Standard deviation for SNG\'s and sample size for CASH games
 

I'm not sure if anyone has written extensively on this topic in regard to heads up games but I'll try to explain things as simple as possible .

Your standard deviation for heads up sng's is a function of your win rate . If you assume that your win rate is ~60% , then your standard deviation will remain fixed for any player who shares the same win rate . A simple calculation proceeds as follows :

Var(x)=E(x^2)-E(x)^2 where E(x) is your win rate(or mean) for a random variable x . For this particular case , we regard the variable x as +1 for when we win 1 unit and -1.05 for when we lose one buyin which also includes the rake .

So assuming you win 60% of the time , your s.d is simply the square root of your variance or var(x) .

E(x^2)=1^2*0.6 +(-1.05)^2*.4
E(x^2)= 1.041

Also E(x)=1*0.6-1*0.4 -0.05
E(x)=0.15

Var(x)=1.041-0.15^2
Var(x)=1.0185

S.D(x)=sqrt(1.0185)~ 1.0092

This means that if your win rate is 60% , then your s.d is approximately equivalent to 1 buyin , no matter what !

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

Our standard deviation for cash games may be different for two players who share the same win rate . As I explained earlier , this is not the case for sng's .

To compute our standard deviation for cash games , we need at least 30 cash game sessions to ascertain that our s.d converges to a steady number . We need not even know our win rate which is why the central limit theorem is so very useful in situations like this . Using the numbers that Jakeduke provided , I will calculate the number of hands needed to determine with 95% confidence, our win rate interval .

Lets say after 50k hands , our win rate is (10 bb)/100 hands and bb is not to be mistaken for big bets .

Our standard deviation , using jakeduke's numbers is ~ 11 big blinds/100 hands .

xbar is our sample mean (10 bb/100 hands)
z= our confidence level which is approximately 2 s.d's above and below the mean .
sigma bar is our sample standard deviation .

In this case , sigma bar is 11/sqrt(500)~ 0.4919 per 100 hands .

10 +- 2*0.4919 Which means that we are 95% confident that Jakeduke's win rate lies between 9.0162 to 10.98 BB's per 100 hands .

Using z=3 , we are about 99% confident that Jake's win rate lies between 10 +- 3*0.4919 or between 8.52 to 11.47 BB's per 100 hands .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

but per how many hu sngs?

it's not 1 per 1000 games.

does my question make sense?

Re: Standard deviation for SNG\'s and sample size for CASH games
 

You lost me right after "simple math".


But I dont even understand the concept of SD so I guess... yeah.

Re: Standard deviation for SNG\'s and sample size for CASH games
 

One correction .

Instead of S.D =11BB/100 hands I meant to put S.D=110bb's/100 hands . No wonder why things weren't adding up .

Jake's true s.d in terms of big blinds is 110bb/100 hands .

Sigma bar becomes 110/sqrt500 = 4.919/100 hands .

Our confidence interval becomes :

10+-2*4.919

and so we're 95% confident that his true win rate lies between 0.162 to 19.838 . All this tells us is that we're pretty confident that he's a winner.

This is one of the reasons why they say 50k hands is sufficient to determine that you're a profitable poker player .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
but per how many hu sngs?

it's not 1 per 1000 games.

does my question make sense?

[/ QUOTE ]

For each 1 sng , your standard deviation is +1 unit from your mean . For a 9 player sng , your standard deviation will be close to 1.7 for each 1 sng .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

Tnixon , here is the day you've been waiting for [img]/images/graemlins/smile.gif[/img]

Jakes s.d for NL100 is 110 bb's/100 hands .
Lets make a hypothetical assumption that an average sng lasts 30 hands . So playing a $100+5 buyin sng at a 60% win rate is equivalent to a s.d of about $100/30 hands which is about THREE times higher than your variance playing in cash games !!

Re: Standard deviation for SNG\'s and sample size for CASH games
 

One other minor correction .

If we include rakes into this discussion for an sng , then this means (as a 60% winner) that you will win 0.95 units , 60% of the time , and lose 1.05 units , 40% of the time .

So for my original variance calculation , we should replace +1 with +0.95 but this doesn't change things much .

E(x^2) = 0.95^2*0.6 + (-1.05)^2*0.4
E(x^2)=0.9825

var(x)= 0.9825 - 0.15^2
var(x)=0.96
s.d(x)=0.97979797 or almost 1 buyin per sng .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
Tnixon , here is the day you've been waiting for

[/ QUOTE ]

The day I've been waiting for? You mean the day where you run through the same calculations that I already showed in another thread (although to be fair, my numbers were off, and showed SNGs being lower variance than they really are, because I incorrectly put the winrate inside the squared terms instead of outside, where it belonged), and ended up basically restating the same conclusion that I already came to in another post, without any acknowledgement whatsoever that it even existed, even though I know for a fact that you at least browsed it?

Wow. You're right. I've totally been waiting for that. In fact, my face is blue, I've been holding my breath for so long.

Re: Standard deviation for SNG\'s and sample size for CASH games
 

It's worth pointing out a couple of things.

First of all, we only have one data point for cash game variance, jakeduke.

Second, we only have one data point for average hands per SNG, and jay didn't even use that number, instead picking an even smaller number, which doesn't seem realistic given that I suspect my games are typically shorter than many, because I would guess that I make a rather large number of high-variance plays compared to many other winning players.

But you can help!

http://forumserver.twoplustwo.com/showfl...=1#Post12263671

Re: Standard deviation for SNG\'s and sample size for CASH games
 

Tnixon , I thought I was helping you out and you're actually criticizing me ? What's wrong with you?

You've never arrived at a standard deviation of 1 unit which is a common total for sng's . You cannot compare s.d's unless you have a reasonable estimate for both sng's and cash games which I've clearly shown in my example .

Please do not ruin this thread with your gibberish .In fact , I would prefer it if Leader locket it .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]

For each 1 sng , your standard deviation is +1 unit from your mean .

[/ QUOTE ]

making sure i get this . . .

so the mean is (excluding rake effect) $17.25 for a $115 (because that's the ROI (15%) with a 60% winrate). correct?

so then i just add/subtract one unit ($115) from the mean. 1 SD would result in +$132.25/-$97.75 for me 68% of the time. am i right?

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
You cannot compare s.d's unless you have a reasonable estimate for both sng's and cash games which I've clearly shown in my example .

[/ QUOTE ]

Which I clearly did in the other thread (even though my numbers were slightly off, but still in the right ballpark), comparing s.d in $/hand of both cash and sngs. The actual unit of comparison is unimportant, as long is it matches in both cases.

Let me attempt to explain why I'm so annoyed with you on this whole topic.

Original thread: I theorize that HUCash is lower variance than HUSNGs, and attempt to provide logical arguments for why it seems likely.

You and "he who shall go unnamed" attempt to provide mathematical "proofs" that sngs are lower variance, by a factor of TEN.

When I try to explain the flaws in your math and comparisons in as many different ways as possible (ways in which anybody who had paid attention to the first 2 weeks in high-school chemistry, where they talk about unit conversion, which 90% of chemistry problems boil down to anyway, would have understood the first time, much less the sixth or seventh) you both act like I'm an ignorant retard who doesn't know what the hell he's talking about. Which is certainly true at least half the time, but I was on the upside of the 50/50 this time.

When I finally lay the math out myself, showing once and for all that what I've been saying all along is, in fact, correct, you act as if that was your point of view all along, and that somehow now I have changed *my* point of view so that we're in agreement. Again, if you disagree with any of this, just let me know, and I'll be more than happy to go back and dig out explicit quotes.


Then the discussion starts again. Being impatient, I lay out the variance math myself in the other thread (admittedly with simple calculation errors, and had you simply corrected those errors, I would have accepted said corrections gracefully and without ire, because GOD KNOWS along with the rest of the world that I'm far from perfect), showing that cash, for a *very* limited sample of real-world data, is significantly lower variance than SNGs for equal buyins.

After presumably reading this post where I make this conclusion (you did make replies in the thread after my post, so I can only assume you read it), you take it upon yourself to start a new thread, laying out basically the same math (in a slightly different way, but once my calculation errors are fixed, the results are the same), drawing the same conclusion.

And somehow, throughout all of this, while you're rehashing something that's already been stated in another thread, you manage to maintain a superior air, as if you're the instructor speaking down to all the students here in the forum.

In fact, somehow, you manage to make this claim:

[ QUOTE ]
I'm not sure if anyone has written extensively on this topic in regard to heads up games but I'll try to explain things as simple as possible .

[/ QUOTE ]

As the introduction to a post that is basically a rehash of one that I had made previously, on a topic that has been *extensively* written about by at least 3 people.

Take it or leave it, that is my point of view. If "help" means rehashing things I've already said, without giving credit or even acknowledgement of work previously done, then please don't help me out anymore.

Yours Truly,
Forum Drama King

(Leader, can you please consider this post as a formal application for a custom title?)

Re: Standard deviation for SNG\'s and sample size for CASH games
 

i'm confused.

you said the SD math is for one sng. and that one SD = 1 buy in.

that doesn't make sense, because 2 SDs would be 2 BIs, yet you can't lose 2 BIs in one sng.

Re: Standard deviation for SNG\'s and sample size for CASH games
 

Nothing like a good battle over standard deviation while I eat my morning cereal. Who needs cartoons?

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
[ QUOTE ]

For each 1 sng , your standard deviation is +1 unit from your mean .

[/ QUOTE ]

making sure i get this . . .

so the mean is (excluding rake effect) $17.25 for a $115 (because that's the ROI (15%) with a 60% winrate). correct?

so then i just add/subtract one unit ($115) from the mean. 1 SD would result in +$132.25/-$97.75 for me 68% of the time. am i right?

[/ QUOTE ]

Correct . I've actually included the rake in my examples to make it even more accurate .

So if your a 60% winner at the $(100+5) games , then your ROI is 14.28% . This means that approximately 68% of the time , you will have an ROI between 114.28% to -85.72% after your first sng .

This type of reasoning only applies after you've played multiple games because the results are not normally distributed until after several sng's . After 30 sessions , you can make all sorts of claims about your sample ROI .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

Tnixon , I still like you.

You've done some pretty good work yourself .

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
Tnixon , I still like you.

You've done some pretty good work yourself .

[/ QUOTE ]

--------------------------------------------^

That period is deviating from the end of your sentence!

Re: Standard deviation for SNG\'s and sample size for CASH games
 

lol

Re: Standard deviation for SNG\'s and sample size for CASH games
 

[ QUOTE ]
You lost me right after "simple math".




[/ QUOTE ]

Re: Standard deviation for SNG\'s and sample size for CASH games
 

Let me add more to this discussion .

Let x be the number of hands needed to be within 1ptbb/100 of your hypothetical win rate . So if we believe that we win 10bb/100 hands and our standard deviation is 110bb/100 hands , then the number of hands needed at 2 standard deviations is :

10+-2*(110/sqrt(x/100))
So we want 2*(110/sqrt(x/100)) =1
sqrt(x/100)=+-2*110
sqrtx=+-2*110*10
x=4 840 000

This means that even after 4 840 000 hands , you are 95% confident that you're within 1 bb/100 of your true win rate . This is one of the reasons why nobody really knows their true win rate because they usually change limits before they can draw any conclusions from it .