Just another std dev question

[jasonHoldEm]

Hi guys,

Been going through my first cold streak recently. As a new player I'm still developing an appreciation for the natural swings in the game (and sadly not having this appreciation has caused me to lose some of my edge and make things worse).

I walked away from the tables for a few days after losing my entire buy-in Sunday night in about a half hour. I had three big hands that went sour (and I probably could have gotten away from two of them had I been thinking more clearly), the other hands I played I just threw my money away, so I think a break is the best thing for me at this point.

I've been trying to gain a better understanding for the swings I am to expect playing poker and since I'm not a math wiz, I'm turning to you guys for help.

Soooo here goes:

I have 36 sessions logged so far. Since I'm still new to the game I keep my sessions fairly short (they range from 12 mins to 126 mins). I converted the minutes to hours so that I could use Mason's formula (from his essay on the main page)...will using decimals (ex: 0.73 hr) throw off this calculation somehow?

I followed the example he gives in the essay (my N=36 instead of 10) and I came out with a variance of 620.5 and a sd of 24.91. I should point out that I play 1/2 at this point.

I'm pretty sure I calculated these correctly (if they seem unrealistic please let me know), but now I get to show my ignorance by asking what exactly do these numbers mean? I squeaked through stats 101 with a D, so please use small words. [img]/forums/images/icons/grin.gif[/img]

Thanks in advance,
Jason

[BruceZ]

will using decimals (ex: 0.73 hr) throw off this calculation somehow?

No, that's how it should be done.

I followed the example he gives in the essay (my N=36 instead of 10) and I came out with a variance of 620.5 and a sd of 24.91. I should point out that I play 1/2 at this point.

I'm pretty sure I calculated these correctly (if they seem unrealistic please let me know), but now I get to show my ignorance by asking what exactly do these numbers mean?

That is a realistic standard deviation. It should be around 10-12 big bets, and it is 12.5. Lower is better, and it will become more accurate with more sessions. It will be higher if you play a lot of shorthanded games or aggressive games. If you win more than 1 bb/hr your standard deviation will usually be higher, but it should still be around 10-12 times your average hourly win. Ways to reduce your standard deviation include improving your hand reading skills, tightening up before the flop if necessary, and avoiding chasing in marginal situations.

Divide this number by the square root of the number of hours played, in this case 24.91/6 = 4.15. Now compute the average amount you have won or lost per hour played. This should lie within $4.15 of your true hourly rate 68% of the time (1 standard deviation), and within 2*$4.15 = $8.30 95% of the time (2 standard deviations). This will become more accurate as you play more hours. This gives you an estimate of your true hourly rate or EV.

You can also use these numbers to figure out your chance of going broke and how much bankroll you need by plugging into these formulas:

ror = exp(-2uB/sigma^2)
B = -[ sigma^2/(2u)]*ln(ror)

ror = risk of ruin (going broke)
B = bankroll
u = hourly rate
sigma = hourly standard deviation

The problem is that you have to play enough hours to get accurate estimates of u and sigma, but if your bankroll too small you may go broke several times before you find that out.

[irchans]

Two thoughts about your post:

1) Typically you need at least a hundred hours to estimate your win rate to within 1 BB / hr or 400 hour for an accuracy of 1 SB/ hr.

2) I think it is very hard to win at 1/2 in a B&M casino because the rake is typically a dollar per hand. On line the rake is often half that, but the games are much tougher.

[jasonHoldEm]

Bruce / irchans,

Thanks for the advice and explanations. Actually irchans answered what was going to be my next question (how long before you can "accurately" estimate your win rate). I plan to keep my eye on this now that I understand it better and as my hours pile up I know the calculations will be more valuable.

Thanks,
Jason

[BruceZ]

Actually irchans answered what was going to be my next question (how long before you can "accurately" estimate your win rate).

Notice that you get irchans' numbers from what I said above. If your hourly standard deviation is 10 times your hourly rate, say hourly rate = 1 bb/hr and hourly standard deviation = 10 bb, then if you play 100 hours you divide 10 by sqrt(100) = 10 to get 1. This means that the standard deviation of your hourly rate will be 1 big bet after this amount of time, or that your true hourly rate will be +/- 1 big bet from your actual hourly rate 68% of the time as I explained. After 400 hours it would be within 10/sqrt(400) = 10/20 = 1/2 big bet or 1 small bet. Notice that it will take even longer than this if your standard deviation is greater than 10 times your hourly rate. This standard deviation of your hourly rate is actually called the "standard error" of the hourly rate. It is a measure of how accurate your estimate of your hourly rate is. Note that it takes a very long time to have a very accurate estimate of your hourly rate. If you want the standard error to be 10%, you would have to play 10,000 hours, since 10/sqrt(10,000) = 10/100 = 10%. Note that this doesn't mean you really know what it is to within 10%, just that it will lie within 10% 68% of the time (68% confidence). 95% of the time it will lie within twice this or 20% etc. Note that it takes much less time for your standard deviation to become accurate than for your hourly rate to become accurate.

[Copernicus]

How "accurate" is any of it. Unless you play the same stakes against the same players your "true" (as well as your "sample") mean and SD change. It is somewhat illusory to think that more data means more accuracy when the conditions arent the same.

[jasonHoldEm]

ooooh that's good.

Making much more sense now, thank you. I actually remember the words standard error, but that was like five years ago so things are pretty rusty...if only I hadn't sold my stats book on ebay. [img]/images/graemlins/frown.gif[/img]

Thanks,
jHE

[redsamurai]

This weekend a college kid came into the cardroom I play $4-$8 at with a stats book in hand. The table gave him a $160 lesson in 2 hours with Pedro joking about taking the white boy's tuition money the whole time. If he comes back I can see if he'll sell the book for a few chips.

[BruceZ]

There is a version of the central limit theorem (Liapounov version) that applies even to random variables that are not from the same distribution (changing). Under these conditions, the overall distribution will still be normal with a mean that is the sum of the means, and a variance that is a sum of the variances of the individual random variables. The changes due to the individual distributions average out in time. This explains why so many distribution in nature that are comprised of many different distrubutions are in fact normal. The convergence can take longer than if they are from the same distibution. See DeGroot p. 276.

[BruceZ]

Sounds like an excellent tactic to make good players play incorrectly.