Hi,
Variance is a mathematical quality that measures how much a certain number is expected to vary.
In poker, when you have a low variance play this means that you take a line where you don't risk losing a lot of money.
When you take a high variance play you take a line where generally you may either lose a lot or win a lot.
Variance in StoxEV is a measure for the spread in your EV when you win money in your line (and how much) compared to how much and how often you lose.
Example:
You have 76s in the cutoff and raise to $3.
The button 3bets to $9.
You try to decide whether to 4bet to $25 or fold.
When you 4bet you expect the button to fold in 65% of the cases and push all in in 35% of the cases.
In this example 4betting is a high variance play and folding is a low variance play. When you fold you know exactly what you'll get, you'll have an EV of $0 in 100% of the cases.
The EV and Variance of folding:
EV=$0. Variance=$0.
The EV and Variance of 4betting (ignoring rake):
EV=65%*(3+9+1.5)-35%*22=$1.075
Variance:
When you win you gain $13.5.
When you lose you lose $22.
Some math shows that your variance here is $16.5.
As you can see, variance is a measure here for the spread around the $1.075 average outcome.
Footnote 1:
The formula for variance is
Footnote 2:
Variance in StoxEV is actually standard deviation, which is the square root of variance. Its formula is
.
I'm just calling it variance because in my experience, when people talk about variance they actually mean standard deviation. Second reason is that in monte carlo mode I already use standard deviation as a measure of error. I thought it would be better to use variance as a term to describe the spread to avoid confusion.
The formula for Standard deviation is