[pzhon]

People mean different things by variance. In all cases, variance should mean some sort of deviation from the average, but some people incorrectly blame the effects of a low or negative win rate (poor skills) on luck.

Confusing the issue is that it's not clear which game to compare. Which level of NL is similar to $2-$4 limit? NL $200 has the same size big blind, but no serious poker player thinks that is the analogous game. $2-$4 limit generates smaller pots than NL games of with the same big blind. It might be analogous to something like NL $50. Which game you feel is analogous might influence which variant of poker has more variance.

To a mathematician, variance is defined to be the average of the square of the difference from the average. This average can be taken per hand, per 100 hands (not different from per hand), or per hour (possibly different). You might face different numbers of hands per hour when you play NL than limit, as NL games are usually a bit slower as people size bets and determine pot odds, and decide whether to commit their stacks. PokerTracker can report the standard deviation, usually expressed in PTBB (2 big blinds, even in NL) per 100 hands. For limit, this is usually about 15 BB/100 for full ring, about 17 BB/100 for 6-max, with slight variations due to playing style and game conditions. For NL with a 100 big blind stack, this is often 40-45 PTBB/100 (80-90% of a 100 big blind buy-in every 100 hands) in full ring, with larger differences due to playing style and game conditions. Again, that doesn't mean NL has more variance. It could mean that it has less, if the analogous game has a big blind which is smaller than 1/3 of the limit game.

Many people mean downswings (both magnitude and length of time/hands) when they talk about variance. Downswings are influenced not only by the mathematical variance, but also your win rate. A break-even player will see arbitrarily large swings in both directions. A marginal winner will see larger downswings than a solid winner, since at the point when a solid winner has come back from a downswing, the marginal winner will still be in the red. Small stakes NL generally allows experts to have a mcuh greater win rate in relation to the variance than small stakes limit does. This means downswings tend to be shorter in NL. One reasonable statistic to use is the standard deviation/100 hands divided by the win rate/100 hands. If you square this, you get the number of 100 hand sets it takes for breaking even to be 1 standard deviation away from average. Multiply by n^2 for breaking even to be n standard deviations away from par.

I call the long run the time it takes for breaking even to be 2 standard deviations away from average. For a solid winner in limit with a win rate of 2 BB/100 and a standard deviation of 15 BB/100, the long run is (15/2)^2 * 4 = 225 sets of 100 hands, 22,500 hands. For a solid winner in NL with a win rate of 8 PTBB/100 and a standard deviation of 40 PTBB/100, the long run is (40/8)^2 * 4 = 100 sets of 100 hands, 10,000 hands. In this sense, the solid NL player reaches the long run faster. If you play one variant significantly better than the other, you might reach the long run faster in your preferred variant, regardless of what happens to other players.

Some people focus on the standard error (the standard deviation of a measurement) of their win rates. This does not depend on the average win rate itself, but it causes different descriptions of the variance when you play more hands. Playing more hands means the standard deviation of your total winnings increases, but the standard deviation of your win rate decreases, as you get a more accurate assessment of your win rate with more data.

Neither the standard deviation per 100 hands nor the size of downswings is affected by multitabling, as long as this does not change your win rate. However, the standard deviation per hour will increase, and the standard error of your win rate after an hour will decrease.

These differences in definitions, as well as differing assumptions on specific statistics, are how reasonable, informed people can disagree about variance. However, unreasonable people find many illogical ways to disagree.