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Old 05-11-2008, 12:58 PM   #1
RhoSkyWalk3r
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What is variance and how is it calculated?

What is equity and how is it calculated? Also how should it be applied to a poker game?
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Old 05-11-2008, 01:59 PM   #2
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Re: What is variance and how is it calculated?

I'm not a math major, but here's my take on it.

In any endevor that involves luck, variance is the measurement of the short term luck over a period of time. This variance can be measured and discribed mathmaticaly.

In a pure luck endevor, say flipping coins, the long term expectation is 50:50. Over the long haul the number of heads and the number of tails will even out. However, there will be periods where "heads" results are clustered together, and other periods where "tails" results will be clustered together. Those are the periods of short term varience.

Let's look at something else that is pure luck - Roulette. The house has about a 5% edge on American Roulette tables. If you wager $1.00 on every spin, then for every 100 spins you will lose $5.00 per 100 spins over the long haul. It doesn't matter if all your bet are "red" or "high" or "17 black", in the long run you will lose $5.00 per 100 spins.

Now, say you go to a casino 100 times and each time you make 100 $1.00 bets. There will be times you leave the casino a significant winner, times you go busto, and times you just win or lose a few bucks. You could busto several times in a row, you could banko several times in a row, but in the long run you'll approach -$5.00 per 100 spins. Those periods of extreem results clustered together though, that's variance.

And because the house always have the same 5% edge, then that variance can be predicted mathamaticaly. This is how the house knows how much cash to keep on hand. It could be that 0 & 00 don't come up on the Roulette wheel several spins in a row. Every spin the house has to pay someone off. So on any given day, what is the most amount of money that the house might have reasonably have to pay out to customers due to pure mathmatical variance? This is the casino's "risk of ruin."

I don't know the answer for Roulette, but belive me the casinos have paid some very bright mathmaticians to do the math. And because the long term expectation can be calculated exactly for Roulette, then the casinos know exactly what their likely short term varience is, what their risk of ruin is.

Now let's look at poker. Poker is a mix of skill and luck. In the long run, every player gets dealt the same mix of good and bad cards. It's how the cards are played that deterins who is a winner at the end of the year. As one player's skill level rises above the skill of the other players, his expectation increases. And as expectation increases, variance goes down.

But here's the thing with poker - Poker skill is not a fixed number, because it's always your skill relative the the skill of the other players. Play with noobs & rubes? Your expectation will be high and your long term varience will be low. Play with a few skilled players? Variance goes up. Play a LAG style? Variance goes up again.

And we haven't ever looked at the cards yet. Sometimes the deck will run cold. Over and over your aces will get cracked, you'll miss your flush.

If you keep careful track of your play, you can measure your likely skill advantage and your likely risk of ruin. This is the amount of money, known as bank roll, that you will need to keep playing.
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Old 05-11-2008, 02:00 PM   #3
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Re: What is variance and how is it calculated?

Wow, totally didn't think I'd be the first reply.
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Old 05-11-2008, 02:42 PM   #4
RhoSkyWalk3r
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Thanks so basically I need long term info to determine my variance

Quote:
Originally Posted by phydaux View Post
I'm not a math major, but here's my take on it.

In any endevor that involves luck, variance is the measurement of the short term luck over a period of time. This variance can be measured and discribed mathmaticaly.

In a pure luck endevor, say flipping coins, the long term expectation is 50:50. Over the long haul the number of heads and the number of tails will even out. However, there will be periods where "heads" results are clustered together, and other periods where "tails" results will be clustered together. Those are the periods of short term varience.

Let's look at something else that is pure luck - Roulette. The house has about a 5% edge on American Roulette tables. If you wager $1.00 on every spin, then for every 100 spins you will lose $5.00 per 100 spins over the long haul. It doesn't matter if all your bet are "red" or "high" or "17 black", in the long run you will lose $5.00 per 100 spins.

Now, say you go to a casino 100 times and each time you make 100 $1.00 bets. There will be times you leave the casino a significant winner, times you go busto, and times you just win or lose a few bucks. You could busto several times in a row, you could banko several times in a row, but in the long run you'll approach -$5.00 per 100 spins. Those periods of extreem results clustered together though, that's variance.

And because the house always have the same 5% edge, then that variance can be predicted mathamaticaly. This is how the house knows how much cash to keep on hand. It could be that 0 & 00 don't come up on the Roulette wheel several spins in a row. Every spin the house has to pay someone off. So on any given day, what is the most amount of money that the house might have reasonably have to pay out to customers due to pure mathmatical variance? This is the casino's "risk of ruin."

I don't know the answer for Roulette, but belive me the casinos have paid some very bright mathmaticians to do the math. And because the long term expectation can be calculated exactly for Roulette, then the casinos know exactly what their likely short term varience is, what their risk of ruin is.

Now let's look at poker. Poker is a mix of skill and luck. In the long run, every player gets dealt the same mix of good and bad cards. It's how the cards are played that deterins who is a winner at the end of the year. As one player's skill level rises above the skill of the other players, his expectation increases. And as expectation increases, variance goes down.

But here's the thing with poker - Poker skill is not a fixed number, because it's always your skill relative the the skill of the other players. Play with noobs & rubes? Your expectation will be high and your long term varience will be low. Play with a few skilled players? Variance goes up. Play a LAG style? Variance goes up again.

And we haven't ever looked at the cards yet. Sometimes the deck will run cold. Over and over your aces will get cracked, you'll miss your flush.

If you keep careful track of your play, you can measure your likely skill advantage and your likely risk of ruin. This is the amount of money, known as bank roll, that you will need to keep playing.
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Old 05-11-2008, 03:10 PM   #5
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Re: What is variance and how is it calculated?

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Originally Posted by RhoSkyWalk3r View Post
Thanks so basically I need long term info to determine my variance
Yes. Some people say 10k hands are enough, others say 25k hands is the absolute minimum.
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Old 05-11-2008, 03:36 PM   #6
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Re: What is variance and how is it calculated?

Whether or not your results converge depends on how you define results. Let's say I play hands of poker under the same stakes and same conditions. I'll define X_a to be my average win rate per 100; X_t to be my total winnings; N to be the number of 100 hand sessions I play; and WR to be my true win rate per 100.

As N increases the expected difference between X_a and WR decreases.

As N increases the expected difference between X_t and N*WR increases.
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Old 05-11-2008, 06:11 PM   #7
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Re: What is variance and how is it calculated?

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Originally Posted by RhoSkyWalk3r View Post
Thanks so basically I need long term info to determine my variance
Your variance (the actual numerical quantity) actually converges pretty quickly. 10K hands should be enough to determine it with a reasonable accuracy. Your winrate, on the other hand, converges really slowly.
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Old 05-11-2008, 06:15 PM   #8
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Re: What is variance and how is it calculated?

wow at all those ridiculously complex replies.

variance is when you have an 80% chance to win a hand and you lose
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Old 05-11-2008, 10:10 PM   #9
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Re: What is variance and how is it calculated?

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Originally Posted by pairtheboard69 View Post
wow at all those ridiculously complex replies.

variance is when you have an 80% chance to win a hand and you lose
That's an example of variance but not its definition or essence. I think the first person who replied defined it a little more accurately and I appreciate his efforts. However your reply is valuable if you only have time for a general idea and not a full explanation.
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Old 05-12-2008, 12:41 AM   #10
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Re: What is variance and how is it calculated?

It sticks in my head that the the mathematical calculation for the variance under the same set of conditions is the standard deviation squared. In other words, if your average winrate over an infinite number of hands is 10 big blinds per 100 hands and your standard deviation over the sample is 50% (5bb) then you should expect your variance to swing your average results either up or down by 25bb. Over any given set of 100 consecutive hands pulled from that infinitely large sample you should expect your results to have a bottom limit of -15bb and an upper limit of +35bb. Keep in mind that I'm pulling this all from memory and haven't verified if I am recalling this information correctly.
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Old 05-12-2008, 01:23 AM   #11
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Re: What is variance and how is it calculated?

--------------------------------------------------------------------------------

Assuming you know all the possible outcomes and their probabilities you can calculated both EV and variance. This is unrealistic in a poker game, so you'll only be estimating.

Say the random variable that represents your results is X. E[X] is the expected value of X, E[X^2] the expected value of X squared (this is not the same as E[X]*E[X]) and Var[X] is the variance of X. Let's say X has 3 possible values.
-It will be 0 with probability of 0.5
-It will be 1 with a probability of 0.25
-It will be 2 with a probability of 0.25

Note that 0.5+0.25+0.25 = 1.

E[X] = 0.5*(0)+0.25*(1)+0.25*(2) = 0.75

E[X^2] = 0.5*(0^2) + 0.25*(1^2) + 0.25*(2^2) = 1.25

Var[X] = E[X^2] - E[X]*E[X] = 1.25 - 0.75^2 = 0.6875

Let's say your outcomes are measured in $. E[X] will have units of dollars. E[X^2] will have units of dollars squared. Var[X] will have units of dollars squared.

This is an example of a discrete random variable. Finding the EV and variance of a continuous random variable would require calculus.

Let's say the random variable Y is the sum of 10,000 independent random variables that all came from the above distribution. Y would follow a normal distribution with mean 10,000*E[X] and variance 10,000*Var[X].
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Old 05-12-2008, 01:29 AM   #12
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Re: What is variance and how is it calculated?

I looked it up and verified that yes, the value of mathematical variance is the square of the standard deviation. Out of curiosity I pulled up my info in PokerTracker and saw that the standard my standard deviation at 100NL is 84 big blinds every 100 hands, which makes the variance 7056bb/100. If we treat these numbers as constant (which we should keep in mind that they are not, this is only as an example) and my winrate is 10bb/100 (once again, not constant) then in any given 100 consecutive hands I should expect a maximum possible loss of $7046 and and amaximum possible win of $7066. The main issue here is that variance is used to determine the outside range of the possible results here. The large majority of the time my results will fall within the range set by the standard deviation which would mean I can expect my results the vast majority of the over any give 100 consecutive hands to be within the range of -$74 to +$84.
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Old 05-12-2008, 01:37 AM   #13
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Re: What is variance and how is it calculated?

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Originally Posted by pairtheboard69 View Post
wow at all those ridiculously complex replies.

variance is when you have an 80% chance to win a hand and you lose
Uh no. Absolutely incorrect. I came to this thread because I expected there would be some wrong definitions, and I hate how often this word is used incorrectly.

Variance is NOT losing an 80/20 hand. An 80/20 hand is GUARANTEED to lose 20% of the time. You hold a set of aces versus a set of queens on the turn. You are GUARANTEED to lose ~2% of the time.

If you were to make a graph displaying the amount you expect to win (or even, how often you expect to win) in certain situations, and then graph your actual results, variance is the amount that your results VARY from the expected amount.

For example, you expect a coin to land heads 50% of the time. You flip is twice and it lands tails both times. What gives? You flip it ten times and it lands heads only 3 out of 10. ??? You flip it 10,000 times and it lands heads 4,987 times. What is happening here?? As your sample size increases, your actual results are getting closer and closer to the expected results. The same holds for poker. You say you lose 80/20 cases more often than you should, but your sample size is only 20 times. Well...the more you play, the more those 80/20 cases will come up, and the closer your average will be to the expected average.

tl;dr Variance is not what most people define it as.
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Old 05-12-2008, 01:43 AM   #14
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Re: What is variance and how is it calculated?

Sean,

Variance is measured in different dimensions than winrate (variance in bb^2 and winrate in bb), so adding the two together is impossible.

If we disregard units, I could come up with examples of distributions where every possible result is over 1 variance from the mean.
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Old 05-12-2008, 07:50 AM   #15
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Re: What is variance and how is it calculated?

Quote:
Originally Posted by Sean Fraley View Post
. Out of curiosity I pulled up my info in PokerTracker and saw that the standard my standard deviation at 100NL is 84 big blinds every 100 hands, which makes the variance 7056bb/100. .
how did u do that? how can i find out what is my standard deviation from my PT?

also, i have a related question hope its ok to ask it here:
i just read the post mr.wookie made on moving up limits according to standard deviation and i have trouble understandig this paragraph:

"My personal metric is, for play at a particular level, to know to within two standard errors (about 98% confidence) that I am a winning player. That is, that my win rate is twice or more as big as the uncertainty of my win rate. To compute the uncertainty of your win rate, take your standard deviation per 100 hands, usually about 15 BB/100, and divide it by the square root of the number of hands you’ve played divided by 100 (the number of 100 hand blocks you’ve played). Playing 20,000 hands with this standard deviation will yield an uncertainty in your win rate of 1.06 BB/100. Thus, you’d need a win rate of 2.12 BB/100 to know with 98% confidence that you were a winning player. Depending on your personal level of boldness or paranoia, you may be satisfied with 84% confidence (uncertainty = win rate) or 99.9% confidence (uncertainty = win rate / 3). Of note is that 20,000 hands at a win rate of 2.12 BB/100 will net you 424 BB, which, combined with the 300 BB you started with, gives you a bankroll sufficient to play at twice the current limit."

will some1 be kind enuf to explain to me what is the bold part mean?
ty.
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Old 05-12-2008, 02:54 PM   #16
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Re: What is variance and how is it calculated?

So..i still don't get what this has to do with
Yes i do
but i don't know what to do with it
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Old 05-12-2008, 04:00 PM   #17
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Re: What is variance and how is it calculated?

Quote:
Originally Posted by TheDataKid View Post
So..i still don't get what this has to do with
Yes i do
but i don't know what to do with it
Poker players, even skilled ones, need a large bankroll because of variance. Statistically it's very easy to go on long runs where you don't get dealt good cards, and the few times you do get dealt good cards your aces get cracked and your sets lose to bigger sets.

Fortunately some very smart and skilled statisticians (of whom Mason is one) have done the hard math for us already.

If you play NL cash games and you have a skill advantage over your opponents, then a bankroll of around 25 buy-ins is enough to get you through the likeliest dry spells. 20 is probably enough if you know you have a skill advantage over your regular opponents, 30 is probably better for new players.

And if you don't have a skill advantage over your opponents, then no bankroll can be sufficient. A losing player's risk of ruin is infinite.
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Old 05-13-2008, 03:11 AM   #18
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Re: What is variance and how is it calculated?

So
if you have 150$ and you play at the 0.1-0.2$ NL, then you playing above your payroll
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Old 08-30-2008, 06:48 PM   #19
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Re: What is variance and how is it calculated?

Quote:
Originally Posted by TheDataKid View Post
So
if you have 150$ and you play at the 0.1-0.2$ NL, then you playing above your payroll
I think you mean to say "above your bankroll".

If we assume the standard 100 BB buy in then your buy-in for .10/.20 NLHE is $20.00. The standard rule of thumb for bankroll management is that you should have a minimum of 20 buyins. $20.00 x 20 = $400. So, depending on how much you chose to buy in for, you may be under bankrolled for .10/.20 NLHE. With your $150 bankroll you could buy in for $10-$15 and be properly bankrolled.

Also, note that the 20 buy-in rule is an approximation. If you like to play a LAG style then you may have higher variance than the typical TAG player and therefore need a larger bankroll. Also, if you are conservative you may prefer to have more than the minimum recommended 20 buy-ins (e.g. I like to have 25 buy-in before moving up).
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Old 08-30-2008, 07:41 PM   #20
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Re: What is variance and how is it calculated?

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Originally Posted by pairtheboard69 View Post
wow at all those ridiculously complex replies.

variance is when you have an 80% chance to win a hand and you lose
I hope this is a level. DrVanNostrin is right. This isn't complex at all. Just basic math.

Var[x] = E[x^2] - E[x]^2
Or in simpler terms the average squared distance from the mean.
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Old 08-31-2008, 04:36 AM   #21
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Re: What is variance and how is it calculated?

Here's one way to estimate variance:

1. Every X number of hands (of X number of minutes), write down how much you are up or down by.

2. Enter this number into an excel spread sheet.

3. Use the "variance" stat function


BTW, standard deviation (square root of variance) has a more intuitive interpretation than variance.
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Old 08-31-2008, 05:40 AM   #22
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Re: What is variance and how is it calculated?

Interesting thread especially Pokey's responses

Quote:
how should it be applied to a poker game?
Well you can estimate your winrate and estimate your 'average variance' and hope fervently you can approximate it all normally and that table conditions etc don't change in order to work out if you need 18 or 19 buyins for this or that, or if your risk of ruin is 2% or 3%.
Or you can just stick roughly to a decentish amount of buyins/big blinds depending on replacement pain (actual $$$ and depositing difficulty etc) and the opportunity to move down/take shots/experiment or whatever.
When GTAOT came out (before my time of course) it seems that the likelihood of prolonged downswings ('bad' variance) was not well understood and was a bit of a revelation to underrolled pros with some gamble in them (if they read the book anyway)
Now it seems that a lot of (often) more statistically educated players want to nail things down to over-exactitude without realising that there are uncertainties and approximations and the end result shifting a little is fine (not a dig at anyone here, it just gets a lot of attention because beginners often play horrendously underrolled and get burnt, and then give it too much importance from then on imo.)
I don't think I'm even disagreeing with anyone itt really but there is a happy medium between sticking your roll on one table and knowing your risk of ruin to three decimal places
While I think of it (again, not aimed at anyone in this thread) there is a propensity amongst beginners to look upon outbreaks of 'good' variance as good play and 'bad' (or even normal) variance as bad luck which can lead both to horribly overestimated winrates and feelings of riggedness when play returns to 'normal'. But maybe that was just me
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Old 08-31-2008, 05:05 PM   #23
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Re: What is variance and how is it calculated?

Quote:
Originally Posted by dismalstudent99 View Post
BTW, standard deviation (square root of variance) has a more intuitive interpretation than variance.
Right...
I dunno, this might be true about some distributions like the normal curve, but in the case of the others variance is far more practical to know and use.
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Old 09-01-2008, 05:48 AM   #24
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Re: What is variance and how is it calculated?

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Right...
I dunno, this might be true about some distributions like the normal curve, but in the case of the others variance is far more practical to know and use.
Like when?
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Old 09-01-2008, 08:36 AM   #25
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Re: What is variance and how is it calculated?

Warning: this is long winded and likely boring for most people. Read at your own risk.

Quote:
Originally Posted by merryber View Post
how did u do that? how can i find out what is my standard deviation from my PT?
If you use PT2, go to the Session Notes tab. In the upper right corner is a button labeled "More Detail...". Click that. A window will pop up giving more information than they could fit into the main window. At the bottom of this popup window is your standard deviation/hour and standard deviation/100.

I don't know where to find it in PT3 but it's in there somewhere.

Quote:
Originally Posted by merryber View Post
also, i have a related question hope its ok to ask it here:
i just read the post mr.wookie made on moving up limits according to standard deviation and i have trouble understandig this paragraph:

"My personal metric is, for play at a particular level, to know to within two standard errors (about 98% confidence) that I am a winning player. That is, that my win rate is twice or more as big as the uncertainty of my win rate. To compute the uncertainty of your win rate, take your standard deviation per 100 hands, usually about 15 BB/100, and divide it by the square root of the number of hands you’ve played divided by 100 (the number of 100 hand blocks you’ve played). Playing 20,000 hands with this standard deviation will yield an uncertainty in your win rate of 1.06 BB/100. Thus, you’d need a win rate of 2.12 BB/100 to know with 98% confidence that you were a winning player. Depending on your personal level of boldness or paranoia, you may be satisfied with 84% confidence (uncertainty = win rate) or 99.9% confidence (uncertainty = win rate / 3). Of note is that 20,000 hands at a win rate of 2.12 BB/100 will net you 424 BB, which, combined with the 300 BB you started with, gives you a bankroll sufficient to play at twice the current limit."

will some1 be kind enuf to explain to me what is the bold part mean?
ty.
The author is simply talking about how confident you can be that the winrate reported by Poker Tracker (or Holdem Manager, or your own calculations, or whatever) is a true reflection of reality. The formulas he gives really only make one assumption, and that is that win rates for poker players have a "normal" distribution. A normal distribution looks like a bell when it's graphed.

As it turns out, win rates do have a normal distribution. If you made a graph with the X-axis showing win rate and the Y-axis showing percentage of the population with that win rate, and then you compiled all the win rates of every poker player on the planet, the resulting shape of the curve would look like a bell, with most players clustered around a slightly negative win rate and the curve dropping off rapidly in either direction. There would be a few very bad players showing up on the left and a few very good players on the right. The curve itself could look elongated and tall or short and flat, but it will look like a bell. It's the standard deviation that determines its "flatness".

Now, if you haven't played many hands at a level, you won't be very confident that the PT win rate is a true reflection of your win rate at that level. The more hands you play, the more confident you can be, but there is still a chance that you've been on a downswing or a heater.

How likely is it that your true win rate deviates a "lot" from the PT win rate? Well, that's what the author is showing. If your win rate is a full "standard error" (as calculated by the author) above zero, you can be 84% confident that you really do have a positive win rate. Basically, this means that if you took all players with your win rate and your standard error, you would find that 84% of them have win rates within one "standard error" of the PT win rate, but 16% are either losers who have been on a heater or bigger winners who have been on a downswing.

If your win rate is two standard errors above zero, then you can be 98% sure you're a winning player. If you're a complete paranoid nit and you want to be 99.9% sure you're a winning player before moving up, you'll have to play enough hands to get your standard error down to 1/3 of your win rate. How many hands that will take depends on your actual win rate and your standard deviation, which depends on your style of play, your consistency, the consistency of your table conditions, etc.

If you're a loose aggressive player prone to fits of rage and tilt and you sometimes play drunk while watching porn videos, your variance (standard deviation squared) is going to be extremely high, and it's going to take a lot of hands to determine if you are actually a winner with any confidence. If you are a tight player, can control your emotions, and manage to avoid distractions for the most part, your variance will be lower, and it won't take nearly as long.

Keep in mind, though, that all these formulas really tell you is how confident you can be that your style of play over the course of the hands in your database was a winning style of play when compared to the others in your database. It can't really tell you if you are, right now, a winning player when compared to the current population of poker players. You might have improved, or you might have got in a fight and suffered brain damage in the Poker Skill part of your brain. Or you might have switched sites recently and your new site has fishier (or tougher) players. Or CTS might have come out with a video for microstakes players last week and everybody saw it and understood it but you. Or the U.S. Congress might have decided to regulate poker, and now a million new and clueless players are playing all over the place and your slightly negative win rate is about to ramp up to a hugely positive one.

In other words, things might have changed.

And if you've read this far, you just might be masochistic enough to succeed at this game.

Last edited by Weevil99; 09-01-2008 at 09:00 AM.
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