I have two real utility ideas for it

1) results convergence. Like say you've averaged 2 BB/100 for awhile and want to answer "am I a winner?" With 95% confidence (z=1.96). It takes 78% longer to answer that question at a SD of 20 BB/100 versus 15 BB/100. Obviously if you're seeking this answer to determine whether full time play is an option, quicker answers is better.

2) tilt prevention. Say you've 2 guys, one (TAG) has the aforementioned 15 BB/100 SD and the other (LAG) is 20 BB/100.

TAG wins at 1.5 BB/100, while LAG wins at 2.5 BB/100. However, using the z table, one can see there's a 1% chance the TAG loses 33.45 bets or more in a 100 hand stretch, while the LAG loses 44.1 with the same probability. So maybe the TAG can't emulate the LAG, because the extra risk exposure erodes his play to a point where he makes himself far too likely to tilt, and may opt to avoid marginal, high variance scenarios that the LAG can maneuver (and not tilt regardless of result).

A lot of us just got used to having SD as a stat in PT or HEM. Go play a couple hundred thousand hands, look at your WR and SD by game type. Know stuff about planning bankroll and limits for next year, assuming you can hold even against the field learning-wise. Won't most of the live tracking software just give you SD or variance? You don't have to keep even sweeps of time to calculate it, the hourly SD comes out of the math. IIRC, variance converges relatively quickly.

It should converge in 10-20 sessions, regardless of session length.

It has been forever since I have seen the math, but don't you just gather up the session data (which includes time played) and out comes SD in BB/HR? That's fast, because you'd have no real idea of WR in 20 sessions. You'd be way better off as an experienced player to ignore results and do as Mason suggested and just guess WR based on mistakes that your opponents are making. You'd still have a good chance of being wrong, but no better than using a 3K hand sample as a WR estimate -- that's assuming twenty 5 hour average sessions of 30h/hr. Show an online player a 3K hand sample and claim something about WR and he'd be giggling uncontrollably.

OK I used google, I think this is BruceZ going through the derivation of Mason's formula from Gambling for a Living(?).
Basically it is a way to weight session results of different time to get better results. If you just pretended that a $500 swing in a 5 hour session is equal to 5 sessions with $100 swings, the result looks too smooth? I think so. We always looked at variance as mean square error of servo position, and since servo sample is very regular, this never came up.

No, this isn't correct, at least not if you're using the formula above. The formula accounts for session length, and it should be unbiased regardless of session length.

This is true.

Yeah, that's way low. Even in a tight game, you shouldn't get an hourly SD less than 10 BB/hr or thereabouts.

Yes.

Right, but for a small number of sessions, the confidence interval for your WR estimate based on your SD estimate will be huge, properly reflecting the large amount of uncertainty in your WR estimate.

In other words, your estimate of your hourly SD can be spot on, but you still have a lot of uncertainty in your WR estimate if you only have a handful of sessions. How many sessions or hours you need to get an accurate WR depends on the ratio of the WR to the SD -- the higher the ratio, the sooner you can get a decent WR estimate (i.e. one with a narrow confidence interval). A rule of thumb is that if you're a solid winner (1 BB/hr) in a typical game with typical variance, you want around 1,000 total hours or more to get a decent level of confidence.

Here's how to think of it statistically: The standard error (SE) of your hourly WR is SD/sqrt(total hours).

So let's say your hourly WR estimate is 1 BB/hr, and your SD is 12 BB/hr. At 1000 hours, the SE of your WR estimate is 12/sqrt(1000) = 0.38 BB/hr.

So your 95% confidence interval is 1 +/- 1.96*0.38 = (0.25 BB/hr, 1.75 BB/hr).

By contrast, if you have, say 20 8-hour sessions (160 hours total), your confidence interval is: 1 +/- 1.96*0.95 = (-.862 BB/hr, 2.862 BB/hr), which is awfully wide.

That's all assuming a WR of 1 BB/hr and SD of 12 BB/hr. The only thing I'm changing is the total number of hours used for the estimates.

Adding: To be safe about it, you should also take these estimates with a grain of salt since the underlying assumptions (independent, identically distributed random variables) are undoubtedly violated.

I can try to post an Excel spreadsheet with all the entries set up to do the calculations, if people are interested.

If SD/WR is high, uncertainty in WR won't affect your certainty in SD.

Right, I'm talking about the level of certainty in your estimated WR (i.e., what statisticians refer to as the "standard error" of the estimate).

And it's WR/SD -- the greater that ratio, the fewer the number of hours you need to be reasonably sure your WR is positive.

If I calculate my 30/60 SD with my actual winrate, it is 13.20.

If I calculate my 30/60 SD with a winrate of 0, it is 13.24.

If I calculate my 1/2 NL SD with my actual win rate, it is 83.2.

If I calculate it with a win rate of 0, it is 83.9.

Those are my two biggest WR/SD ratios, so this pretty much shows how bad it can get.

That sounds reasonable.

You're supposed to calculate it using the WR based on the data. You say "actual winrate" above, but I think you mean "empirical winrate" (i.e. the estimated winrate based on your data). If so, that's the right way to estimate your SD.

I don't know why you'd calculate your SD with a winrate of 0, unless that's the estimate based on your data.

Under the usual statistical model, your "actual winrate" is an unknown parameter. Your empirical winrate (i.e. based on the data) is an estimate based on sampling from the underlying distribution (the parameters of which are unknown).

So happy I instigated math in the NC/LC

Let's say you're playing a new game, e.g., OFC, where your SD is a total unknown. You can know your SD a lot faster than your WR, and the uncertainty in your WR typically won't matter much.

Yeah, thanks a lot for the uptick in my inferiority complex.

I think people on this board care to much about stats and data sometimes, or at least don't prioritize it correctly. I talk about it to an extend but only bexusse I don't want to see somebody on here read people saying $18k is a solid 40 roll and then go broke.



Even bb/hour or $/hour can be an extremely misleading stay and is only really good for helping yo decide which game you should play if you have options. For example: I'd be willing to prop bet I could beat 20/40 for $60 hour+ over the next 500 hours. While it's possible I could do hat it's more possible that I go broke from living expenses first sitting out all the games where I'm winning $20-30 an hour in waiting for those 1.5BB/hour 7 std def games to pop up

But how can you "know" your SD without estimating it based on the formula above (which incorporates your WR as estimated based on the data)?

If you just plug in some assumed WR, your estimated SD will be biased, unless you happened to get lucky and your assumed WR is equal to the true WR.

Well I have to get some kind of mileage out of my high-level stats education. I don't often use it in my law job, so might as well use it for dick-sizing on the internets.

Don't get me wrong, I think it's very valuable and wish I could the calcs that soem of you guys can (I can't), I'm just also pointing out that the results of some calculations can be extremely misleading

No, you're right, its value is limited in this context. Frankly, I'd probably trade my stats degrees for your poker skillz.

But only because I already have another career. My R skills are supposedly in high demand.

Sure, it will be biased, but biased by a very small amount. That's why I calculated my SDs with WR = 0.

I'd call it "misinterpreted" rather than "misleading."

The same calculations are used in the financial industry to calculate risk/reward ratios or in pharmaceutical trials to determine whether a drug is "effective." And it's pretty universally misinterpreted.

I think you're absolutely right about that. And the misinterpretation can be very dangerous.

A lot of the quants responsible for the financial meltdown in the 2000s did not understand the limits of their models. I think it's really important to understand what you're doing, to know the limits of your assumptions, and to build in a huge safety factor in interpreting your results.

But that takes a solid theoretical understanding of the models (and hence their shortcomings), combined with real world experience in how they perform with real world data. As well as a good understanding of the processes that generate the data.

It's a form of quantitative wisdom, in a sense. Not so easy to come by.

Speaking of BJ, just got back from Vegas a bit ago. I call shenanigans on some casinos now paying 6:5 instead of 3:2 when you hit a blackjack. Don't play BJ that much so it may have been going on for some time at more and more places. Not sure, just happened to notice it this trip. Wasn't very high stakes, $10 or $15 min bet.

Cliff notes: Shenanigans was called at 10:16 PM PST