Standard Deviation and z-score in plain English

[Bayes]

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.

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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[VLS]

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 

[Bayes]

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?  

[Blood Angel]

I enjoy your posts bayes.

[VLS]

I enjoy your posts bayes.

Count 2 with me.

There you go dear Bayes; you've got audience

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

[MrJ]

One of the best threads I have seen in years!  Ken

[mr.ore]

Really, really nice thread 

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?

[mr.ore]

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.

[Bayes]

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.

[mr.ore]

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.

[mr.ore]

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.

[mr.ore]

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.

[Blood Angel]

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 

[Blood Angel]

Hi Bayes

Is SD and Z score the same thing?

[Bayes]

Ah, you must have read my mind, I was going to cover that next.  

Not exactly, but they're very closely related. Details to follow...