Variance uncovered

The term variance gets bandied around flippantly in poker circles, with people using it as an excuse to play badly and blame it on their losing streaks. It often has negative connotations, being connected to players going bust. But when you really understand variance, you should welcome it with open arms, embrace it and recognise what it means to you specifically – a student of poker.

Variance doesn’t affect everyone in the same way. Odd you might say, given that it is purely a mathematical measurement used for analysing data sets. But with regard to poker, the environment must be taken into account. Winning at poker is about the application and reapplication of advantages over a long period of time. If you have a positive expectation versus your opponents every time you sit down, over the long run you will win. Simple as that.

Variance is important because it directly affects your results. It controls your profits and losses. A bad player might get all his money in as a 40% underdog and ‘suck out’ a couple of times. His strong opponent knows that he got his money in as a favourite, but with a sample size of two he can’t complain.

The important thing is to keep getting your money in as the favourite, regardless of how ‘good’ or ‘bad’ you are running. On the opposite page are two graphs (taken from www.tworags.com) showing how poker profit is not earned in a linear fashion. Short-term luck means that actual results can differ significantly from expectation.

ODDS WILL PREVAIL

The difference a lot of players have is their approach to variance. A lot of players play very well for long periods, lose a hand where they are a big favourite and begin to lose track of the decision process that gave them the positive expectation in the first place.

For example, someone playing a lot of high-stakes Omaha could well get into five or six big pots during the day where he was a 60% favourite and lose them all. After becoming disillusioned and losing heavily, he might try to break even, thinking that it’s fine to get his money in as the 40% underdog on a few occasions, as he’s obviously running bad and somehow ‘owed’ some good fortune. Wrong. Poker has no memory of how well we run, how much we lose and how badly we get sucked out on.

Each result is an individual event, made up of specific probabilities getting a result dispersed around the mean. This is variance, and allows bad players to keep playing badly and strong players to keep pressing home strong statistical advantages. You can’t expect to win every day, every week or every month.

Poker money does not move in such a simple way – results deviate from the norm, and luck can prevail. Bad luck can destroy not only players that play too high for their bankroll, but an average player’s game mentally. Losing is part of winning in poker. Good players know how their edge makes them money.

HOW TO CALCULATE VARIANCE

According to Mike Caro, variance is: ‘A measure of the spread of statistical distribution about its mean or centre. With respect to poker, the distribution of your results over a set of hands or sessions, or the swings in a positive or negative direction of cash flow. The greater the variance, the wilder the swings; the lower the variance, the more likely a given session’s results will be close to one’s average result.’

So, variance is a measure of the distribution of a data set around the mean or average.

Variance in poker is best expressed as a measure of distribution around the win rate. So here’s a simple example showing how to obtain your own variance. Let’s say you play five sessions of 100 hands of $100 no-limit hold’em and your results are:

1 $10 2-$6 3$14 4-$12 5$9

So your win rate is $3 per 100 hands, i.e. (10 – 6 + 14 – 12 + 9)/ 5 = 3. Now, to figure out the variance, we look at each session and how far away from the average win rate that result is:

1 $7 2-$9 3$11 4-$15 5$4

To determine the variance we square these values, add them up and divide by the number of sessions (five). This gives us a variance of $98.4/100. To get the standard deviation (the value used by statisticians), we find the square root of the variance. In our example it’s $9.9/100.

STYLE ISSUES

From looking at software programs like Poker Tracker, we can see how the standard deviation differs from game to game. The average no-limit hold’em player has a standard deviation of 35 big blinds/100 or thereabouts. The average limit player has a standard deviation of 15 big blinds/100.

So no-limit hold’em has greater variance than limit hold’em. Tell us something we don’t know, I hear you cry! Well, it might surprise you to hear that the style of no-limit hold’em you play can have dramatic effects on your variance.

Good, tight-aggressive players enjoy a relatively stable income, with standard deviation being a lot lower than their loose-aggressive counterparts. Because they always have solid hand values, their achieved results are a lot nearer the mean. Going to a showdown regularly with strong cards helps here.

On the other hand, their loose-aggressive friends must put up with much bigger swings, a fact of life if you are playing good, loose-aggressive poker. But often, loose-aggressive players have a better win rate in the long term.

There is something of a trade-off between variance and win rate. Playing ABC solid poker can grind you out a respectable, reliable income at a lot of levels online – very useful if you are operating on a shortish roll of say, 15 buy-ins or so.

Very good loose- aggressive opponents can enjoy far greater win rates, but they have to stomach the occasional heavy downswing that accompanies this style of play. If you are adequately bankrolled this makes little difference to your overall earning capacity of course, but it is something you should always bear in mind.

The stakes you play obviously have a huge effect on your variance. Compare player A, playing $10/$20 hold’em with 20 buy-ins, with player B, playing $1/$2 hold’em with 100 buy-ins. The difference in variance can be quite brutal. Player A is playing high-stakes poker in a game where even the best players only have a small edge on their opponents.

He is likely playing on one or two tables to maintain focus on betting patterns, timing tells and complex levels of thinking. If he drops five buy-ins in a week he has just lost 25% of his roll.

This affects him financially (he probably will know he’s playing too short for that game now) but could also affect him mentally. He could start to play scared poker, lack heart and conviction when the pots get big or pick up some other disadvantage.

Player B has the luxury of playing in a game where he knows he has a big edge over the long-term, can afford to play numerous tables at a time to further reduce his variance and knows that dropping five buy- ins won’t hurt him in the least. By being over-rolled, he can relax and play the game.

This can be a bad thing for some players too though. I know many players who can’t play their A-game when the money at stake doesn’t hurt.

RESULTS MAY VARY

The fact remains however, that if both players have the same expectation, let’s say a 15% edge on their foes, player B is going to have a much better ride with regard to variance. The losing streaks won’t hurt as much and the increase in hand volume will help to force achieved results to conform more closely to expected results.

Another thing to keep in mind is that certain types of poker have an even higher variance. Short-handed hold’em has a slightly higher variance due to the fact that you will be involved in a greater number of pots per orbit and will be relying less on good cards.

The 60/40 match-ups mentioned earlier are very familiar to those who play Omaha, where often the money going in belongs to the big made hand and the big draw. Limit hold’em has the lowest standard deviation of the staple games, while dealer’s choice games can give huge fluctuations in results.

Take six- card Omaha high/low for instance. In this game you not only have to be highly skilled in pot odds, but you also have to understand that you are rarely going to get your money in as a big favourite.

So hopefully you now have a slightly better understanding of what variance actually is. By being adequately bankrolled for your game, you should be able to make variance work with you rather than constantly battling it.

Playing high-stakes Omaha on a short roll is not embracing variance – it’s pitting you against it. Remember, playing good poker is about knowing the game inside and out and never letting ego get in the way. Variance is a key part of the game, so get to grips with it and your game is sure to improve.