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View Full Version : Standard Deviation and Confidence Intervals

weevil
08-17-2005, 03:03 AM
In the process of building a new bankroll, I've logged 7k hands at 2/4 in the last week. My standard deviation is 18.6BB/100, and my BB/100 is at 8.5 (yeah, I've been running pretty well, even admist two, thousand + breakeven spells, these games are just damn soft). Obviously, something is incorrect with my standard deviation, since it would seem to imply I should have larger negative swings as well (I've only had one 100 hand session &gt; -25BB, most losing sessions average around 4-5BB/100). Using a link (http://www.svenskpoker.com/math.php?hands=7000&amp;bb100=8.5&amp;std100=18.6&amp;ci=95&amp;bt nCalc=Submit?Query) I found in one of the forums for figuring your true BB/100 with a certain confidence interval, it thus appears there's a 95% chance I'm between 4.14 and 12.85BB/100 winner. I don't know the math behind these calculations, but they seem rather crazy, especially given that my strd/dev shouldn't be that high given that most of the deviation is in the positive.

To conclude, this isn't a brag post, but I am curious if I can have any confidence from these numbers and stats in being at least a 4BB/100 winner in the party 2/4 games? If so, since I can log 200 hands/hour easily, that's over \$30/hour without rakeback, and a hell of a lot more than I can expect to earn in any current prospective job. Please don't repond with sarcastic illusions to my deity /images/graemlins/tongue.gif, helpful comments on the relavance of this sample size given the confidence interval calcuations involved is what I'm curious about.

BruceZ
08-17-2005, 04:34 AM
[ QUOTE ]
In the process of building a new bankroll, I've logged 7k hands at 2/4 in the last week. My standard deviation is 18.6BB/100, and my BB/100 is at 8.5 (yeah, I've been running pretty well, even admist two, thousand + breakeven spells, these games are just damn soft). Obviously, something is incorrect with my standard deviation, since it would seem to imply I should have larger negative swings as well

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Not necessarily. Your results for 100 hands may not be normally distributed, so they can be skewed positive. This is especially true if you had a few hot sessions which exaggerate your mean relative to your median (mid-point of the distribution). Your long term results and averages will be normally distributed, and you use this mean and standard deviation to analyze the long term results. Your standard deviation looks reasonable, and excellent for that win rate. How did you compute it?

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(I've only had one 100 hand session &gt; -25BB, most losing sessions average around 4-5BB/100). Using a link (http://www.svenskpoker.com/math.php?hands=7000&amp;bb100=8.5&amp;std100=18.6&amp;ci=95&amp;bt nCalc=Submit?Query) I found in one of the forums for figuring your true BB/100 with a certain confidence interval, it thus appears there's a 95% chance I'm between 4.14 and 12.85BB/100 winner. I don't know the math behind these calculations, but they seem rather crazy, especially given that my strd/dev shouldn't be that high given that most of the deviation is in the positive.

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That is the correct 95% confidence interval for the numbers you gave. Note that it is

95% CI = 8.5 +/- <font color="red">1.96*</font>18.6/sqrt(70)

where 70 is the number of 100 hour sessions. But it doesn't mean that there is a "95% chance" that your win rate is in this range. Only that there is a 95% confidence that it is in this range. See this post (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&amp;Board=probability&amp;Number=258 7704&amp;Forum=,,f11,,&amp;Words=confidence&amp;Searchpage=2&amp;L imit=25&amp;Main=2571009&amp;Search=true&amp;where=bodysub&amp;Nam e=&amp;daterange=1&amp;newerval=3&amp;newertype=m&amp;olderval=&amp;ol dertype=&amp;bodyprev=#Post2587704) for the important distinction.

You can do this analysis for only 7K hands, but remember that the conclusions only apply to the same game conditions that existed for those 7K hands, and if they change significantly, then the analysis will no longer apply to future sessions. In other words, 7K hands may not be a large enough sample to represent the game conditions that you will experience in future sessions.

weevil
08-17-2005, 04:57 AM
[ QUOTE ]
[ QUOTE ]
In the process of building a new bankroll, I've logged 7k hands at 2/4 in the last week. My standard deviation is 18.6BB/100, and my BB/100 is at 8.5 (yeah, I've been running pretty well, even admist two, thousand + breakeven spells, these games are just damn soft). Obviously, something is incorrect with my standard deviation, since it would seem to imply I should have larger negative swings as well

[/ QUOTE ]

Not necessarily. Your results for 100 hands are not normally distributed, so they can be skewed positive. Your long term results and averages will be normally distributed, and you use this mean and standard deviation to analyze the long term results. Your standard deviation looks reasonable, and excellent for that win rate. How did you compute it?

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Well, I guess I really don't know much about the distinction between mean and standard deviation, looks like your linked post is a good place to start. I got the std. dev. numbers from the sessions tab in PT. It seems like everyone in the small stakes forum thinks 14-15BB/100 is a good normal standard deviation, though I suppose if I'm running well and/or at a higher average BB/100 rate it will be higher.

[ QUOTE ]
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(I've only had one 100 hand session &gt; -25BB, most losing sessions average around 4-5BB/100). Using a link (http://www.svenskpoker.com/math.php?hands=7000&amp;bb100=8.5&amp;std100=18.6&amp;ci=95&amp;bt nCalc=Submit?Query) I found in one of the forums for figuring your true BB/100 with a certain confidence interval, it thus appears there's a 95% chance I'm between 4.14 and 12.85BB/100 winner. I don't know the math behind these calculations, but they seem rather crazy, especially given that my strd/dev shouldn't be that high given that most of the deviation is in the positive.

[/ QUOTE ]

That is the correct 95% confidence interval for the numbers you gave. Note that it is

95% CI = 8.5 +/- 18.6/sqrt(70)

where 70 is the number of 100 hour sessions. But it doesn't mean that there is a "95% chance" that your win rate is in this range. Only that there is a 95% confidence that it is in this range. See this post (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&amp;Board=probability&amp;Number=258 7704&amp;Forum=,,f11,,&amp;Words=confidence&amp;Searchpage=2&amp;L imit=25&amp;Main=2571009&amp;Search=true&amp;where=bodysub&amp;Nam e=&amp;daterange=1&amp;newerval=3&amp;newertype=m&amp;olderval=&amp;ol dertype=&amp;bodyprev=#Post2587704) for the distinction.

You can do this analysis for only 7K hands, but remember that the conclusions only apply to the same game conditions that existed for those 7K hands, and if they change significantly, then the analysis will no longer apply to future hands.

[/ QUOTE ]

By game conditions, do you mean the quality of play of both myself and my opponents, or the distribution of cards dealt (not quite sure what this latter would mean /images/graemlins/crazy.gif)

EDIT, for some reason your last sentence got cut off in my quoted post, so I didn't see that. Ignore my confusion, I see what you're saying.

BruceZ
08-17-2005, 05:58 AM
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Well, I guess I really don't know much about the distinction between mean and standard deviation, looks like your linked post is a good place to start.

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Was that a typo, or did you mean to say that you don't know the distinction between the mean and the standard deviation? The mean is the average, and the standard deviation is a measure of the "spread" of your results, but you seemed to know that.

[ QUOTE ]
I got the std. dev. numbers from the sessions tab in PT. It seems like everyone in the small stakes forum thinks 14-15BB/100 is a good normal standard deviation, though I suppose if I'm running well and/or at a higher average BB/100 rate it will be higher.

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That would be excellent. <font color="red">EDITED FROM:17 times the win rate in bb/100 is good.</font> <font color="blue"> EDITED TO: 6 times the win rate in bb/100 is good, or 18 bb for a win rate of 3 bb/100.</font>

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You can do this analysis for only 7K hands, but remember that the conclusions only apply to the same game conditions that existed for those 7K hands, and if they change significantly, then the analysis will no longer apply to future hands.

[/ QUOTE ]

By game conditions, do you mean the quality of play of both myself and my opponents, or the distribution of cards dealt (not quite sure what this latter would mean /images/graemlins/crazy.gif)

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The play of yourself and your opponents.

uuDevil
08-17-2005, 02:47 PM
Nitpicking: You left out a constant in your 95% confidence interval arithmetic. Should be:

95% CI = 8.5 +/- 1.96*(18.6/sqrt(70))

[ QUOTE ]

[ QUOTE ]
I got the std. dev. numbers from the sessions tab in PT. It seems like everyone in the small stakes forum thinks 14-15BB/100 is a good normal standard deviation, though I suppose if I'm running well and/or at a higher average BB/100 rate it will be higher.

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That would be excellent. 17 times the win rate in bb/100 is good.

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Wondering: are you saying here that 14-15 BB/100 is excellent and that anything &lt;17xWR is good? Or is that greater than?

/images/graemlins/diamond.gif The same players reporting SD ~15 BB/100 are also reporting WR ~3BB/100. Well, I assume they are the same or similar players-- here are some links:

SS WR (http://archiveserver.twoplustwo.com/showflat.php?Cat=&amp;Number=1753522&amp;page=&amp;view=&amp;sb=5&amp; o=&amp;fpart=1&amp;vc=1)

!Edit: hard to tell what the actual average is. Probably between 2 and 3 BB/100.

/images/graemlins/diamond.gifAlthough I'd like a low SD and high WR, a lower than normal SD seems correlated with running bad. See pzhon's post here. (http://archiveserver.twoplustwo.com/showflat.php?Cat=&amp;Board=probability&amp;Number=1903458 &amp;Forum=All_Forums&amp;Words=pattern&amp;Searchpage=0&amp;Limit =25&amp;Main=1902619&amp;Search=true&amp;where=bodysub&amp;Name=91 48&amp;daterange=1&amp;newerval=1&amp;newertype=y&amp;olderval=&amp;ol dertype=&amp;bodyprev=#Post1903458)

BruceZ
08-17-2005, 04:18 PM
[ QUOTE ]
Nitpicking: You left out a constant in your 95% confidence interval arithmetic. Should be:

95% CI = 8.5 +/- 1.96*(18.6/sqrt(70))

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Thanks, I fixed it.

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[ QUOTE ]

[ QUOTE ]
I got the std. dev. numbers from the sessions tab in PT. It seems like everyone in the small stakes forum thinks 14-15BB/100 is a good normal standard deviation, though I suppose if I'm running well and/or at a higher average BB/100 rate it will be higher.

[/ QUOTE ]

That would be excellent. 17 times the win rate in bb/100 is good.

[/ QUOTE ]

Wondering: are you saying here that 14-15 BB/100 is excellent and that anything &lt;17xWR is good? Or is that greater than?

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I fixed that too. 17-18 bb should correspond to a win rate of 3 bb/100, so the SD should be ~6 times the win rate.