two columns at once. But I also juxtapose that two doz/col with betting flat or progression with a single dozen or column bet. It all comes down to my ability to read the current conditions. I also just focus on the doz/col and leave all the rest to go off and do whatever it does. I don't watch the EC's anymore. I have my reasons.
Title: Re: Standard Deviation in plain English
Post by: albertojonas on Jun 08, 06:07 PM 2011
may you elaborate on that?
Title: Re: Standard Deviation in plain English
Post by: Gizmotron on Jun 08, 08:21 PM 2011
Quote from: albertojonas on Jun 08, 06:07 PM 2011
May you elaborate on that?

I look for singles or sleeping dozens occurring in the doz/col sections of the table. If they are dominating then I bet with them. I do this by betting the other two dozens that are not sleeping. Or I bet that another single will hit. If these conditions are absent and repeating doz/col are running strong then I bet single doz or col bets. Sometimes I bet flat, sometimes I bet my progression or my Up & Pull Money Management technique. I figured these things out on my own.  I don't see a compelling reason to share good MM techniques on an open forum. People are smart enough around here to figure out good balanced methods on their own. It's fun to watch them try too. (Look up "John Patrick, Up & Pull, and regression.")
Title: Re: Standard Deviation in plain English
Post by: hanshuckebein on Jun 29, 12:41 PM 2011
hi Bayes,

thanks a lot for this tremendous piece of work you've done for and presented to us all.  :thumbsup:

I must admit that reading and trying to understand it really keeps the blood in my brain circulating.  :)

my schooldays were over about 30 years ago and math was never one of my most beloved subjects. I never listen too well to what the teacher tried to tell me. it seems now it's all coming back to haunt me.  :o

may I ask one question: can I use the z-score also to measure the runs and the changes of an even chance?

say I have a result of R R R B R R B B R B.

so I have a run of  R first, then change to B, change to R, run of R, change to B, run of B, change to R, change to B.

and now I'm not interested in R or B as such but interested in how often the ball produces a run or a change no matter what colour

well, I hope I've made myself halfway clear and hope for an answer that suits a math-dummy like me.  :)

thanks and cheers

hans






Title: Re: Standard Deviation in plain English
Post by: Bayes on Jun 29, 03:56 PM 2011
Hi Hans,

You can indeed use the z-score on runs and chops. The 'law of series' says that a double  is half as likely as a chop, a run of 3 is half as likely as a double, a run of 4 is half as likely as a run of 3 etc. This means that you can use the standard formula with p = 0.5 (ignoring the zero effect). So to get the z-score of the singles vs other series (2,3,4...) just count the number of series as though they were red and the number of chops as though they were black (or vice-versa) and plug them into the formula. You can also use it for series vs longer series, so for example, ignoring all the chops, you could just look at the runs and calculate the z-score of the runs of 2 vs the remaining series (3,4,5,6...).

Hope this is clear, if not let me know and I'll post an example.
Title: Re: Standard Deviation in plain English
Post by: ZeroBlue on Jun 29, 07:06 PM 2011
Quote from: Bayes on Jun 29, 03:56 PM 2011
Hi Hans,

You can indeed use the z-score on runs and chops. The 'law of series' says that a double  is half as likely as a chop, a run of 3 is half as likely as a double, a run of 4 is half as likely as a run of 3 etc. This means that you can use the standard formula with p = 0.5 (ignoring the zero effect). So to get the z-score of the singles vs other series (2,3,4...) just count the number of series as though they were red and the number of chops as though they were black (or vice-versa) and plug them into the formula. You can also use it for series vs longer series, so for example, ignoring all the chops, you could just look at the runs and calculate the z-score of the runs of 2 vs the remaining series (3,4,5,6...).

Hope this is clear, if not let me know and I'll post an example.


Bayes do you presently apply Marigny de Grilleau »Le Gain scientifique d'une seule unité« ?
[/size]That is a very cool subject i would love a thread on that as it is part of the not very well told roulette myths.
[/size]
[/size]nice and concise explanation by the way.
[/size]Congratulations, Bayes
Title: Re: Standard Deviation in plain English
Post by: Bayes on Jun 30, 02:17 AM 2011
Hi Zeroblue,

Member 'ego' has discussed Marigny's methods in detail on the forum. Search for 'cut point methodology'. He's the expert on it so you should contact him if you have any questions.
Title: Re: Standard Deviation in plain English
Post by: hanshuckebein on Jun 30, 04:03 AM 2011
hi bayes,

thanks for your explanation. an example would be great.  :)

this is why I asked my question:


link:://rouletteforum.cc/roulette-and-gambling-framework/one-more-time-the-layout/ (link:://rouletteforum.cc/roulette-and-gambling-framework/one-more-time-the-layout/)

cheers

hans
Title: Re: Standard Deviation in plain English
Post by: Bayes on Jul 01, 03:24 AM 2011
Ok, the 'master' formula for calculating the z-score is <drum roll>

z = (w - np) / √(np(1 - p))

n = number of spins or events.
w = number of wins.
p = probability of the 'event' you're interested in.

Using your example:

Quotesay I have a result of R R R B R R B B R B.

Whenever you're using the formula, you have to be careful to define the number of events (n) and wins (w) in terms of what you want to know. For example, if you wanted to know the z-score for red vs black, you would count the total number of spins to find n, but in this case, the so called 'sample space' is divided into not red and black, but streaks and chops, so the total number of streaks and chops (n) is 5. Note that I've ignored the final B in the sequence because we don't know whether it's a chop or the first element of a streak.

It doesn't matter whether you choose wins to be chops or streaks, as long as you're consistent when interpreting the results. Let's take w to be the number of chops, in that case w = 2, and p = 0.5.

so z = (2 - 5 x 0.5) / √(5 x 0.5 x 0.5) = -0.447

The essence of Marigny's method is to track the chops and streaks until you get a z-score of  ± 3.0. You then bet for the 'correction'.
Title: Re: Standard Deviation in plain English
Post by: monaco on Jul 11, 07:20 AM 2011
hi Bayes, fascinating thread!

I read one of your posts in another thread, you wrote something along the lines of ‘not all bet selections are equal’ â€" by this, are you saying that different bet selections deliver different z scores? & so need to be chosen & tailored very specifically in their application to any system/method of play?
Title: Re: Standard Deviation in plain English
Post by: hanshuckebein on Jul 20, 10:48 AM 2011
hi bayes,

sorry for my very late response to your answer regarding the z-score. I took a few days off from my pc and from thinking about roulette.  :)

well, now I'm back and want to say "thanks a lot" for the formula and your explanation.  :thumbsup:

cheers

hans
Title: Re: Standard Deviation in plain English
Post by: MrJ on Aug 13, 01:48 PM 2011
I read this thread twice a month......GREAT STUFF.

Ken
Title: Re: Standard Deviation in plain English
Post by: Blood Angel on Aug 13, 04:16 PM 2011
Hi Bayes
So the expectancy is that a number to show is 1/37 then what's a repeats and a trebles please? Maths isnt my strong point!!

Thank you in  advance for your time.
Title: Re: Standard Deviation in plain English
Post by: Bayes on Sep 13, 10:14 AM 2011
Quote from: Blood Angel on Aug 13, 04:16 PM 2011
Hi Bayes
So the expectancy is that a number to show is 1/37 then what's a repeats and a trebles please? Maths isnt my strong point!!

Thank you in  advance for your time.

Sorry, I missed this one. For a double it's 1/37 × 1/37 and a treble is 1/37 × 1/37 × 1/37.
Title: Re: Standard Deviation in plain English
Post by: Blood Angel on Sep 13, 10:29 AM 2011
Quote from: Bayes on Sep 13, 10:14 AM 2011
Sorry, I missed this one. For a double it's 1/37 × 1/37 and a treble is 1/37 × 1/37 × 1/37.
Thank you for your answer.
Title: Re: Standard Deviation and z-score in plain English
Post by: Drazen on Oct 20, 12:17 PM 2012
What is wrong with the forum code so that all formulas here are incorectly shown?

Who can/has to fix that?

Cheers

Drazen
Title: Re: Standard Deviation and z-score in plain English
Post by: Ralph on Oct 20, 04:24 PM 2012
Do not forget 1/37 * 1/37 *1/37 is the chance a specific number will show three times, before any spin. That any number will show three times is 1/37 * 1/37.

A double hit on any number is 1/37.
Title: Re: Standard Deviation and z-score in plain English
Post by: Blood Angel on Oct 20, 04:44 PM 2012
Thank you Ralph :)
Title: Re: Standard Deviation and z-score in plain English
Post by: rouletteKEY on Aug 15, 10:37 PM 2013
I just thought I would hit this thread and bring it back up to the forefront...it has alot of good info and is worth a read for some of the newer people here and a re-read for some that have been around a little longer.
Title: Re: Standard Deviation and z-score in plain English
Post by: bigmoney on Feb 28, 04:23 AM 2022
outstanding
Title: Re: Standard Deviation and z-score in plain English
Post by: bigmoney on Feb 28, 06:17 AM 2022
i feel this thread need more posts
Title: Re: Standard Deviation and z-score in plain English
Post by: Roulettebeater on Mar 01, 03:32 AM 2022
Math is cool but it won’t change the odds.
The game from a pure mathematical perspective is a losing game, the only two ways to win are : increase accuracy or play hit/ run
Title: Re: Standard Deviation and z-score in plain English
Post by: Taotie on Mar 01, 04:41 AM 2022
Quote from: Roulettebeater on Mar 01, 03:32 AM 2022the only two ways to win are : increase accuracy or play hit/ run

So what do we do if we play don't hit/    ?

Do we still run? Do we keep playing until hit/run? Or hit/ recover a bit?

And what's that, a little bit, a big bit?

You know hit/run is bullshit don't you?

Do the math.
Title: Re: Standard Deviation and z-score in plain English
Post by: ego on Mar 05, 06:02 AM 2022

Here is an excel sheet from Bayes, American, Europa wheel and Baccarat Z-score calculator.
See attachment
Title: Re: Standard Deviation and z-score in plain English
Post by: ego on Mar 05, 06:06 AM 2022

I can explain how, where and when to bet for regression toward the mean (( if anyone is interested ))

That 68.3% of the time the divergence would be one SD or less. Either side of the MEAN.
That 95% of the time the divergence would be 2 SD's or less. Either side of the MEAN.
That 99.7% of the time the divergence would be 3 SD's or less. Either side of the MEAN.
That only 0.3% of the time would the divergence exceed 3 SD's
Title: Re: Standard Deviation and z-score in plain English
Post by: poluvolo on Mar 05, 08:33 AM 2022
Yes  I am very serious interest