Variance and standard deviation, help me out in understanding them

Could somebody explain these to me? I looked up variance on Wikipedia, but it's so freakin complex. I read that variance is the square of the standard deviation I think? Why have both then? I understand how to read the numbers, such as a variance of 20 on video poker is very low, and 80 is very high. I know high variance means the swings will be bigger. I understand that, but what exactly do the numbers mean exactly?

Oh, and how would n-play video poker make the variance go up? (See link below.) Shouldn't it make it go down? I could swear I remember reading that n-play video poker will smooth out the ups and downs, which makes total sense. This is why I need the actual number explained to me.

https://wizardofodds.com/videopoker/appendix3.html

Something about a standard deviation being either negative or positive, but if you square it you will always be dealing with a positive number. I guess you can tell I really don't know the answer as to why there are both terms.

Both variance and standard deviation are a measure of how broad distributed you can expect random results.

Mathematicians love variance, which is the average squared distance from the mean. Variance provides easy properties for calculation of random results (most important you can sum up variances from different bets) , however - since it is a squared property (in terms of units) it is not easy to imagine.

Standard deviation, as the root of the variance, has the same unit as expectation value, and hence both standard deviation and expectation value are comparable. This is nice for calculating probabilities (i.e. you have a 68% chance that your results are within 1 standard deviation around your expectation value if you have a large number of plays). However, from the mathematical view standard deviation is quite messy, and not much useful for any serious calculations.

Most people confuse standard deviation and variance, and use the term "variance" for any kind of randomness. In common conversations this is not a big problem, when variance is low so is standard deviation (and vice versa). But if you are up to actual numbers, you need to understand.

The thing about multi-play video poker and variance is a different story. You need to introduce the concept of correlation.
Different rounds of video poker are fully independent, and as independent rounds their variance simply add up. (the standard deviation doesn't add up, as it is a square root). If you were to play 100 rounds of video poker, you would get 100 times the variance of a single round.

Now, what happens if you play a 100-hand video poker machine ? Now each hand is not independent, as you select your cards for all 100 cards simultaneous. Hence all 100 hands are correlated, which INCREASE variance for that 100-hand round. However, since you can play 100-hand video poker at a lot less stake per hand, one round of 100-hand VP at betsize 1/100 unit each hand has LESS variance than one round of a single hand VP at betsize of 1 unit.
The expectation value however is the same (as the stake is the same each round), but the variance is greatly reduced.

I don't think I'll ever understand beyond simply knowing what values are high and what are low, like knowing to stay away from something with a variance of 80 if you have a small bankroll. Is a variance of 80 the same across everything, or is it relative to something?

And I just want to know what the actual number means. 5 "deviations", or 45 "variance" OK, what is a deviation/variance? To go outside the norm, etc., I know, but 1 standard deviation is what? Within 68% of the average or something? OK well where did that 68% come from? Frustrating.

I should probably just use Google on this, but give me an example. Like a variance of 30 (or SD of 5 or something), in terms of flipping a coin, would mean...

Quote: SilentBob420BMFJ

I should probably just use Google on this, but give me an example. Like a variance of 30 (or SD of 5 or something), in terms of flipping a coin, would mean...

If examples help then you can ponder this WoO webpage.

But don't over think this. For example recently I came to realize that the statement in that webpage that says "The standard deviation of the final result over [so many] bets is the product of the standard deviation for one bet (see table) and the square root of the number of initial bets made in the session" outlines something that is actually quite easy for anyone to calculate. The word "product" indeed means multiplying these items... I was far enough removed from my days of taking math to have forgotten that at first.

What you wind up with is something that has some meaning. OK, one standard deviation should have been this much won or lost, I can compare that to what happened or what I expect to happen.

Just 2 cents from somebody who struggles with this too.

Variance of 80 is relative. There is another measure called coefficient of variance, which can be compared across games (although it isn't used often)

The 68% is an area under a pdf curve for the normal distribution.

There is a difference between the variance (or standard deviation) of a distribution, and an event being a specific # of standard deviations away from the average for that distribution.

If video poker has a variance of 80, it means that the sum of the squares of each (result - average) is 80. But when you actually play some hands of video poker, you will have a specific outcome (like, down 20 units). You can the calculate how likely it was that you got the specific outcome in terms of deviations from the norm.

so, variance of 30 for coin flipping doesn't mean anything, until you start talking about flipping a coin a certain number of times. If you flip a coin 22 times, and it is heads all 22 times, that event is 5 standard deviations away from the norm.

Quote: dwheatley

Variance of 80 is relative. There is another measure called coefficient of variance, which can be compared across games (although it isn't used often)

The 68% is an area under a pdf curve for the normal distribution.

There is a difference between the variance (or standard deviation) of a distribution, and an event being a specific # of standard deviations away from the average for that distribution.

If video poker has a variance of 80, it means that the sum of the squares of each (result - average) is 80. But when you actually play some hands of video poker, you will have a specific outcome (like, down 20 units). You can the calculate how likely it was that you got the specific outcome in terms of deviations from the norm.

so, variance of 30 for coin flipping doesn't mean anything, until you start talking about flipping a coin a certain number of times. If you flip a coin 22 times, and it is heads all 22 times, that event is 5 standard deviations away from the norm.

So, are you saying that variance in video poker is based on a certain number of hands? How many? Kinda like how mph, mpg, etc. are based on going a certain distance/amount.

And if that's not it, and what it means is that with SD of 4, it's saying at any given time, you shouldn't be more than 4 SD away, well then I have a question about that. With what % chance are they saying you will be within a SD of 4?

"This statistic is commonly used to calculate the probability that the end result of a session of a defined number of bets will be within certain bounds."

What number of bets and what bounds? For instance, for JOB VP it's 19 variance, which would make it a SD of 4.36. OK, so explain what that's saying. And I'm confused, because SD is a %, so why not just say the %? If 1 SD is 68%, why not just say you have a 68% chance? All this is confusing as hell.

Please forgive me if this starts out not being directly gambling, so here goes.

Suppose someone is making pencils, filling coke bottles, cans of food, whatever. It is almost impossible to make a pencil exactly 6" long. Typically there will be an average and, due to manufacturing, different length pencils will actually be produced. Most natural things approximate to the normal curve/distribution.

If we wanted to ensure that a certain percentage of pencils were 6" or longer (99% 95% or whatever), we would have to set the average length to be manufactured at more than 6". On the other hand, we might want to lower the variance (i.e. better quality production) and reduce the average error. The SD gives a measure of how far out we usually are. So for instance if it was .1", we might aim to produce pencils of average 6.2", so usually they would be 6.0" to 6.4". However if we were much better at manufacturing, we might be able to aim towards an average of 6.04".

Back to gambling. The usual question is each time I make a bet of (say) £1 what will happen. The average tells us that in the long term we'll get back (say) 98p per £ (or c per $ if you prefer). Obviously, normally, we'll either win multiples of £1 or nothing, and after (say) making 1000 bets we'll have won back something (or possibly nothing!) The SD gives an indication of how near £980 we're likely to be - like the pencils how far out we usually are. If the SD is large then we might make a lot of money, or have lost a lot. If the SD is low (such as even money roulette or punto banco bets) it is likely we'll be (say) between 900 and 1100 - so we would tend not to lose a lot nor win a lot - it's less "risky".

You may have heard risk associated with investments - essentially they are bets (albeit with positive expectations). A (good!) bank pays the same, called interest - little risk - no SD you know what you'll get back at end of year (X + Y%). A Dow Jones share has some risk - medium SD. A minor oil drilling company, or similar, might have major risk - the upsides are good - but the risk of losing everything is quite high! As they say you pays your money and takes your choice.

PS The reason for the 68% is that is the chance of being within 1SD.
In our exaggerated example 68% of pencils will be 6.1" to 6.3", 95% will be 6.0" to 6.4" (2 SDs), 99% will be 5.9" to 6.5" (3 SDs) etc. By setting the average to 6.2 we are 97.5% "confident" the pencil will be >6" (assuming we're happy with the 2.5% pencils that are too long!)

Quote: charliepatrick

Please forgive me if this starts out not being directly gambling, so here goes.

Suppose someone is making pencils, filling coke bottles, cans of food, whatever. It is almost impossible to make a pencil exactly 6" long. Typically there will be an average and, due to manufacturing, different length pencils will actually be produced. Most natural things approximate to the normal curve/distribution.

If we wanted to ensure that a certain percentage of pencils were 6" or longer (99% 95% or whatever), we would have to set the average length to be manufactured at more than 6". On the other hand, we might want to lower the variance (i.e. better quality production) and reduce the average error. The SD gives a measure of how far out we usually are. So for instance if it was .1", we might aim to produce pencils of average 6.2", so usually they would be 6.0" to 6.4". However if we were much better at manufacturing, we might be able to aim towards an average of 6.04".

Back to gambling. The usual question is each time I make a bet of (say) £1 what will happen. The average tells us that in the long term we'll get back (say) 98p per £ (or c per $ if you prefer). Obviously, normally, we'll either win multiples of £1 or nothing, and after (say) making 1000 bets we'll have won back something (or possibly nothing!) The SD gives an indication of how near £980 we're likely to be - like the pencils how far out we usually are. If the SD is large then we might make a lot of money, or have lost a lot. If the SD is low (such as even money roulette or punto banco bets) it is likely we'll be (say) between 900 and 1100 - so we would tend not to lose a lot nor win a lot - it's less "risky".

You may have heard risk associated with investments - essentially they are bets (albeit with positive expectations). A (good!) bank pays the same, called interest - little risk - no SD you know what you'll get back at end of year (X + Y%). A Dow Jones share has some risk - medium SD. A minor oil drilling company, or similar, might have major risk - the upsides are good - but the risk of losing everything is quite high! As they say you pays your money and takes your choice.

PS The reason for the 68% is that is the chance of being within 1SD.
In our exaggerated example 68% of pencils will be 6.1" to 6.3", 95% will be 6.0" to 6.4" (2 SDs), 99% will be 5.9" to 6.5" (3 SDs) etc. By setting the average to 6.2 we are 97.5% "confident" the pencil will be >6" (assuming we're happy with the 2.5% pencils that are too long!)

Good post in explaining what SD is, but I already know basically what it is. I just want to know how the numbers are used. Your last paragraph starts to talk about it I suppose. Like what does 1 SD mean. That it's a 68% chance it will be exactly the mean? So like I said in my previous post, why talk in SD, why not just talk in %? So instead of saying something is 1 SD, just say it has a 68% accuracy rate or something like that.

When the realm of data is known (Say, a Blackjack game of a given set of rules or a slot machine with the expected frequency of results known), the standard deviation helps identify how expected a given result is. I'll give an example that's less number-oriented, but I think it makes the point.

To wit, if I told you I lost 10 hands of BJ consecutively on XXX online gambling site and was whining that the site was rigged, you would all give me the finger and tell me to f--- off (some more politely than others ;)). You would all do this because, while unfortunate, a -10 streak in blackjack isn't terribly uncommon.

Now, if I told you (and properly documented) a 200-straight hand losing streak in that same BJ, people would start to raise eyebrows. Perhaps the game isn't actually being played as it's advertised. And the reason one might think that is because the results are just so extraordinarily unexpected in this game.

Similarly, if I told you I played 200 spins of a "Hundred-or-nothing" slot machine and didn't win once and made similar complaints, you would all tell me to shut up again. That result - even 200 losses in a row - is just downright expected on this game because it's more variant. It might take 1,000 straight losing (or even more) spins of this game before anyone reasonably suspected foul play.

On the flip side, the casinos leverage the same logic. If I hit 3 major jackpots in mere hours from each other, maybe I'm doing something that isn't on the up-and-up to "entice" these outcomes that would otherwise take weeks, months or years to happen. Or, maybe I'm just damned lucky. Just because something is significantly deviant from what's expected doesn't make it impossible. Merely, as Marvin the Paranoid Android would say, improbable.

So, back to your 2 video poker machines. Your machine with a SD of 20 will "win" more often, but the wins will be smaller. Over a stretch of time, you'll lose, lose, win, lose, win, win, lose, win, lose, lose, lose, lose, win, win... On the machine with an SD of 80, you won't win as often, but when you do, it'll be big wins: lose, lose, lose, lose, win, lose, lose, WIN, lose, lose, lose, lose, lose, lose, lose, lose, lose, win, lose, lose, lose, lose, lose, lose, lose, lose, lose, lose, OMFG WIN

Over a billion hands, these two machines might have the same house edge (the same as expected value or weighted mean of all possible results from the machine), but if you have a single Benny in your pocket and a limited number of hands to play, you have a choice. You can pick the machine that will probably let you play for a longer time, sacrificing the possibility of that big hit. Or you can play the machine that will probably drain you of your money pretty fast. But maybe, just maybe you'll be the one who will hit it big. Standard deviation differentiates these two machines.

Is that helpful?

To add - there are two ways one might want to look at things.

(1) The first is before the event occurs and asks what are the likely outcomes. As you understand the average gives the expected value (e.g. 98% payback) and the standard deviation the range or variety of results.

To give a similar example to the fruit machine, suppose there was a FAIR roulette table (it makes the maths easier): even money bets at roulette are similar to tossing a coin - at the end of 1000 spins the average is win 500 lose 500. It is intuitive that you are unlikely to win or lose a lot - here is the maths.

Using 2SDs (if I've got it right n=1000 p=1/2 q=1/2 SD=SQRT(npq)) means 95% the result will lie 470-530 (i.e +/- 30 units). Using 3SDs 99% 455-545. Thus knowing (a) the SD AND (b) setting your risk (i.e. only 1 in 100 will I do worse) gives SD=15 #SDs=3, hence range = 455-545. What you need to know is 68%=1SD, 95%=2SD, 99%=3SDs etc. I think these are known a confidence levels. (The actual number about 15.811 but I've used 15 to keep numbers simple.)

(2) Now let's turn the question around and say someone has come along AFTER the event and said there have been n reds and (N-n) blacks. Using reverse logic we can work out how likely this should have been.

So if you said I lost 200 units in 10000 spins - the maths N=10000 S(Npq)=S(2500)=50. The number of wins was 4900 (cf 5100) and implies 100 was 2SDs away, so while unlucky, it lied within a 95%, so is not suspicious.

So if you said I lost 1000 units in 10000 spins - this is 10SDs away from the average and hence is highly suspicious.

(3) Finally let's pose this as a question. Suppose you were only betting one number and said it had only come up a certain number of times, what might we expect before being suspicious. N = 1000 p=1/36 q=35/36.

SD = Sqrt(N p q) = 5 (rounded), Average = 1000/36 = 28 (rounded). So one would expect 99% of the time 28 +/- 3*5 or 13 to 43. I think you'd be really unlucky if it never came up, but possibly not enough that you could say for certain the wheel was not true. However if you waited say 5000 spins with no number, then it would be suspicious.

Also note that playing 1000 red/black our 99% confidence was 455-545 wins, ie bank=910-1090. Playing single numbers, 13 to 43 wins, i.e. bank= 468-1548 (the numbers are not symmetrical due to rounding assumptions).

The "Average" will be the same (in our case 1000), but you can see the range is about 100 either way for Evens and 500 either way for Straight-Up. This is what a larger SD tells us.

Hope this helps.

I completely understand SD and variance as far as your post goes. I already knew all this. When you said "I'll give an example that's less number-oriented", I was like *facepalm*. Thanks for trying, but I want the opposite. I want to know what the numbers mean. I don't think anybody is understanding me. I know that a variance of 80 is far worse than a variance of 20, and that you'll have to play longer to achieve the mean. I know all that stuff. What I want to know is what does a variance of 20 mean? In WinPoker, it tells the variance, and it says "In single coin bets", so what does that mean, that I will be +/- 20 coins over a certain amount of hands? What amount?

I'm currently reading Charlie's post, which from a glance looks like what I'm looking for.

Quote: charliepatrick

To add - there are two ways one might want to look at things.

(1) The first is before the event occurs and asks what are the likely outcomes. As you understand the average gives the expected value (e.g. 98% payback) and the standard deviation the range or variety of results.

To give a similar example to the fruit machine, suppose there was a FAIR roulette table (it makes the maths easier): even money bets at roulette are similar to tossing a coin - at the end of 1000 spins the average is win 500 lose 500. It is intuitive that you are unlikely to win or lose a lot - here is the maths.

Using 2SDs (if I've got it right n=1000 p=1/2 q=1/2 SD=SQRT(npq)) means 95% the result will lie 470-530 (i.e +/- 30 units). Using 3SDs 99% 455-545. Thus knowing (a) the SD AND (b) setting your risk (i.e. only 1 in 100 will I do worse) gives SD=15 #SDs=3, hence range = 455-545. What you need to know is 68%=1SD, 95%=2SD, 99%=3SDs etc. I think these are known a confidence levels. (The actual number about 15.811 but I've used 15 to keep numbers simple.)

(2) Now let's turn the question around and say someone has come along AFTER the event and said there have been n reds and (N-n) blacks. Using reverse logic we can work out how likely this should have been.

So if you said I lost 200 units in 10000 spins - the maths N=10000 S(Npq)=S(2500)=50. The number of wins was 4900 (cf 5100) and implies 100 was 2SDs away, so while unlucky, it lied within a 95%, so is not suspicious.

So if you said I lost 1000 units in 10000 spins - this is 10SDs away from the average and hence is highly suspicious.

(3) Finally let's pose this as a question. Suppose you were only betting one number and said it had only come up a certain number of times, what might we expect before being suspicious. N = 1000 p=1/36 q=35/36.

SD = Sqrt(N p q) = 5 (rounded), Average = 1000/36 = 28 (rounded). So one would expect 99% of the time 28 +/- 3*5 or 13 to 43. I think you'd be really unlucky if it never came up, but possibly not enough that you could say for certain the wheel was not true. However if you waited say 5000 spins with no number, then it would be suspicious.

Also note that playing 1000 red/black our 99% confidence was 455-545 wins, ie bank=910-1090. Playing single numbers, 13 to 43 wins, i.e. bank= 468-1548 (the numbers are not symmetrical due to rounding assumptions).

The "Average" will be the same (in our case 1000), but you can see the range is about 100 either way for Evens and 500 either way for Straight-Up. This is what a larger SD tells us.

Hope this helps.

How did you get +/- 30 units?

And I understand that if something is 10 SD away, it means 99.999 certainty it shouldn't have happened, but then what does it mean if a game has a SD of let's say 4? In your example, SD of x is saying how rare it was, but when you say a game has a SD of 4, I don't see how you can turn that into a sentence. After the fact, something was x SD away, OK got it, but before hand, what does it mean? It basically sounds like SD is a measurement of how rare something was, so I don't see how it applies to something before the fact. Saying a game has a SD of 4, well it could have an SD of 10 after the fact, or 1, or 13, etc.

Quote: SilentBob420BMFJ

I know all that stuff. What I want to know is what does a variance of 20 mean?

Variance - by definition - is the mean squared distance of results from their average.

A fair even bet (i.e. a coin flip paying 1to1) hence has a variance of 1 units^2. Since it is fair, the average is 0.
The possible results are -1 and +1, and hence the variance is (1/2) * (-1 - 0)^2 + (1/2) * (1 - 0)^2 = 1.
If the min bet for this coin flip is $5, the variance is $²25:
(1/2) * (-$5 - 0)^2 + (1/2) * ($5 - 0)^2 = $²25.

A variance of 20 means, the average squared distance from the mean value is 20. Whatever that means to you. A more intuitively measure is the square root of variance, the standard deviation. It tells you the rough spread of results around the mean, but you know that already - right ?

Quote: SilentBob420BMFJ

How did you get +/- 30 units?

cp used the formula "square root of n*p*q" in his binomial example to get SD=15.

His 4th paragraph is what you need to understand. He laid out the example that is easy to follow.
It is for only a binomial standard deviation where there is only success or failure as for one trial outcome.

mean is 500
sd=15
So, a final result between 485-515 is 1sd from the mean or average (plus and minus) as a "range of possible outcomes".
2sd and 3sd and 4sd etc.

This is simple math. Just play with it a bit and it will make sense.

Now the standard deviation of a set of data (population) is done differently like the average height of 100 women.
check out this tutorial starting at: http://stattrek.com/Lesson3/Variability.aspx?Tutorial=Stat
maybe it will help more

Another way of looking at variance:

A sum of a large number of i.i.d. random variables (i.i.d. means independent, identical distributed), i.e. a session of roulette, approaches a Gaussian distribution (the bell-shaped curve).
The only parameters of a Gaussian distribution are the average value of that distribution (the mean result), and it's width (the variance).
Hence if you plan to play the same game a large number of times, regardless of the payout structure, your result will end up in a Gaussian distribution.
One can calculate the average and width of the Gaussian distribution in principle by the paytable and rules of any game.

What people mean if they say "result X is Y SD away", is: According to the calculated Gaussian distribution of results, your specific result X is far off. With "far", it is relative to the intrinsic width of that distribution (the SD). For any specific result, it is highly unprobable (but not impossible) to be many many SD's away from the mean. If you are, it is very likely that either the calculation of this distribution (from the theory of the game) is wrong, you play not optimally, or you're being cheated. Of course there is also the element of bad luck. But with the Gaussian distribution, you can calculate how likely it was "bad luck".

As it's already said: behind more than 2 SD's is a "bad luck" of about 2.5%. (You are within 2SD's within 95%, ahead 2.5%, and behind 2.5%).


I honestly don't know why you always say "you know all that", but then on the other hand ask the same questions whose answers are already given to you.

Quote: charliepatrick

To give a similar example to the fruit machine, suppose there was a FAIR roulette table (it makes the maths easier): even money bets at roulette are similar to tossing a coin - at the end of 1000 spins the average is win 500 lose 500. It is intuitive that you are unlikely to win or lose a lot - here is the maths.

Using 2SDs (if I've got it right n=1000 p=1/2 q=1/2 SD=SQRT(npq)) means 95% the result will lie 470-530 (i.e +/- 30 units). Using 3SDs 99% 455-545. Thus knowing (a) the SD AND (b) setting your risk (i.e. only 1 in 100 will I do worse) gives SD=15 #SDs=3, hence range = 455-545. What you need to know is 68%=1SD, 95%=2SD, 99%=3SDs etc. I think these are known a confidence levels. (The actual number about 15.811 but I've used 15 to keep numbers simple.)

maybe a few pics of the above example: one can see the probability of falling within a range next to the probability button.
1SD range 485-515

2SD range 470-530

3SD range 455-545

There is still a slight probability of seeing 0 to 454 successes as is 555 to 1000. Rare but not improbable.

So, even before the 1000 trials begins we have an idea of the possible number of successes that can happen using EV and SD values.
(EV: 1000 * 50% = 500.)
Of course we do not know their order of successes and failures.

This stuff is over my head, I will end it like that. For instance I don't know what intrinsic or binomial means without looking them up. Most people won't admit this, but I just did, so give me a break. And what I meant was that people were explaining SD and variance in an extremely basic way, such as what it generally means, when in fact I'm trying to understand what it means on a more specific level, one which is too far for me to reach apparently.

Quote: SilentBob420BMFJ

This stuff is over my head, I will end it like that. For instance I don't know what intrinsic or binomial means without looking them up. Most people won't admit this, but I just did, so give me a break. And what I meant was that people were explaining SD and variance in an extremely basic way, such as what it generally means, when in fact I'm trying to understand what it means on a more specific level, one which is too far for me to reach apparently.

I agree with you that some of this is just intimidating and not helpful. Having some of these folks help me with these things, I can't think they are trying to make it more difficult, but they have forgotten what it is like not to have certain concepts down. They'd probably have to go back to being 9 years old and just can't remember what it was like.

I still say you may want to keep going at this WoO webpage [the standard deviation section for sure] until you get it down, but then again maybe I don't know what you are after exactly either.

Without resorting to too much maths consider you walked into a very generous casino that offered three games. The first game allows you to bet $1 on either Heads or Tails, whence you win $1 if you are correct (or lose $1 if you're wrong). The second game allows you to bet $1 on any of the dice numbers 1 to 6 (your choice) and pays $5 if you are correct (or lose $1). The third game requires two dice and pays $35 if you thrown two sixes, else you lose your $1. I'm not sure if it's obvious, if not trust me, but all the games are entirely fair in that in the long run you'll neither win nor lose.

However the is one key difference between the games might be phrased such as: (i) which game gives me a better chance of winning a large amount of money or (ii) which game gives me a better chance of lasting the entire evening.

In the Head or Tails game you are unlikely to lose a lot and similarly unlikely to win a lot.

In the two dice game, it is perfectly possible you may never thrown a 66 before your money runs out or it may be your night and you thrown quite a few 66s and make a lot of money.

Thus the first game has a narrower range of results (unlikely to lose/win a lot) whereas the third game has a wider range (money runs out/make a lot of money). The second game is somewhere in-between. The variance or Standard Deviation is merely a mathematical method of calculating and expressing in a quantifiable way how wild or different results are likely to be and help answer questions such as if I start with $100 what are my chances of ....?

The important thing to recognize is the heads/tails is less risky whereas the 66s has more risk - potentially more chance to win big but also more chance to lose your bankroll. The numbers quoted at the start of this thread express this risk more precisely and allow a more informed choice as to which type of game you'd prefer to play.

A cohesive explanation that even I could understand. Well done SIR !!

Quote: charliepatrick

Without resorting to too much maths consider you walked into a very generous casino that offered three games. The first game allows you to bet $1 on either Heads or Tails, whence you win $1 if you are correct (or lose $1 if you're wrong). The second game allows you to bet $1 on any of the dice numbers 1 to 6 (your choice) and pays $5 if you are correct (or lose $1). The third game requires two dice and pays $35 if you thrown two sixes, else you lose your $1. I'm not sure if it's obvious, if not trust me, but all the games are entirely fair in that in the long run you'll neither win nor lose.

However the is one key difference between the games might be phrased such as: (i) which game gives me a better chance of winning a large amount of money or (ii) which game gives me a better chance of lasting the entire evening.

In the Head or Tails game you are unlikely to lose a lot and similarly unlikely to win a lot.

In the two dice game, it is perfectly possible you may never thrown a 66 before your money runs out or it may be your night and you thrown quite a few 66s and make a lot of money.

Thus the first game has a narrower range of results (unlikely to lose/win a lot) whereas the third game has a wider range (money runs out/make a lot of money). The second game is somewhere in-between. The variance or Standard Deviation is merely a mathematical method of calculating and expressing in a quantifiable way how wild or different results are likely to be and help answer questions such as if I start with $100 what are my chances of ....?

The important thing to recognize is the heads/tails is less risky whereas the 66s has more risk - potentially more chance to win big but also more chance to lose your bankroll. The numbers quoted at the start of this thread express this risk more precisely and allow a more informed choice as to which type of game you'd prefer to play.

Is this calculation correct for the third game - rolling two sixes?

Variance is 1260 and SD is 35.5, which means 95% of the time you should expect to lose or win up to $71.

I think I forgot to divide the 1260 number by the total possible outcomes.

Now I think its a variance of 35 and an SD of 5.92

Quote: charliepatrick

Without resorting to too much maths consider you walked into a very generous casino that offered three games. The first game allows you to bet $1 on either Heads or Tails, whence you win $1 if you are correct (or lose $1 if you're wrong). The second game allows you to bet $1 on any of the dice numbers 1 to 6 (your choice) and pays $5 if you are correct (or lose $1). The third game requires two dice and pays $35 if you thrown two sixes, else you lose your $1. I'm not sure if it's obvious, if not trust me, but all the games are entirely fair in that in the long run you'll neither win nor lose.

However the is one key difference between the games might be phrased such as: (i) which game gives me a better chance of winning a large amount of money or (ii) which game gives me a better chance of lasting the entire evening.

In the Head or Tails game you are unlikely to lose a lot and similarly unlikely to win a lot.

In the two dice game, it is perfectly possible you may never thrown a 66 before your money runs out or it may be your night and you thrown quite a few 66s and make a lot of money.

Thus the first game has a narrower range of results (unlikely to lose/win a lot) whereas the third game has a wider range (money runs out/make a lot of money). The second game is somewhere in-between. The variance or Standard Deviation is merely a mathematical method of calculating and expressing in a quantifiable way how wild or different results are likely to be and help answer questions such as if I start with $100 what are my chances of ....?

The important thing to recognize is the heads/tails is less risky whereas the 66s has more risk - potentially more chance to win big but also more chance to lose your bankroll. The numbers quoted at the start of this thread express this risk more precisely and allow a more informed choice as to which type of game you'd prefer to play.

Again, a great post, but this is too basic. I know what variance is on this level. I know that 2 games with the exact same EV, can be completely different in the short term. Sometimes it's so bad, that it's not worth playing, such as is the case with the St Petersburg coin flip paradox, or when the lottery gets to an astronomical $1,000,000,000. Both games are by far worth playing in the long run, but both are horrible to play in reality.

Quote: odiousgambit

I agree with you that some of this is just intimidating and not helpful. Having some of these folks help me with these things, I can't think they are trying to make it more difficult, but they have forgotten what it is like not to have certain concepts down. They'd probably have to go back to being 9 years old and just can't remember what it was like.

I still say you may want to keep going at this WoO webpage [the standard deviation section for sure] until you get it down, but then again maybe I don't know what you are after exactly either.

What I'm after is to understand exactly what the numbers mean in terms of bets/hands when it comes to the variance of video poker. A variance of 20 in video poker means what? I will give a random example that's clearly wrong, but shows you what I'm looking for:

A variance of 20 in video poker means that for every coin you bet, you will be within 20 coins with a 90% chance after you play 100 hands.

Quote: SilentBob420BMFJ

What I'm after is to understand exactly what the numbers mean in terms of bets/hands when it comes to the variance of video poker. A variance of 20 in video poker means what? I will give a random example that's clearly wrong, but shows you what I'm looking for:

A variance of 20 in video poker means that for every coin you bet, you will be within 20 coins with a 90% chance after you play 100 hands.

First of all, a variance of 20 means a standard deviation of sqrt(20) = 4.5. All your fluctuation estimates will be in units of standard deviations.

Second, the probability of being within a single standard deviation around the mean is 65% only if the corresponding total distribution is normal.

Third, although any finite payout game will turn into a normal distribution when considering the total of all those plays, the open question still remains: What number of plays will be sufficiently close to infinite? Clearly, 100 hands will not be enough.
This question you cannot answer in general, as it will depend on the specific game distribution. If the distribution is very wide (having rare events that pay high), not only the standard deviation (or the variance) will be high, but also the number of plays untill you can consider being close to a normal distribution will be high. (This will not depend on variance at all, but in leading order on kurtosis - basically the deviation of the individual game distribution from a normal distribution).


To put that in order:
You start with a game, which has a given probability of results and a given payout. This distribution will be not normal (that means has the Gaussian bell shape). In case of VP (and slots) this distribution has a very long tail.

Then, for a large number of games, the distribution of your total performance (summing up all outcomes of this game) WILL be normal, guaranteed if you play infinite times. For finite times, there will be a certain number N of plays, above which you can reasonable assume your total performance will be normal.

Then: If your distribution is normal (or reasonable close), you can compute probabilities of certain total performance with the EV and standard deviation of the original game.

Nothing more, nothing less.

" Nothing more, nothing less. " Silentbob, are you more or less confused than before reading this answer. Manjo, I am sure your answer is accurate, just way over my head !

Going to attempt to put this somewhat differently than others have tried. A high variance is usually associated with games of staggered pay scales. Video poker is perfect: the four of a kind, straight flush and royal flush account for the bulk of their return, yet are rare. Forget SD for the moment and see if I can explain away variance first.

The odd/even bet on Roulette will have a lower variance than the Straight Up wager. The larger the variance, the more trials we will need to see before "normal" expected results show up. Perhaps a better way would be to substitute Bankroll swings for Variance. In video poker, say holding four to a royal is superior to holding a high pair, say 10, J, Q, K... we have flush draws, straight draws, pair draws and the royal, over an extended sample size, we are vastly better off playing the 4 cards vs. say a pair of Jacks. BUT, if this is the last of our funds for the foreseeable future, we may choose to hold the Jacks thereby insuring a second chance.

The same holds true in Blackjack, we can (and should) double the 11 vs. a dealer 10, yet if our bankroll is short, to reduce variance we may elect to simply hit (this will needlessly always result in a 21 from experience). The superior play over a massive number of trials is to double, but in this instance we chose to preserve bankroll and simply hit. The same theory holds true with Even Money... take the guaranteed win (reduce variance) or go for the larger win and risk gaining nothing. Variance simply put, is how long can I go before realizing the expected results? The higher the variance, the longer one may go before seeing what is expected.

I hope this helped. Now Standard Deviation is what we go to when after 100+ hands, I am down 60 units vs. the expected 44 units. WTF is up???!!! We run our results vs. the SD to see if something is amiss. This leads to a multitude of things. 1 sd will cover all results 68% of the time, two will cover all results 95% of the time and three is 99%. So attempting to simplify, I should be up 15 units with a SD of 10, means 68% of the time I will be up five units or up 25, 95% of my results will have me up 45 or down 15 and three is up 60 or down 30. The larger our sample size, the more reliable our results will be. Over the course of 10 hands we could easily be 5 standard deviations away from the norm, even though this is a 100,000 to 1 chance. Over 1000 hands, should we find ourselves 5 standard deviations away from expectation, we should rethink what is happening. Either the house edge is greater than we thought, IE, Single Zero vs. Double Zero roulette (double the house advantage) or something is seriously amiss!!

So now lets put these numbers into real life practice. We judge which game we wish to play in the casino. SD does not (instantly) apply. We want to know variance... If my casino offers a better craps game than Blackjack, should I play craps? Yes, yet, with a short bankroll, don't take the odds as this will increase your variance much higher than Blackjack. Or the major problem, should I split 8,8 vs. a 10 or Surrender? Over 1000 plus trials splitting will yield (depending on game) superior results, and yet to reduce our variance in the short term we would probably surrender.

Soo tired, and hope this helped...

Quote: dwheatley

so, variance of 30 for coin flipping doesn't mean anything, until you start talking about flipping a coin a certain number of times... .

Of course it does since it means one has flipped a fair coin 120 times. No other number of coin flips will produce a variance of 30.

Quote: LVJackal

Now Standard Deviation is what we go to when after 100+ hands, I am down 60 units vs. the expected 44 units. WTF is up???!!! We run our results vs. the SD to see if something is amiss. This leads to a multitude of things. 1 sd will cover all results 68% of the time, two will cover all results 95% of the time and three is 99%. So attempting to simplify, I should be up 15 units with a SD of 10, means 68% of the time I will be up five units or up 25, 95% of my results will have me up 45 or down 15 and three is up 60 or down 30. The larger our sample size, the more reliable our results will be. Over the course of 10 hands we could easily be 5 standard deviations away from the norm, even though this is a 100,000 to 1 chance. Over 1000 hands, should we find ourselves 5 standard deviations away from expectation, we should rethink what is happening. Either the house edge is greater than we thought, IE, Single Zero vs. Double Zero roulette (double the house advantage) or something is seriously amiss!!

Like I've said before, I completely understand variance. I could give countless examples of variance, easily explained in different ways. However the above is much closer to the answer I was looking for. What I'm curious about is what game are you talking about where you're expected to win much more than lose? You said "up 60 or down 30" for one part, so that's quite a game there. And I got confused right after you said "attempting to simplify", because before that you were saying you were down 60 instead of the normal 44, so let's see what's up, but then you went on to say that you should be up 15 units with a SD of 10? What are you saying, that if you were up 10 units in that game where you're supposed to be down 44, that that's a SD of 10? That can't be true, I must be misunderstanding. SD of 10 would be damn near winning every hand I'd imagine.

I understand that "x SD covers y% of results", but I don't understand "It has a SD of x". Basically in hindsight after you lose, I get that if you had a SD of 2, that means that you're within 95% of the norm, which really means you were in that 5%, since it gets rare the higher up you go. Makes me wonder why it's listed as 8/95/99/etc. and not 32/5/1. But yes, I get that, but I don't get "it has a SD of x". Is that saying that if it's a SD of 3, that you are in that 1% (99%, however you want to put it) of results? I guess so. Just the 2 different ways it's worded confused me. I also wonder why we even talk in SD, instead of just saying "there was a x% chance of that happening".

Quote: SilentBob420BMFJ

I understand that "x SD covers y% of results", but I don't understand "It has a SD of x".

Then maybe you don't "completely understand variance". Variance (as EV) is a number describing any random variable. The best description of a random variable is it's full probability distribution, but most often (especially when talking about the "long run") EV and variance is fully sufficient.
EV and variance sums up the most important features of any random variable, namely the center and the width of its underlying probability distribution.

Say you have a game X where you win 40% of the time with even payout. You lose 50% of the time, and you push 10% of the time.
The probability distribution is simply a table: P(X = -1) = .5, P(X = 0) = 0.1, and P(X = 1) = 0.4 (and P(X = ...) = 0 for all others)
Instead, especially if the probability distirbution is more complex, you can also state EV(X) = -0.1, and SD(X) = 0.94
This X is about the game.

Now comes something different. You play the game X mentioned above, say, N=100 times. The result after those 100 plays, is a _different_ random variable, let's call it Y. The probability distribution of this Y is much more complex. If you would want to tabulate it, it would have 201 entries P(Y = -100), P(Y = -99)... P(Y = +99), P(Y = 100), meaning the probability of losing all 100 games, lose all but 1 game (which is a push), up to the probability of winning all 100 games.
If you did play the game 100 times, and you are up +10, then knowing the exact probability P(Y = +10) is of very little practical value. The probability of a +10 result is pretty low - not only it is a losing game, you have 201 different possible results. So P(Y = +10) will be something like less a percent.
What is more important is EV(Y) and SD(Y). You could calculate EV and SD from the P(Y =...) table itself, but since Y is a very complex random variable, this is difficult. However, you can quite easily calculate EV(Y) and SD(Y) from the game X itself:
EV(Y) = N * EV(X) = 100 * -0.1 = -10
and
SD(Y) = sqrt(N) * SD(X) = 10 * 0.94 = 9.4

So your performance Y of the game X, after 100 plays, has a SD of 9.4. Your expected result was EV(Y) = -10, and your actual result was +10.
So you are +20 units up compared to EV. Since the SD of your performance is 9.4, your result is +20/9.4 = 2.1, usually stated as "2.1 SD away".

Now comes the bell curve into play, this is the part you seem to understand. While the result of exact 10 units up is neglectable, a result happening 2.0 SD away (or better) happens with probability of 2.2%.

Quote: MangoJ

Then maybe you don't "completely understand variance". Variance (as EV) is a number describing any random variable. The best description of a random variable is it's full probability distribution, but most often (especially when talking about the "long run") EV and variance is fully sufficient.
EV and variance sums up the most important features of any random variable, namely the center and the width of its underlying probability distribution.

Say you have a game X where you win 40% of the time with even payout. You lose 50% of the time, and you push 10% of the time.
The probability distribution is simply a table: P(X = -1) = .5, P(X = 0) = 0.1, and P(X = 1) = 0.4 (and P(X = ...) = 0 for all others)
Instead, especially if the probability distirbution is more complex, you can also state EV(X) = -0.1, and SD(X) = 0.94
This X is about the game.

Now comes something different. You play the game X mentioned above, say, N=100 times. The result after those 100 plays, is a _different_ random variable, let's call it Y. The probability distribution of this Y is much more complex. If you would want to tabulate it, it would have 201 entries P(Y = -100), P(Y = -99)... P(Y = +99), P(Y = 100), meaning the probability of losing all 100 games, lose all but 1 game (which is a push), up to the probability of winning all 100 games.
If you did play the game 100 times, and you are up +10, then knowing the exact probability P(Y = +10) is of very little practical value. The probability of a +10 result is pretty low - not only it is a losing game, you have 201 different possible results. So P(Y = +10) will be something like less a percent.
What is more important is EV(Y) and SD(Y). You could calculate EV and SD from the P(Y =...) table itself, but since Y is a very complex random variable, this is difficult. However, you can quite easily calculate EV(Y) and SD(Y) from the game X itself:
EV(Y) = N * EV(X) = 100 * -0.1 = -10
and
SD(Y) = sqrt(N) * SD(X) = 10 * 0.94 = 9.4

So your performance Y of the game X, after 100 plays, has a SD of 9.4. Your expected result was EV(Y) = -10, and your actual result was +10.
So you are +20 units up compared to EV. Since the SD of your performance is 9.4, your result is +20/9.4 = 2.1, usually stated as "2.1 SD away".

Now comes the bell curve into play, this is the part you seem to understand. While the result of exact 10 units up is neglectable, a result happening 2.0 SD away (or better) happens with probability of 2.2%.

I never said I completely understand variance, at least not the way you think. Imagine you wanted to learn how a car engine works, or the transmission, etc. Well then what would you say if more than just one person comes along and keeps explaining in different ways that cars have 4 wheels, a gas pedal, a brake, and gears. They tell you cars are a form of transportation. Then they gave you an example of how you start a car and shift it into gear. Wouldn't you say "I know how cars work" in that respect? That's exactly what's going on here.

I'm not, however, referring to you with that example. What you say is too complex for me, as well as that other poster who said he was more confused. But we both know your answers are good if you understand them. I do not have the math knowledge you have, which is probably the issue. I know basic algebra and that's it. Functions and how to place them on a graph, I don't know. And if you're thinking it's gonna be hard to grasp more than the basic stuff of variance without more than basic algebra, you're probably right.

I just figured maybe it was more clear cut than it is. I didn't want to know how to calculate my own variance given xyz, I just wanted to be able to know exactly what a variance of x means in a certain game, but it's just not that simple. Can't just say that it means you'll be up/down 20 units after 100 hands or something like that if the variance were 20.

Quote: SilentBob420BMFJ

I never said I completely understand variance, at least not the way you think. Imagine you wanted to learn how a car engine works, or the transmission, etc. Well then what would you say if more than just one person comes along and keeps explaining in different ways that cars have 4 wheels, a gas pedal, a brake, and gears. They tell you cars are a form of transportation. Then they gave you an example of how you start a car and shift it into gear. Wouldn't you say "I know how cars work" in that respect? That's exactly what's going on here.

If I would want to learn how a car engine "really" works, I would (no joke) first take a course on advanced thermodynamics - before even looking at a car engine sketch.
Being able to draw pressure/temperature diagrams is much more fundamental understanding of the engine, than just "ok here the gas enteres, it get's pressurized there, then is a ignition, and after that this part is moving ..."

If it's either to trivial (all those who give examples) or to complex (all those who give you formulas and ideas), you should define a level of detail you are comfortable with - and try to tell us. The best thing you could do is, try to explain us what variance is in your own words. We will surely notice all gaps you are missing, and can fill you in with details. If you cannot describe what variance is, you probably understand less than you think. Again, waving with "all this I already know, tell me something else" doesn't help you, you should ask yourself "although I know all those points, I cannot figure why it is important to understand variance".

If would like to start from the beginning, you should understand the concept of a random variables. Random variables are not just the outcome of simple dices, they are also more "abstract" random variables, like the performance of your next week's session. Unless you figure out what the outcome of a simple dice has in common with your next week's performance, you won't understand variance. If you would say something like "I know dices, but I don't play craps. So why does it matter for my performance ?" - then you may know about dices, but you definetly don't know about random variables.