Relevance of Standard Deviation (Mason?)

Ok, so I've calculated the standard deviation for my last 30 sessions according to MM's formula and got a result of approximately 80. 80 what? What does that tell me? I'm sure it's something useful, but I really need somebody to help me understand what that number means and how I can use it to my advantage.

Thanks,

MD

Standard Deviation tells you how far your real, actual results are supposed to fluctuate around your theoretically expected result.


Your theoretically expected result (call it EV), expressed in units or dollars, is what you have calculated to be the average result of your play over the period you monitored your results, namely 30 sessions.


If that average result has been so far let's say a win of about $10 per session and you have also calculated the Standard Deviation of all those sessions to be $80, that means that your real, actual results in the future will, most probably, average between EV+$80 and EV-$80. In other words, you will most likely be bouncing between +$90 and -&70 per session, in the future.


Of course, there are many assumptions at work here : That the quality of your own game and your opponents will remain constant over time, that "past results guarantee future performance", that each session lasts the same amount of time, etc.


(This is a basic explanation. For more on what "most probably" means, for example, look up a Stat book.)


Hope this helps, as they say.

"If that average result has been so far let's say a win of about $10 per session and you have also calculated the Standard Deviation of all those sessions to be $80, that means that your

real, actual results in the future will, most probably, average between EV+$80 and EV-$80. In other words, you will most likely be bouncing between +$90 and -&70 per session, in the future."


This isn't right. Since it's hard to type in my car, I'll let others elaborate.

If I remember correctly from stats class (very unlikely) about 66% of your sessions will be within $80 dollars of your $10 dollar EV. This is assuming all session lengths are an equal time interval and your results are normally distributed.

The standard deviation you calculated will tell you how much fluctuation you can expect in future game sessions, assuming similar conditions and play.

First, 30 sessions might be a little light for significant stats, but it's good enough to get an idea. Everything below assumes a normal distribution which you might not have with 30 sessions.

Now, I'm going to assume the "80" you calculated to be in dollars and not BB.

So let's say you found your average win rate to be $10. With a standard deviation of $80 then 68% of the time you will see a net between -$70 and +$90.

95% of the time you will see a net between -$150 and +$170.

99.7% of the time you will see a net between -$230 and +$250.


This can be helpful for wrapping your head around what you can expect out of playing poker and hopefully you will avoid going on tilt because you will see it is statistically reasonable for you to loose $X.

Most of the time if you lose you'll lose $70 or less. A vast majority of the time if you lose you'll lose $150 or less.

The converse goes for winning too of course. Most often though we tend to forget the wins and remember the losses.


Finally, I'm not sure how this part works yet, (though I would love to know), but you can use your standard deviation information to determine an ideal bankroll to avoid risk of ruin.

Basically the idea behind this calculation is there is a certain probability based on the standard deviation and average win rate you calculated that you won't loose more than $X before you gain your money back.

Just for reference, the limits I've been playing are from 2-4 to 6-12. Also, I figured out the units for my result: dollars / SQRT(sessions * hours). Unfortunately, these units are meaningless to me. I would like to know how this number can be of use to me in terms of affecting decisions that I make, during play or otherwise. How does this number have a relationship to bankroll requirements?


-MD

Well, according to a blackjack site that zooey posted a link to a couple weeks ago, the formula for risk of ruin is as follows:


Let W = win rate, S = std. dev., N = number of hours played, BR = bankroll.


Let A = (1 + W/S)/(1 - W/S)


Then R = [A^(WN/S) - 1]/[A^((BR+WN)/S) - 1]


So for W=1, S=20, N=100, if we want a risk of ruin of 1%, we would have:


A = 21/19


.01*(A^(5 + BR/20) - 1) = A^5 - 1


.01*(A^5)*A^(BR/20) + .99 - A^5 = 0


(A^5)*A^(BR/20) + 99 - 100*A^5 = 0


A^(BR/20) - 100 = -99/A^5


(BR/20)*ln(A) = ln(100-99/A^5)


BR = 20*ln(100-99/A^5)/ln(A)


So BR = 737


Which means that if you have a 1bb/hr win rate, a 20 bb/hr std. dev., and you want to play for 100 hours with a 1% chance of going broke, you need 737 bb. (Mason's bankroll formula would give 900 bb in this instance, but that assumes a longer playing time.) If you change N from 100 hours to 1000 hours, you would need 919 bb to have a 1% risk of ruin.


Hope this helps,

Lenny

Yeah, it relates to bankroll requirements and the swings you should expect, see my post above.

. . . write while driving a car! It leads to accidents, in both writing and driving.


Mason, I tried to be as simple as possible "but not more so". I would invite you to point out what is "not right" about the simple explanation above - only you're driving. (If you meant to quibble about SD "per session" - don't.)


Aaah, memories of "Alone Together".

"Aaah, memories of "Alone Together"."


Only you know and I know.

95% of your results will be within 2 standard deviations, and 99.7 (i think) will be within three.


Pat

can a player with a SD as large as 20 BB(that's HUGE) really be winning 1 BB per hour?

--"Can a player with a SD as large as 20 BB(that's HUGE) really be winning 1 BB per hour?


Yes, why not? The expected value of a certain player may be 1 BB per hour, or less, but his results may very well be fluctuating wildly around that EV, to the tune you describe - or worse! The wild fluctuations give a measure of the risk the player undertakes. The weaker the fluctuations, the more close to the EV his actual results are, and, therefore, the less risk he runs to be wiped out of all his money. When the fluctuations get dangerously wild, the player runs the risk of going overboard and getting (his bankroll) killed.


And that is why one should not look only for games with a "good EV" but for the games with the best Expectation/Risk ratio. Not just in Poker but in any game.

well, of course, mathematically, its possible, its similar to blackjack.


i meant, has anyone heard of results like that? it seems to me a standard deviation that high would imply playing very loose, and an EV that high would imply playing fairly tight and very well. it just doesnt seem like those types of results fit the nature of structured limit texas hold'em.


generally the players who make the 1 BB per hour have their game pretty solid and their SD buckled down to 7-10BB.


if my SD was 20 BB, I wouldn't be very comfortable with my EV estimate, or my game.