Guys,

I've read the great article by Angel Largay on this site about Std. Deviation and calculating your bankroll. I've also downloaded a copy of the Excel spreadsheet that someone posted in the forum and used it to great effect myself. But I still have one nagging problem.

What exactly is Std. Deviation? Why is it expressed in $ terms sometimes, and in BB/100 in other articles?

Is it the dollar range I should expect to win/lose by per session? Or is it the amount of dollars I expect to win on average every 100 hands?

I understand what variance is from this article...
just not Std. Deviation

Cheers,

SS

Standard deviation is defined as the square root of the variance. What does this mean? Well, basically, it's a measure of the spread of a set of numbers. So, for instance, if you take an average of your wins and losses, the SD is an estimator of how wide your swings are, in either direction. How is this different from variance? Well, it's because SD is actually used to describe the "spread" in the context of a normal distribution - which is your famous Bell Curve. Unfortunately, I can't draw a picture of a bell curve on BTP, but I've provided some links below, in case you're not familiar with this concept. If you can picture a bell curve, the mean of the distribution (or the average) is the point that reflects the middle. Another way of thinking of this is that it is a reflection of an event that is most likely to occur. Back to poker - this means that if you are winning, on average, +$5, then it is the mean, or the "center" of a distribution of your wins and losses. It means that you are most likely to end up +$5 in any given session. The standard deviation, on the other hand, tells you how far away you have to move from the center before you hit a less-likely event. For instance, if your SD is $20, it means that whenever you have a winning session that is $25, you are actually +1 SD out from the mean, which corresponds to (in a perfect normal distribution) an event (in this case, winning an extra $20) that is one-third as likely to happen as an event at the mean. Accordingly, it means you have an equal probability (which is one-third less likely to occur than the "average" case of winning $5) of moving -1 SD away from the mean, which translates to a net loss of $20.

So what does all this mean? It really just goes back to the idea of how volatile your style of play is. I'm not sure if this is entirely an accurate statement, but you might consider it one measure of how loose-aggressive your play is. (If anyone else as a better concept of how it directly relates to poker, feel free to clarify/correct me )

Hopefully this makes some sense to you. Now, let me address your other question about why it's sometimes expressed in $s and sometimes in BB/100 - this is quite simple. Ultimately, it doesn't matter *what* units you use for calculating your SD - it is merely an expression of the "spread" of a set of numbers, right? So if all of your numbers are in terms of BB/100, then it's an expression of your spread in terms of BB/100. If, on the other hand, if your 'numbers' (wins/losses) are measured in $s, then your SD reflects your volatility in actual dollars.

Now since PT calculates means and SDs in terms of straight dollars, BB/100, *and* BB/Hr, I find that looking at SD in terms of pure dollars is what's of most interest to me, since all I am really concerned about is my true win rate in terms of dollars. This is important to me, because I can use the SD (in dollars) as well as my current bankroll (in dollars) to calculate my current risk-of-ruin, to see if my true win-rate justifies the level of play in which I am involved. I've found that since PT is perfectly happy calculating a weighted mean for your win rate and, correspondingly, your SD, these stats in terms of dollars are significantly more useful in determining what my BR *should* be based on the risk-of-ruin I'm willing to accept.

Am I off base here, guys?

Here's some links to articles about SD:


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Hey Thanks musicman80,

That cleared things up so much.

That first link listed, is the best explanation I've come across on the web. And I didn't know there was a function in Excel that calculates Std. Deviation already. I'll try it to see if gives the same results as my current spreadsheet.

Cheers,

SS

p.s.

One more question please...?

In your example, you imagined an hourly/average win rate of $5 and a std. deviation of $20. then also you said if you had a session of +$25, you'd be 1 SD out of your normal range. (Which if you look at that graph in that link you sent , this +$25 session would be in the right hand green section of that graph.) That's crystal clear with me now.

But what happens, in this far-fetched example, if that EVERY session ends at +$25. 100 or more in a row. What happens to your std. deviation? I'd imagine that the graph would start 'skewing' to the right? But what happens to std. deviation value? I'd guess that would decrease? Because you are no longer having large swings, the graph would start to thin out and become narrower.

Is this correct?

Thanks again.

Standard deviation and variance is used to describe the spread of a sequence of number around the sequences mean value. I think that the standard deviation is the interesting concept. I just use the variance to calculate the standard deviation. In words you can define the standard deviation as "The average distance of the numbers from the average of the sequence". You don't have to use normal distributions to define standard deviations. I basically agree with everything that musicman80 says but just wanted to point out that since there was some doubt if your poker earn rate is normally distributed if you follow the link that SideSwipe supplied. Lets look at some examples. (If you want to you can imagine that you play poker for one hour four time and these are the outcome of your sessions. If you don't like the outcome of your sessions feel free to change the numbers ) We have the sequence

4, 0, 4, 0

What is the mean of this sequence? It is (4+0+4+0)/4 = 8/4 = 2. What is the standard deviation? In other words, what is the average distance of the numbers from 2. If you look at the sequence you see that the average distance is 2. Now lets see if we can calculate this. You cannot just calculate the average of the difference between the numbers and the mean. That is

((4-2) + (0-2) + (4-2) + (0-2))/2 = 0

is not correct. What went wrong? Well, distances are always positive. So instead lets calculate the average of the distance squared from the mean of the sequence. Then we get

((4-2)^2 + (0-2)^2 + (4-2)^2 + (0-2)^2) / 4 = (4+4+4+4)/4 = 4.

Not quite what we wanted. This is the variance of the sequence. To get the standard deviation take the square root of this number. Then we get 2.

Look at the sequence

6, -2, 6, -2

instead. If you do the same calculations here you will see that the mean is still 2 but the standard deviation is 4. So standard deviation is a measure of the spread of the sequence.

So what does the standard deviation say about your poker play? I'm not sure but close decisions will increase your standard deviation. Lets say that you make the same type of decision a number of times (lets say n times) every session. The probability that you win $1 is p and that you end up with $0 is (1-p). The expected value (or mean) of these decisions will be n*p. The standard variation will be the square root of n*p*(1-p). If you try different values of p will see that the standard deviation will be maximal when p = 0.5. If you want to read more on this you should be able to find much material on the web if you search for "binomial distribution" and "standard devation" using some search engine.

Hope this post adds something to the thread.

/CipherJr

Thanks CipherJr,

Now I'm more confused than when I started this! Only kidding.

I think your explanation is beyond my comprehension at the moment. I'm happy with musicman80's explanation. So once I've passed my noddy, beginners level maths exam, I'll look into your information further.

But it definetly has added greatly to the thread.

Cheers,

SS

Hope I didn't make you too confused

Basically I'm just saying that it is much better to be all-in pre-flop with AA against QQ (p approximately 0.8) rather than with QQ against AK (p approximately 0.55). Not only is the expected value on the first situation higher the standard deviation is also smaller. Hence you will grow your bankroll fast and safe in the first situation and slower and more unsafe in the second situation. So as you can see just math jargon for something rather obvious. However the math is nice since we get some quantative information. But the real difficult part, and I guess thats where poker skills enter, is to decide what p is when you make your decision.

/CipherJr

Musicman, Cipher,

thanks for your great explanations!

I still have one last question, how do I get from my SD to the prop. that i will become broke with a specific BR size?

I have seen example numbers in some Books/Articles but never how to calculate this by myself.

Thanks Again,
Simon

Simon,

It's pretty straightforward to calculate, as long as you don't mind solving an exponential equation (it's actually a lot easier than it sounds, trust me... ) I figure it's probably best if I simply share the formula I used:

risk of ruin = Exp( -2*BR*WR / (SD^2) )

BR = bankroll
WR = win rate (I like to use the hourly win rate in $ that PT tells me)
SD = standard deviation (again, I use the hourly SD in $)

Note that the 'Exp' function (which is also builtin to Excel, and you'll like this if you are a spreadsheet junkie like me ) is the exponential function, or the base of the Napier Logarithm, e (a number whose value is approximately 2.718).

So let's try an example.

Suppose your bankroll is $1000, you have a win rate of $20/hr, and your SD is $90/hr. First, calculate the square of the SD. SD^2 = 90^2 = 8100. Then, we have the following:

risk of ruin = Exp( -2 * 1000 * 20 / 8100) = Exp( -40000 / 8100) = ~0.007

This means that with the bankroll you have, and with your win rate and SD, you will have a 0.7% likelihood of losing your entire bankroll.

What I have yet to figure out is why this formula or its inverse (the formula that takes a risk-of-ruin and gives you a required BR) makes mathematical sense... If anyone's got a quick answer to this, I'd be interested to hear it. If not, there's always Google...