I see on this forum and on 2+2, people complaining about the huge amount of downswings and they always wonder "why me?" People win big amounts and then inexplicably lose big. The answer lies in two magical statistical numbers: ROI (for SNGs) or WR (for cash) and SD (standard deviation).

I'll try to give an intuitive meaning of what these mean. ROI and WR are basically the average (or mean) amount you will be making over a sample set of tourneys or cash games you played. So if your PTBB is 4/100 that means you are making 8BB per 100 hands over the data sample you have. This may or may not be your true win rate. Now, SD tells you how much deviation that sample has. From [url=http://en.wikipedia.org/wiki/Chebyshev%27s_inequality:207bor2g]Chebyshev's inequality[/url:207bor2g], there are some simple facts.

At least 50% of the values are within 1.41 standard deviations from the mean.
At least 75% of the values are within 2 standard deviations from the mean.
At least 89% of the values are within 3 standard deviations from the mean.
At least 94% of the values are within 4 standard deviations from the mean.
At least 96% of the values are within 5 standard deviations from the mean.
At least 97% of the values are within 6 standard deviations from the mean.
At least 98% of the values are within 7 standard deviations from the mean.

I highlighted the second because I think that's the most common in Poker. So if you are a 2PTBB winner with an SD of 10PTBB/100 (mine is actually 30PTBB/100 in my life time database of >100K hands), you can be easily be within -8PTBB to +12PTBB with 75% certainity. Imagine if you were a break-even player, this is like -10 to +10 with 75% certainity.

Another way to look at it is: what is the chance that you have a winning session? I asked this question in theory forum on 2p2 and got some [url=http://forumserver.twoplustwo.com/showflat.php?Cat=0&Board=genpok&Number=12041787&Se archpage=1&Main=12041787&Words=%2Bstandard+%2Bdevi ation&topic=&Search=true#Post12041787:207bor2g]great answer[/url:207bor2g] as to how to calculate this stuff using [url=http://en.wikipedia.org/wiki/Standard_score:207bor2g]z-score[/url:207bor2g]. In short, as your ROI starts dropping and sd increasing, your chance of winning session drops significantly. Of course, this can be reduced by playing large sessions and longer time.

Overall, if you are seeing some variance, say +500 to -500 at 100NL in like 1000 hands, your true win-rate is actually small, sd is high and that's why you are seeing this. It's NOT because online poker is rigged or some sites are against you. The key to understanding this phenomenon is that every one sees same kind of variance, but for some the swings seem huge. Why? because the true win-rate is much lesser and the sd is very high.

In summary, if you are seeing big swings, first thing you should blame is your play. True, bad stuff happens, but bad stuff gets magnified if you are playing bad.

Hope this helps some one to look at the game of Poker differently.

SPAM

What does the anachronym PTBB stand for? It looks like the number of big blinds won per 100 hands is equal to twice PTBB.

In another post you mentioned that a good player should make at least 4PTBB at low stakes, which would make a win rate of 8BB per 100 hands the minimum level to be considered a pretty good or successful player at the lower levels. 8BB per hour; seems to me that this level of proficiency would have a good edge over the majority of players at the level. I assume would require a pretty good level of skill to achieve, and it appears to me that relatively few online players achieve this.

As for me, I do not have a large enough sample size in SNGs or ring games to know my actual level. I twice lost my stack on ultimate bet trying to bet $1 on c-bets, and having tables jump around, and accidentally going all in instead. Unfortunately stuff like that, which cannot happen live, counts in your win rate.

I strongly agree that if you have a large downswing or are losing money, the first assumption is to blame your own play. I would add, don't become over confident if you have a large upswing, unless the sample size is very large. Also, it can be as easy to become under confident, as well over confident, and both can be serious problems.

PTBB = 2 * normal_BB

At low stakes, 4PTBB is not that difficult to achieve especially with good game selection. However, I think it's tough for a casual player to achieve that with out studying the game.

sfustsh

From what I remember about standard deviation, I thought the probability of falling within two SD was something like 96%.

Obviously I don't remember as well as I remember, or all the data I've worked with had been in tighter samples.

__________________
johnny350h: that was in his range yes yes

Ugignadl

Hey Linux. I agree with the point of your post: if you are having big swings it is most likely due to the way you play. For me, I take a lot of marginal situations and play them hard, so I end up with big swings in that respect. So good post on that account.

That said, the way you have used Tchebysheff's inequality is a little misleading. You don't know the required information to infer trends about future data. In this context, it is a result which implies information about the data set for which you have already calculated the mean and variance (or standard deviation). Further, the use of the theoretic mean and the actual mean for your example of a `winning session' is even more misleading, due to the nature of how people play poker from session to session. In a lot of ways, it is nonsensical to apply Tchebysheff's inequality to future *single sessions* of poker using *all previous hands* as your past data.

So I wouldn't be basing an argument with your conclusion here on Tchebysheff's inequality. It's a nice result, true, but it is in a lot of ways a `dumb' result as well. It is far from sharp in applications (this is usually when 0 < k < 1) and is, at its heart, a statement about the properties of functions of numbers. If you did want to apply it to poker, you would have to do it in a much more general way than you have above. And when you did that, you would see that it doesn't give you the same `startling' results answering questions like ``what are the odds I will have a winning session?''

Nitpicky I know.

__________________
I have been blogging. Last updated April 29th. http://www.feltpoker.com/blogs/ugignadl/

Have you read the 2p2 thread I linked to where I asked the question? The z-score method (which I believe is another application of Chebyshev's inequality) is more appropriate.

I am not sure what you are trying to say about Chebyshev's inequality. I didn't use it directly rather the derived facts. What do you mean a general way?

I agree that if you have small data set (meaning you played a few hands), it's tough to find your true win rate for future sessions anyway. That's what misleads so many people, even those who make an effort to calculate these numbers. However, I believe that if you have 100K hands, the probability of a winning session is going to be more precise. I have seen it for myself many times repeated over and over.

Heids

Sfustsh - you're about right on. Assuming the data follows a normal distribution, you can expect the following:

+/- 1 Standard Deviation = 68.3% of all data,
+/- 2 Standard Deviations = 95.4% of all data,
+/- 3 Standard Deviations = 99.7% of all data,
+/- 4 Standard Deviations = 99.994% of all data,
etc.

For those who care, the above probabilities are accepted to be more exact than the Chebyshev's inequality probabilities. Quoted from an article at the below link,

"Although Chebychev's therom is true for any data from any distribution, it is limited in that it only gives the smallest proportion of observations within k standard deviations of the mean. In the case of when the distribution of a random variable is known, a more exact proportion of observations centering around the mean can be computed. For instance, if certain data follow a normal distribution, approximately 68%,
95%, and 99.7% of the data are within 1, 2, and 3 standard deviations of the mean."

http://etd.library.pitt.edu/ETD/availab ... ed/Seo.pdf

I do believe that if you've played many sessions over the long term, you
could predict the probability of your future win rates and identify your expected swings. This may help you better understand what is normal and what is not normal for your style of play so you can quickly identify whether or not you have developed a leak in your game. It would also help you identify what your bankroll needs to be in order to absorb the typical variance you'll face in your game. The ability to predict, of course, assumes you don't change the way you play the game dramatically - in other words, you see about the same proportion of hands preflop and raise with the same range of hands in similar situations. If you permanently change your game (see more hands or play more aggressively or passively) the new way you play will be part of a different distribution (you'll likely generate a new mean and new standard deviation) and the past won't reflect the future. While statistics are often a statement of the past, under controlled conditions they can pretty reliably predict the future.

Good post Linuxrocks. Expected variance is an important concept that people often forget about.

I am honored to have you reply, Heids. I see the difference in the percentages now. Chebyshev's inequality extends it to all kinds of random distributions. I guess we can take Poker as a normal distribution and take the percentages you mentioned.

Ugignadl

Heids, all due respect, but I don't think poker fits a normal distribution. Certainly not for some people, and they may well be the people who are the target audience of this post.

The overriding problem here is that the expected value of the random variable determining our profit or loss for a given hand will not in general exist. Mathematically, it is most common in industry applications in this kind of scenario (stock trading, or even gaming, yes the university will consult on gaming) that the expected value depends itself on the number of observations taken. This is a much more realistic assumption.

With regard to your question Linux, I am asserting that is is much more meaningful to apply Tchebysheff's inequality to a collection of data structured in a more relevant way relative to our desired conclusion. Here is an example. We want to conclude something about a future contiguous block of data points. Say we have a collection of K hands (data points). We fix a number of hands we will play in the next session: say x. Break the K hands up into contiguous blocks of x hands. Now, calculate the mean, variance of this data set of ~K/x data points (which are now blocks, intuitively sessions). NOW use Tchebysheff's inequality on this collection of blocks of hands. Now, your expected value is in fact the expected value of a session. The problem mentioned above still holds, but its effect on our results is lessened by some small amount.

As an appendix to this post, and perhaps the thread, I should point out that the wording on the Wikipedia page is also very misleading. As opposed to a theoretic argument I will provide an example of what I mean. Take the sentence "As an example, using k = √2 shows that at least half of the values lie in the interval (μ − √2 σ, μ + √2 σ)." This is certainly true. However, what they mean by this is that at least half the values of the random variable itself, as it ranges over every possibility, are in a certain interval. (If you take a subinterval, then this will no longer be true.) It is definitely not what most people think of, and it is not clear at all what you would mean applying this to a poker type scenario. Eliminating this confusion, and similar other sources of confusion, is the motivation behind my example above.

Now that was a huge waste of time, right?

__________________
I have been blogging. Last updated April 29th. http://www.feltpoker.com/blogs/ugignadl/

SPAM

This thread should probably be copied and posted on 2plus2.