There is no such thing as a confidence interval for sit-n-go's.

[Big Limpin']

HOW I THINK OF STATS:

I find i can grasp the significance of statistcal data/preditions more easily if i can make a mental picture.

Picture a multi-sided dice (die). There is one side for every result you have logged.

If you play 10 tourneys, and get 1 win, 1 second, 1 3rd, you will have a 10 sided die, with one side a 1, one side a 2, one side a 3, and SEVEN sides are 0.

If you want to know the best statistical "guess" over your next 20 games, roll this 10-sided die 20 times. Sum up the results.

Now, lets say you have 100 games logged. There are 100 sides on our new die. Lets say there are 17 sides for 1st, 13 sides for 2nd, 15 sides for 3rd. And 55 sides for loss.

This is a much better predictor, to roll this 100-sided die.

Now, picture a 1000-sided die. This is better still.


I think the relevance of predictions increases as the dice evolves closer and closer to a sphere.

Good/bad?

[bball904]

Limpin'

You absolutely nailed it.

[ QUOTE ]
This is a much better predictor, to roll this 100-sided die.


[/ QUOTE ]

You just came across the key word in this whole discussion: predictor. Much like we predict the Patriots will roll the Eagles, and the Mets will still suck, and Sammy Sosa will struggle to hit 30 bombs without his juice; that's all the more we can hope to do with sng results because we are talking about a subject matter of skill based performance. Has anyone ever seen a confidence interval on the number of yards Tom Brady will throw for in the Super Bowl?

[jcm4ccc]

[ QUOTE ]
HOW I THINK OF STATS:

I find i can grasp the significance of statistcal data/preditions more easily if i can make a mental picture.

Picture a multi-sided dice (die). There is one side for every result you have logged.

If you play 10 tourneys, and get 1 win, 1 second, 1 3rd, you will have a 10 sided die, with one side a 1, one side a 2, one side a 3, and SEVEN sides are 0.

If you want to know the best statistical "guess" over your next 20 games, roll this 10-sided die 20 times. Sum up the results.

Now, lets say you have 100 games logged. There are 100 sides on our new die. Lets say there are 17 sides for 1st, 13 sides for 2nd, 15 sides for 3rd. And 55 sides for loss.

This is a much better predictor, to roll this 100-sided die.

Now, picture a 1000-sided die. This is better still.


I think the relevance of predictions increases as the dice evolves closer and closer to a sphere.

Good/bad?

[/ QUOTE ]

I don't agree with this. Your 1000-sided dice might have a little more precision. For example, your 100-sided dice has 17 sides for 1st, 13 sides for 2nd, and 15 sides for 3rd. If your 1000 sided dice has 170 sides for first, 130 sides for 2nd, and 150 sides for 3rd, then there is no difference. If your 1000 sided dice has 172 sides for first, 135 sides for 2nd, and 148 sides for 3rd, then that dice is a little more precise. But not much.

It's not the size of your dice that matters. It's how often you throw it.

[jcm4ccc]

[ QUOTE ]
Limpin'

You absolutely nailed it.

[ QUOTE ]
This is a much better predictor, to roll this 100-sided die.


[/ QUOTE ]

You just came across the key word in this whole discussion: predictor. Much like we predict the Patriots will roll the Eagles, and the Mets will still suck, and Sammy Sosa will struggle to hit 30 bombs without his juice; that's all the more we can hope to do with sng results because we are talking about a subject matter of skill based performance. Has anyone ever seen a confidence interval on the number of yards Tom Brady will throw for in the Super Bowl?

[/ QUOTE ]

I don't agree that it is more difficult to predict something based on past experience just because there is a skill element involved. You wouldn't want to calculate a confidence interval on the number of yards Tom Brady will throw for in the Super Bowl because that is such a unique situation. But look at these stats for his throwing yardage by year:

2002: 3764 yards
2003: 3620 yards
2004: 3692 yards

You don't think you can make a meaningful prediction of his 2005 stats?

The only thing to worry about when skill is involved is if the skill level changes appreciably over time. But you can certainly make meaningful and fairly accurate predictions of future performance based on past performance.

[bball904]

[ QUOTE ]
You don't think you can make a meaningful prediction of his 2005 stats?


[/ QUOTE ]

You just made my entire point. Yes, I absolutely can make a meaningful prediction, but it is just that. Science is not involved in making that "prediction".

[MWC]

Interesting discussion. The use of confidence intervals to evaluate the uncertainty of your estimated average win rate (per SNG) is clearly valid, as AgentSq thoughtfully describes in a post that should win this week’s award for Best First 2+2 Post. There is no need for the distribution of finishes to be normally distributed. Indeed, as jcm4ccc shows empirically, the estimated average win rate is already converging nicely toward a normal distribution with only 25 SNGs.

In the example bball904 gives, there’s nothing wrong with the CI the poster provides. (That poster uses the t distribution instead of the z [standard normal] distribution, but as the sample size grows, the t distribution converges to the z distribution – for the sample sizes we’re talking about it’s safe to essentially ignore the distinction.) While bball904 is largely off base in arguing that the calculations are misguided, he’s right on target that they are often over-interpreted - let’s look closer at the example he quotes:

win rate: $30.67, sd=76.63, n=43

95% CI: 7.09 to 54.25
But if we calculate some other CIs:
98% CI: 2.41 to 58.93
99% CI: -0.86 to 62.20

So even at the $31/SNG win rate, we’re not 99% confident that the player is even a winning player. This confirms our intuitive feeling that results of 43 SNGs are insufficient to draw useful conclusions (like, the guy should quit his day job), because the CI is always going to be too wide.

OTOH, if we had a guy with half the win rate, the same SD, and 10 times as many SNGs, we’d be much more confident that he’s a winner –

win rate $15.33, SD=76.63, n=430:

95% CI: 8.06 to 22.59
98% CI: 6.70 to 23.96
99% CI: 5.77 to 24.89

There are challenges, of course. The die-rolling examples of stillnotking and BigLimpin are useful analogies, but the SNG situation is more variable, in all the ways AleoMagus describes. But most of the ways the SNG situation departs from the ideal of “each SNG is an independent trial from the [static] population of all SNGs you might play” come out in the wash. Some of your past SNGs have been 4-tabling while drunk on Monday mornings with 12,000 players on Party and some have been sober, 1-tabling, on Saturday nights with 65,000. But that’s probably the distribution you’ll have in the future, too. The SNG situation is much more like the die-rolling scenario than the # of Cubs wins or # of Tom Brady passing yards situations are. (Why this is true is left as an exercise for the reader. [img]/images/graemlins/smile.gif[/img] ) And the information is much more useful, too.

There’s not much useful application to knowing the CI for Tom Brady’s long-run average SuperBowl yardage. For one thing, you don’t have a stake in it, and for another, Brady’s not going to play in enough more Super Bowls that you could profit from it if you did.

But it is useful to know have an idea of the CI for your SNG avg. win – the confidence interval doesn’t directly answer specific yes/no questions you might have, like, am I a winning player, or should I move up a level, but it does give some insight. And if the player in the 2nd example above (+$15/SNG) wants to ask himself, "Should I spend 40 hours/week over the next month to play 1000 SNGs or do something else with that time?" then it’s relevant for him to know that he can expect to be up somewhere between $6000 and $25000 ahead at the end, assuming he’s playing in the same general conditions as his last 430.

[Big Limpin']

[ QUOTE ]
It's not the size of your dice that matters. It's how often you throw it.

[/ QUOTE ]

I beg to differ. Its not that the bigger dice has more significant figures, its that the sample size to CONSTRUCT the dice is bigger, and therefore is more reflective of true probabilities.

For dramatic effect, say you play 2 games, and make a 2 sided dice (coin). It doesnt matter how many times you flip it.

All im saying is that as the dice evolves into a sphere, it likewwise evolves to a better predictor.

Theres a chance i could "make" (read: sample) a 100-sided die that has 35 first place finishes on it. Doesnt mean i am gonna finish 1st 35 times out of the next 100.

However, a 10000 sider is going to have to be damn close to true likelyhood.

[bball904]

[ QUOTE ]
The use of confidence intervals to evaluate the uncertainty of your estimated average win rate (per SNG) is clearly valid, as AgentSq thoughtfully describes in a post that should win this week’s award for Best First 2+2 Post

[/ QUOTE ]

I agree the AgentSq has provided the most enlightening and most spot on correct post of this entire thread. I second his nomination. However, you didn't state his complete message. The CI's you can construct around your estimated win rate are really only valid for past results. The unresolved issue is that the uncontrolled conditions of all your future experiences are not relative to your past experiences for any confidence intervals you can construct.

[ QUOTE ]
jcm4ccc shows empirically, the estimated average win rate is already converging nicely toward a normal distribution with only 25 SNGs.


[/ QUOTE ]

I agree that jcm4ccc proved that sng results can be grouped so that they will effectly approximate a normal distribution over a large number of data sets. However, he showed this using 30,000 x 25 = 750,000 sng's. That's a few more than 25.

[Marcotte]

[ QUOTE ]
There is something called the central limit theorem that is terribly important in most stats but is generally brushed over. What it says is that the distribution of an average is normal irrespective of the distribution it is drawn from.

This is a fairly strange and counter intuitive result but is true.

Applied to SNG's what this means is that the observed average $ rate per game is an estimate for your true average rate. And what's more, this estimate is normally distributed having a mean of the observed average and a standard deviation of the standard deviation of the observed results divided by the square root of the number of obs.

[/ QUOTE ]


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That is a post going over my head. Anyone have the appropriate animated smiley? [img]/images/graemlins/laugh.gif[/img]

[Strollen]

I am 99% percent confident that average math SAT on these forums is at least one standard deviation above the average. [img]/images/graemlins/smirk.gif[/img] and equally confident that mine (720 taken 20+ years ago) is not.

Very interesting discussion.