Question About Standard Deviation

[ZManODS]

Once i figure out my standard deviation, what is it good for?

[ThaSaltCracka]

figuring out if the amount you win or lose in a session falls into your mean. you would probably want to be within 2 standard deviations(+/-) of your mean.

[Homer]

With knowledge of your standard deviation (SD) and mean (win rate, EV, whatever you want to call it), you can determine your risk of ruin (ROR) for a given bankroll (or vice versa). Also, you can do things such as assess how likely a losing or winning streak was.

Sources

Risk of ruin equation (Thank you BruceZ)

Computing your standard deviation (Thank you Mason)

Examples

Let's say you have an EV of 1 BB/hr and an SD of 10 BB/hr (which you solved for with the assistance of Mason's article). Here are some things you can do:

<font color="blue">1)</font> You have a bankroll of 300 BB's, and would like to know what your ROR is:

r = exp(-2uB/sigma^2) = exp(-2*1*300/10^2) = exp(-6) = ~.25%

<font color="blue">2)</font> You would like to have a risk of ruin of 1%, and are wondering what size bankroll is required:

B = -(sigma^2/2u)ln(r) = -(10^2/2*1)*ln(.01) = ~230 BB's

<font color="blue">3)</font> You have just suffered through a 150 BB losing streak over your last 50 hours of play. You would like to know how often you should expect to lose that much or more:

a) Solve for your EV over a 50 hr stretch:

EV/hr = 1 BB
EV/50 hrs = 50*1 = 50 BB's

b) Solve for your SD over a 50 hr stretch:

SD/hr = 10 BB
SD/50 hrs = 10*sqrt(50) = 70.7 BB's

c) Determine how many negative standard deviations you have suffered through to lose 150 BB's:

50 + 70.7*x = -150
70.7*x = -200
x = -2.83 SD's

d) Look up N(-2.83) in a probability table or use Excel (go into a cell and type <font color="green">=normsdist(-2.83)</font>):

The value is .234%, or 1/427.

-- Homer

[Copernicus]

Once you digest all of the academic answers, the practical answer, since you arent going to change your play one iota by knowing your SD, is, "not much".

[BruceZ]

Once you digest all of the academic answers, the practical answer, since you arent going to change your play one iota by knowing your SD, is, "not much".

How do you know he won't change his play? Perhaps he will determine that he should play lower stakes because his risk of ruin is too high for his bankroll. Or perhaps he will determine that he can win more by playing for higher stakes and playing very tight, so that he can afford the stakes at a sub-optimal win-rate. Or perhaps he will determine that he should play more than one table at lower stakes to reduce his variance. Or more than one table at the same stakes. Perhaps he will decide to play a different form of poker, play at a different site, or that he would do better playing tournaments. Or maybe he will determine that he is pushing too many marginal hands, causing his standard deviation to be too high.

Most people have no idea about these things because they do not know their standard deviation, risk of ruin, and bankroll requirements. This is why most people are surprised by the magnitude of their swings, and why they find themselves broke. In addition to the items that Homer Simpson listed, it also allows you to determine how long it will take you to break even, or to win a given amount, with a certain confidence. It can help you determine if a bad run is more likely due to bad luck or bad play. Standard deviation is at least as important as your win rate, because you need your standard deviation to know how accurate your win rate is after a given number of hours. Standard deviation is the one statistic that you can be compute to high accuracy after a fairly small number of sessions.

[bigpooch]

Absolutely agree with your post. SD estimators converge
much more rapily than EV estimators. Players need to
have a lot of data to have any confidence in their win
rate: playing online, I suppose it is possible to keep
data for every 15 minutes or half an hour which would be
much better than session by session results. This way,
the win rate could be determined with more confidence.

Also, it is quite possible to compare SD for various kinds
of games and for various table compositions (weak-passive,
tight-aggressive games, etc.) and thus determine if a
game is too big relative to BR and RoR.

[psychprof]

Once you figure your SD, is there a way to tell if it is "good" or "bad" (this assumes that there is such a thing as a good and/or bad SD).

I'm probably wrong, but my impression is that any given SD is neither good nor bad. For example, consider the following players:

Player A (6 sessions)
+1
+100
+50
+5
+150
+10

Player B
-1
-100
-50
-5
-150
-10

Won't both players have the same, high, SD?

Let's say a person's stats for 1/2 stakes were a win rate of 4.5 bb/hr and an SD of $30. I understand how to use this to determine RoR, but how does one use this information to determine:

"that he can win more by playing for higher stakes and playing very tight, so that he can afford the stakes at a sub-optimal win-rate. Or that he should play more than one table at lower stakes to reduce his variance. Or more than one table at the same stakes. Perhaps he will decide to play a different form of poker, play at a different site, or that he would do better playing tournaments. Or maybe he will determine that he is pushing too many marginal hands, causing his standard deviation to be too high."

To be very specific, I'm not sure how to use SD to evaluate how I play and to determine what changes I should make in my play.

Thanks for any advice!

[mosch]

Well, let's say you calculate your SD for a bunch of games, and wind up finding it's:
Stud/8: 14BB/hr
Stud: 19BB/hr
Holdem: 22BB/hr

Now one day you find your bankroll hurting and you're debating what to play, assuming similar win rates, Stud/8 becomes the obvious choice, since you can safely play it on a much smaller bankroll.

Or you might know that you love playing hold'em, but you aren't sure where you should play. You could take a look at your std dev calcs and see:
PartyPoker: 17BB/hr
PokerStars: 21BB/hr
UltimateBet: 24BB/hr
and realize that it's probably "safest" for you to play at partypoker.

I've found it interesting and useful to look at the deviation as I take shots at higher limits, compared to lower limits. It's obvious from a glance that I play them far more conservatively, as every time I try a new limit I have a lower standard deviation (and lower win rate) at it. It was these statistics which helped me determine that I'd probably be fine in my local 10/20 despite the fact that I didn't initially think I could play well enough to survive in that game.

Does this help answer your question at all?

[psychprof]

Does this help answer your question at all?

Yes it does, to some degree. I understand the value of comparing SDs to each other to determine which games and/or where to play. However, I still am not sure how to use SD (if it can be used at all for this) to determine how to evaluate my play at a given stake and given location.

In the example I used before, does the SD of $30 mean that he is playing too tight or too loose, or is that something that SD can't indicate?

If $30 is considered a small SD for those stakes, does that mean the player is playing weak-tight and should be more aggressive? If $30 is considered a large SD, could it be due to playing too many marginal hands? And how does one know, for any given stake, what is considered a large or small SD?

Those are the kinds of things that I question with regard to my own SD. Win rate is very clear as to what it means and what I want it to be, but every time I look at my SD I just wonder if it is good or bad.

I'm probably being to "results oriented" and should just focus on improving my play more directly through hand analyses, but after going through the hassle of putting Mason's formula into my spreadsheet, I want to use that data for something! [img]/images/graemlins/grin.gif[/img]
Thanks