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Originally Posted by DJAbacus
I realise this, but you are saying that a sample of 500 hands is too small with regards to an accurate VPIP/PFR read.
VPIP/PFR stats of 13/9 of course doesn't mean that a Villain raises the top 9% of hands, this only means that a Villain raises 9% of the time and we assume that this is the top 9% of hands. It is highly likely that these 9% of hands are strong hands.
Obv. we assume villain has the top 9%, because if we're wrong, then we have even more equity than we had estimated.
This is completely beside the point.
The math says that 9% is only as "accurate" as ~6.8% to ~11.8% @95% CI - which is a pretty wide difference in poker ranges.
6.8% = { 88+,ATs+,KJs+,AQo+ }
9.2% = { 88+,ATs+,KTs+,QJs,AJo+,KQo }
11.5% = { 77+,A9s+,KTs+,QTs+,JTs,ATo+,KQo }
and that's just at 95% CI... which means you expect to be wrong 1 out of 20 times you assume the range is this tight.
Do you see what I mean by grain of salt? Yes, the odds of Villain's actual range being 11% is less likely than 10%, but both are still well within the realm of possibility that is predicted by the math.
The tighter the bounds you set, the lower the CI, and the higher the CI, the wider the bounds.
When you say, "The stat says 9%, so the range is 9%." then you have a 0% CI. (Well, you have very low CI, since we're talking discreet math).
If you want a 100% CI, then it always extends from 0% to 100% (non-inclusive).
These are not helpful.
For any meaning from the stats, you have to allow for uncertainty. No, not uncertainty. It's certainty. When we choose a 95% confidence interval, we are certain that our error bars will be too small 5% of the time. That's the point. We accept that there will be error, then we quantify how much we are willing to accept. Once we know that, we cut off the appropriate amount of the "tails" of the distribution.
The variance is the same, no matter where we cut the tails, but the level of confidence we have in the results will choose what parts of the distribution count as the "tails."
We can't get less than 100% CI without cutting off the tails. When we cut off the tail, we allow/expect to have results that fall outside our confidence interval. It is by choosing a good confidence interval that we make practical use of a statistic.
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