[AaronBrown]

Without disputing your argument, the rationale for standard deviation is generally based on the Central Limit Theorem rather than the symmetry of the underlying distribution. If you add up enough independent observations, whether Poker hand outcomes or daily portfolio returns, only the mean and variance matter. The shape of the distribution washes out.

Say your successful player with the positively skewed distribution loses $10 on nine hands in ten and wins $200 on the tenth hand. The unsuccessful one has the opposite pattern, making $10 on nine hands in ten and losing $200 on the tenth hand. All hands are independent. The successful one makes $10 per hand, the unsuccessful loses $10, but both have the same $63 standard deviation.

After 1,000 hands, the distribution of total profit and loss is almost indistinguishable from Normal in both cases. Average profit per hand will be plus or minus $11 with a standard deviation of $2. The probability that the successful player is between $9 and $11 is 34.7%, versus a Normal approximation of 34.1%. The probability that the unsuccessful player is one standard deviation below his mean (-$13 to -$11) is 33.6%. The probability that the successful player is between $7 and $9 is 14.1%, versus 13.6% in a Normal, and also 13.6% for the unsuccessful player being two standard deviations below his mean ($15 to -$13). So knowing the standard deviation per hand, and the mean, tells you almost all you need to know about the long term distribution of outcome.

I think the more important issue is independence. Poker hands are not independent, neither are hedge fund returns. Even small deviations can skew these calculations.