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Variance: Part Two

by Steve Zolotow |  Published: Apr 22, 2020


The two most popular statistical tools in use are the mean and standard deviation. They are the most common way in which large masses of data are summarized, but neither was developed with poker players in mind.

The mean is commonly called the average, and it is calculated by adding up all the observations and dividing that by the number of observations. Many natural phenomena follow a pattern referred to a normal distribution. The normal distribution gives rise to a ‘bell shaped’ curve. The center and highest point of a normal curve is the mean. The rest of the curve is symmetrical around the mean. (In a true normal distribution both the mode (most common observation) and the median (most frequent observation) are the same as the mean.

The further an observation is from the average, the less likely it is to occur. Variance measures how far each observation is from the mean. Standard deviation measures variance, but it is ‘standardized’ so that the same percentage of observations occur within any number of standard deviations.

Here is an internet example:

Notice that 64 percent of the observations are within one standard deviation of the mean. Virtually all of them, 99.5 percent are within the area from -3 to +3 standard deviations. Statisticians have developed some wonderful tools for analyzing data that follows this pattern.

Unfortunately, poker results don’t follow anything close to a normal distribution. I don’t care if you track your results by hand, by hour, by session, or by tournament. Look at the results for a typical tournament player. You will see a huge number of small losses representing buy-ins that didn’t result in cashes, a moderate number of small cashes, and a few big wins.

If you average winning $50 in a 220-entry fee tournament, no observations will occur at the mean. There is huge variance. Even a cash game player’s results will seldom resemble a normal distribution. Yet over and over, I see poker results analyzed using tools designed for data that is normally distributed. (One of the causes of the last financial crisis was using models that treated credit defaults as normally distributed, when they weren’t anything close to that.)

Every poker session is different. In a cash game with three weak players, the hero has a great win rate. With one weak player, his win rate is small. With none, he loses the rake.

This points up the importance of finding good games (with a lot of bad players.) It also highlights how futile it is to use statistical tools, designed for analyzing normal data, to analyze a series of cash game wins and losses.

There are some online variance calculators or analyzers that purport to simulate your best, worst, and most likely results over any number of hands, given a few simple inputs. These include your win rate in big blinds per 100 hands and the standard deviation per hand. Since win rates aren’t as regular as we’d like to believe and standard deviations don’t measure poker results, the output of these calculators is suspect. It might give you some very rough idea of your risk of ruin in a specific game if you really know your win rate over a large sample of hands. (Online players who play a high volume under unchanging conditions might end up with somewhat meaningful results.)

I have found players often start tracking results when they are on a good streak, which obviously distorts their win rates. There are also a lot of players who begin to play significantly worse when they are on a losing streak. Over time, the average ability of players is improving. You must keep improving to maintain the same win rate.

I have written several columns about bankrolls, and I don’t want to start rehashing this material. One size fits all bankroll management rules make no sense. A young single man with an income producing profession needs a lot smaller bankroll than a full-time pro with a family and no outside income. I will just state, you almost certainly need a bigger bankroll than you think.

Reducing variance is a popular goal. I frequently read or hear discussions of the importance minimizing variance of results on any given hand or session. In the next column, I will debunk this misguided notion. Takeaways from this column:

Don’t use tools that treat data as being normally distributed unless you are very sure your data is at least approximately normal.

One of the most important things you can do is play in good games.

Play smaller or build a bigger bankroll if you want to avoid tapping out. ♠

Steve ZolotowSteve ‘Zee’ Zolotow, aka The Bald Eagle, is a successful gamesplayer. He has been a full-time gambler for over 35 years. With two WSOP bracelets and few million in tournament cashes, he is easing into retirement. He currently devotes most of his time to poker. He can be found at some major tournaments and playing in cash games in Vegas. When escaping from poker, he hangs out in his bars on Avenue A in New York City -The Library near Houston and Doc Holliday’s on 9th St. are his favorites.