Why would variance be higher at 6max?

[Aaron W.]

[ QUOTE ]
I have read it here repeatedly that your variance will be higher at 6max tables than at full ring. Why would that be? It seems to me that at short handed games, you will win more pots per hour, and that the average pot size will be smaller, so I would think this would lead to lower variance.

My standard deviation based on my pokertracker data is 15/100 hands at 6max, and 21 at full ring. Is this common for everyone else? Doesn't this mean lower variance?

Imagine player at a poker table with 100 people, if this were possible. Would not your variance be much higher? During most hours you wouldn't even win a single hand, but when you won a big hand, it would be enormous. If you were a winning player there at 3BB/100, you might have a standard deviation of 100/100 hands.

Am I missing something here?

[/ QUOTE ]

Okay, I'm going to bite on the math thing.

Variance is the sum of the squared distance to the mean divided by the number of samples. So let's say you play 100 hands, folding 99 of them (no blinds... since we're playing imaginary poker) and winning 1. To make the numbers match, you need to win 300 BB in the one hand to have a 3 BB/100 winrate.

Then your variance is going to be...

[(0-3)^2 + (0-3)^2 + ... 97 more of these ... + (100-3)^2)/100 = (891+9409)/100 = 103.

Your standard deviation is the square root of variance, which is
10.1 or about .1 BB/100. So unless I'm mistaken in the calculation, your variance and are incredibly *TINY* in this situation.

I don't know how strongly I believe that variance is bigger in 6-max.

There are two factors going into the variance:
1) The size of the pots that you win matter. In a full ring game at these limits, the pot is something around 12 BB on average. So when you win a pot, you're winning about 9 BB over your average. In shorthanded games, the pot is smaller, say 9 BB, which is only 6 BB over your average. This means that the variance for full ring games might tend to be bigger than shorthanded.
2) However, when you play full ring games, you spend a lot more time folding, or investing just 1 SB into the pot to see a flop with a marginal hand, which means you are very consistent in the amount you put in. In shorthanded games, you play more pots, so you don't "hover" around zero as much, and so your pot size statistics are more spread out with wins and losses. This contributes to a larger shorthanded variance and smaller full ring variance.

#2 is what people are usually talking about when they say variance is larger at 6-max. But it's not necessarily a mathematical truth that variance is actually larger. You're not allowed to just sit out most of the hands and wait for a monster pot to collect -- you've got to get involved in hands, and so you see more ups and downs.

The truth is that it just doesn't matter at all. Go play poker and let the statistics do their own thing.

[BatsShadow]

My thought is that while you do have many more marginal decisions, that means that against the truly horrible player, you have that many more chances to make good decisions and your variance should go down while your win rate goes up. -- I guess that means that your decisions aren't really that marginal.

If your opponents are better, then your marginal decisions become more marginal and variance goes up.

[Bankuri]

But variance doesn't have much to do with making good or bad decisions. Variance is the result of random processes in the game itself. If you only play hands that are 60-80% sure to win, your variance will be much smaller than if you are playing hands that win only 30% of the time. You aren't making poor decisions to play the latter if you have 2 or more opponents playing the hands with you, but your variance will be higher.

[Shillx]

In the example you give...

Figures are in BB/hand which would make the mean .03 BB/hand

s = root ((.03*.03*99 + 2.97*2.97)/99) = ~ .3 BB/100

Now let's say that you win 5 pots/100 @ 15 BB each and lose 9 pots/100 @ 8 BB each. The other 85 pots you don't factor into. Again your EV is .03 BB/hand or 3 BB/100.

s = (.03*.03*85 + 14.97*14.97*5 + 7.97*7.97*9)/99) = ~ 4.12 BB/100

Now try it when you win 15 pots/100 @ 15 BB each and lose 20 pots/100 @ 11.1 BB each. Don't factor in the folded hands. Again, our hero's win rate is .03 BB/hand or 3 BB/100.

s = root ((14.97^2*15 + 11.07^2*20)/99) = ~ 7.66 BB/100

The thing is that 6 max pots tend to be about the same size as full ring pots. You just win about twice as many in the same 100 hand peroid (and you lose many more bets in this span as well). This really shoots the SD through the roof. They don't calculate it on a per hand basis though. Say that after 500 hands, your +/- per 100 goes as follows...

+30 BB, -9 BB, +17 BB, -42 BB, +19 BB

mean = + 3 BB/100
s = root 2738/4 = 26.2 BB/100
sigma = root 2738/5 = 23.4 BB/100

This is why you need a biggish sample to get a SD figure from PT. Since you have to divide by (n-1) it takes time to get a somewhat accurate figure.

Brad