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How Accurate Are Your Statistics?

Livetracker
Poker Office 5's Live Tracker provides users with plenty of statistics

Poker trackers, such as Poker Office 5 and Poker Tracker, present the user with a plethora of statistics based on players' historical actions. These include Saw Flop (SF), Voluntary Put Money In Pot (VPIP) and Flop-Turn (FT).

These statistics can be useful for identifying weaker players. For example, weak players are often too loose, and so the SF and VPIP statistics can be used to identify them.

However, how confident can we be of an opponent's future performance matching their historical behaviour? And if we assume that an opponent's current playing style will match their past, how accurate will the statistics presented by your Poker tracker be?

As an example, consider player A, for whom you have 50 hands of data. They have seen the flop in 25 of these hands, so their SF statistic is 50%. However, did they play 50% of their hands because they hand a good run of cards? Or did they play 50% of hands because they are loose?

We can estimate the accuracy of these sample percentages using Binomial Proportion Confidence Iintervals, which are justified by the central limit theorem. These will give us bounds for where the true value is likely to lie.

The best estimator for a population proportion is:

p = (number of occurances) / (number of samples)

The 95% confidence interval is given by

p ± 1.96 x sqrt(p x (1 - p) / n )

The number of samples is denoted by n, which in our case is the number of hands for player A. This estimate will tend to hold when n x p > 5. However, it will totally fail when p=0.0 or p=1.0.

So for our example, the bounds are:

0.5 ± 1.96 x sqrt((0.5 x (1 - 0.5) / 50 )

= 0.5 ± 0.14

= (0.36 , 0.64)

This means that we can be 95% certain that the true proportion of hands that player A sees to the flop is between 36% and 64%. 36 % is only marginally above the 30% level that is commonly used to characterise loose players. While 64% is very weak.

So while we can categorise the player as being weak, the margin of error is still pretty high. If we had 100 hands of data, then the 95% confidence intervals would be 40% and 60%, which tightens things up slightly.

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