Variance and standard deviation, help me out in understanding them

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".