What Does It Mean in Poker? Variance is the difference between expected and actual results. Also known as luck. For example, AA is an 80 percent favorite over KK preflop, but you could lose this battle with AA two or three (or more) times in a row, due to variance.
In reality, if you aren’t ready to deal with the variance of playing poker over a long stretch of time, then it just isn’t the game for you. Also just to note, variance is the root cause of all bad beat stories around the world, so let’s face it, it has a lot to answer for!

Casino Games Online › Card Games › Poker › Poker Variance Explained Using a Simple Example

A simple and illustrative example of variance calculation and its practical meaning and utilization. We will calculate expected profitability (in a way that can be applied to operation of any game or to the game of poker) and variance as possible deviations from the expected profit. After that you will be e.g. able to answer the question 'How many hands do I have to play not to lose a dollar with the 99% likelihood?'

What Is Variance

While variance is technically a statistical term to measure the spread of a sample from its expected value, the term is more broadly used in poker to mean fluctuations in luck and the corresponding effect on a player's bankroll. Even more generally, poker players often just refer to extreme good luck or bad luck as 'variance.'

Variance is a statistical measure that shows how the short-term earnings deviate, or may deviate, from the expected long-term average profitability. The more hands are played by a player, the more his profitability (measured e.g. by ROI – Return of Investment) gets crystalized.

Hence the poker player may expect some rate of profitability with some sort of likelihood. There may be a period of bad luck, when his profitability is lower than expected (that is lower than his long-term average). The phenomenon is called the downswing. However he may also be lucky and earn more than the expected average. In this case we talk about the upswing.

The frontiers between which the player's profit moves are called the variance (VAR). The basic information on variance, but also on the way how the good or bad luck may be measured, you can find on the page Variance & Dollar EV Adjusted. This article will show the step-by-step calculation of variance and give you a better understanding of its meaning and use in practice. If this is a new topic to you, then you will be certainly enriched.

The Example or How to Get the Variance Right

Imagine that we took a simple coin and played the following two simple games with it.

Game #1: I will flip the coin. If the 'heads' comes out, I will give you $1, and if the 'tails' comes out, you will give me $3. At first sight it is obvious that it is a good business for me, but we will get to the point later on.

Game #2: It is coin flipping again, but this time, to make it more interesting, I will have to toss the tails three times in a row. If I manage to do so, you will give me $15, and if I fail, I will give you $1 for each unsuccessful attempt (that is when the heads comes out straight away or after one tails or after two tails in a row).

1. Calculation of Expected Profit or Value (EV)

As for the Game #2 we can see that it is not so easy to determine by eye-balling whether the game is advantageous or not. The decision making, which of the games is better from the long-term prospective, will be aided by the Expected Value (EV), which is a sort of an estimate of a mean profitability of both games (a long-term average). To arrive at the EV we also need to determine the probabilities of winning and losing in both games. The probabilities serve as the weights for possible dollar profits or losses.

Very briefly: the odds of winning and losing in the first game are 'Fifty-Fifty', thus the probabilities are equally (1/2) = 0.5. As for the 2nd game, I need to toss the tails 3 times in a row. The probability of managing so is 0.5 × 0.5 × 0.5 = 0.125 (or 12.5%, if you like). The probability of losing is the remainder from one (100%), thus 0.875 (or 87.5%). Now we have everything we need to calculate the expected profit (EV) of both games:

Game #1: EV = 0.5 × $3 + 0.5 × (−$1) = $1.5 − $0.5 = $1.

Game #2: EV = 0.125 × $15 + 0.875 × (−$1) = $1.875 − $0.875 = $1.

One more comment on the calculations above: the probability of winning times the profit + the probability of losing times the loss.

Several things arise from the calculations. First, the expected value is positive (+EV), hence both games are advantageous for me (or their operator) in the long run. If I found somebody willing to play them with me, then I would earn in average a dollar out of each toss. (Please note that in terms of the Game #2 we can talk about one attempt or one round that consist in the successful toss of the three tails in a row and in any other failure to do so.)

2. Variance Computation (VAR)

It was the intention of this example that the expected value or profit (EV) of both games was $1. Although we can feel that the Game #1 is somehow more stable and that the deviations in earnings in the Game #2 are more likely. That is what variance (VAR) is all about. It shows the dispersion of the values around the expected value (EV) on both the negative and the positive side. Let us compute the variance. Again, it is not a too big deal, perhaps the explanatory comments below will be helpful.

Game #1: VAR = 0.5 × ($3 − $1)² + 0.5 × (−$1 − $1)² = $4.

Game #2: VAR = 0.125 × ($15 − $1)² + 0.875 × (−$1 − $1)² = $28.

Let us stop at the variance calculations right away. Why they are calculated like that and what do they mean? First we deduct the expected value (EV) from each individual value—that is we deduct the expected profit from all possible profits and/or losses (we have got only one profit and one loss). It shows how much they deviate from the mean value (EV).

As for the Game #1 those are the expressions ($3 − $1) a (−$1 − $1). Then we square the expressions (it will be explained why): ($3 − $1)² a (−$1 − $1)². And finally we multiply the squared expressions by their probabilities (again they serve as weights—in the first game they are equal and thus could be omitted, but in the 2nd game the weights play their part). Now why have we squared them? Simply to prevent the positive and the negative values from eliminating each other. Look what would happen if the differences between the values and the expected value were not squared:
VAR = 0.5 × ($3 − $1) + 0.5 × (−$1 − $1) = 0.5 × $2 + 0.5 × (−$2) = $1 − $1 = $0.
It would suggest that the variance (or the dispersion around the mean) is null, which is not true.

3. Determination of Standard Deviation (SD)

The deviation from the mean value (EV) is determined by the statistical measure known as Standard Deviation (SD). The standard deviation (SD) is nothing else than a square root of the variance (VAR). Can you notice the connection with the previous chapter? First of all, all differences from EV were squared, hence the negative values turned into the positive values. Now we take the square root of the sum of all these (positive) values. By that we arrive at the standard deviation around the EV on both sides (+/−).

Game #1: SD = square root of 4$ = $2.

Game #2: SD = square root of 28$ = $5.29.

Practical Use of Variance

How can we utilize the general knowledge and specific calculations of the expected profit (EV), variance (VAR) and standard deviation (SD) from the mean value? First let us have a small recap so that we can come back to it if needed.

Table 1 – Summary of the Game #1

Table 2 – Summary of the Game #2

Let Us Play 100 Rounds and See What Happens

Imagine that we want to play 100 tosses (rounds or attempts). If we expect to earn a dollar in average on each round of the game, then after 100 rounds we are supposed to earn 100 dollars in average. 'We are supposed' does not mean that we will earn $100. We can earn more or less and as we will see 100 rounds is not large enough number for our expected earnings to take full effect.

Game #1:

EV = 1$ × 100 rounds = $100 VAR = $4 × 100 rounds = $400 SD = square root of $400 = $20

Note: SD can also be calculated as the square ro

Poker Variance Meaning