## Standard Deviation in Blackjack

As has been discussed many times, blackjack has one of the lowest house edges in the casino and by using perfect blackjack basic strategy, you can lower the house edge to 0.5%. That means that in the long run, you can expect the house to take 0.5% of your money. But that’s not always what happens, is it? Sometimes you win more and sometimes you lose more. Anything that deviates from the average of a 0.5% house edge is called “variance” and you calculate variance using “standard deviation.”

If you think back to math class, you’ll remember that probability can be charted with a bell curve. On that curve, the middle amount is the average. Using that bell curve, you can show the normal distribution, which is the probability of possible outcomes. With that probability bell curve, standard deviation is the number of units to the left and right of the average. 68% of outcomes will fall within 1 standard deviation of the average, 95% of outcomes fall within 2 standard deviations and 99.7% fall within 3 standard deviations.

By crunching the numbers and taking into account all of the rules of blackjack, mathematicians a lot smarter than me have concluded that with basic strategy, blackjack has a standard deviation of approximately 1.14.

So what does that mean? Remember that the bell curve shows odds on both sides (win, lose) of the house edge. If you look at the 0.5% house edge, that means that in a single hand of blackjack you will win or lose lose 1.14 betting units or less 68% of the time, win or lose 2.28 betting units (2 standard deviations) 95% of the time and 3.42 betting units 99.7% of the time.

The possible outcomes and variation from the house edge is also affected by the number of hands you play, with the more hands you play bringing you closer to the average. By including the variable of how many hands you play, you can predict the likelihood of winning or losing certain amounts of units over that number of hands. To calculate that, use the following formula:

Take the square root of the total hands played and multiply by 1.14.

Now let’s say you play 200 hands of blackjack. Using that formula, you get a standard deviation of 16.12. Now let’s say you bet \$1 on each hand for 200 hands. Since there is a 0.5% house edge, you can expect to lose \$1 (1 unit) during that time. But you won’t always have a \$1 net loss when betting \$1 each hand for 200 hands. What about other outcomes? Playing 200 hands, you can expect to win or lose up to 16.12 units 68% of the time, win or lose up to 32.24 units 95% of the time and win or lose up to 64.48 units 99.7% of the time.

How often will you lose \$30, for example?

First, you have to calculate the difference between the expected loss (1) and the actual loss (30). In this case, the difference is 29. You then take the difference and divide by 16.12 (the standard deviation):

29 / 16.12 =   1.799

In this example, 1.799 is a little less than two standard deviations. Taking that into account, you could win or lose \$30 or less 95% of the time. Therefore, if you have a day where you win or lose \$30, it’s not really uncommon.

What about losing \$45, though? 45 / 16.12 = 2.79. That means it falls outside of two standard deviations and is almost 3 standard deviations. Outcomes fall within 2 standard deviations 95% of the time, which means they only fall outside of 2 standard deviations 5% of the time. Therefore, if you lose \$45 when playing 200 hands using basic strategy, it just wasn’t your day, because an outcome like that only happens 5% of the time.