Standard deviation for slot machines can vary from one machine to another. 3.14 was the standard deviation for just one actual machine studied. The house edge is defined as the ratio of the average loss to the initial bet. The house edge is not the ratio of money lost to total money wagered. In some games the beginning wager is not necessarily the ending wager.
For example in blackjack, let it ride, and Caribbean stud poker, the player may increase their bet when the odds favor doing so. In these cases the additional money wagered is not figured into the denominator for the purpose of determining the house edge, thus increasing the measure of risk.
The reason that the house edge is relative to the original wager, not the average wager, is that it makes it easier for the player to estimate how much they will lose. For example if a player knows the house edge in blackjack is 0.6% he can assume that for every $10 wager original wager he makes he will lose 6 cents on the average. Most players are not going to know how much their average wager will be in games like blackjack relative to the original wager, thus any statistic based on the average wager would be difficult to apply to real life questions.
The conventional definition can be helpful for players determine how much it will cost them to play, given the information they already know. However the statistic is very biased as a measure of risk. In Caribbean stud poker, for example, the house edge is 5.22%, which is close to that of double zero roulette at 5.26%. However the ratio of average money lost to average money wagered in carribean stud is only 2.56%.
The player only looking at the house edge may be indifferent between roulette and Caribbean stud poker, based only the house edge. If one wants to compare one game against another I believe it is better to look at the ratio of money lost to money wagered, which would show Caribbean stud poker to be a much better gamble than roulette.
Many other sources do not count ties in the house edge calculation, especially for the don’t pass bet in craps and the banker and player bets in baccarat. The rationale is that if a bet isn’t resolved then it should be ignorred. I personally opt to include ties although I respect the other way defintion.
Element of Risk For purposes of comparing one game to another I would like to propose a different measurement of risk, which I call the “element of risk.” This measurement is defined as the average loss divided by total money bet. For bets in which the initial bet is always the final bet there would be no difference between this statistic and the house edge.