Standard Deviation In Blackjack

Standard Deviation is a measure of how results are distributed within a range of possible outcomes. This score is useful when comparing averages – for example two scores may have the same average of ‘50’ with one comprising of results entirely between 45 and 55 and the other having results ranging from 1 through to 100.  The second set of scores is more widely distributed than the first, which will be reflected in a higher standard deviation score.

In Blackjack the standard deviation can be used to show you what the probability of winning or losing a number of betting units is, based on the number of hands you play. We could use this score to answer questions such as ‘what is the probability of winning more than 20 betting units over the course of 100 hands?’ or “Is winning 30 bets over 200 hands a normal outcome, or did I get lucky?’

This article will show you how to calculate (with the help of a calculator) your standard deviation in Blackjack and how to use this information to your advantage, whether you play live or in an online casino.

Calculating The Standard Deviation In Blackjack – A Note About Distribution Curves

Blackjack Deviation

You get to the probability of an event occurring by comparing your standard distribution score to a ‘bell-curve’ of possible outcomes, known as a ‘normal distribution’. The score is calculated in such a way as to show that 68% of outcomes will fall within 1 standard deviation of your average, 95% of outcomes fall within 2 standard deviations and 99.7% of outcomes fall within 3 standard deviations.

In other words, once we have worked out what the standard deviation score is for a certain set of data we then compare this to the normal distribution curve to arrive at a probability of any single score being within the normal set of expected outcomes.

Blackjack Standard Deviation – How Can We Chart The Distribution Of Blackjack Outcomes?

Outcomes of a single blackjack hand can be mapped precisely – since we know the rules, probabilities and all of the cards in the deck. The most common outcome is to win or lose one betting unit, with splits, doubles and blackjack making it possible to win or lose more. With the larger number of single units dominating it is possible to work out that the standard deviation for a single hand of 6-deck Blackjack is exactly 1.1418 based on card distributions and rules alone.

Net Win in Blackjack
Net win Total Probability Return
8 1079 0.00000063 0.00000506
7 10440 0.00000612 0.00004287
6 64099 0.00003761 0.00022563
5 247638 0.00014528 0.00072642
4 1307719 0.00076721 0.00306885
3 4437365 0.00260331 0.00780994
2 99686181 0.05848386 0.11696773
1.5 77147473 0.04526086 0.06789129
1 540233094 0.31694382 0.31694382
0 144520347 0.08478716 0
-0.5 76163623 0.04468366 -0.02234183
-1 684733650 0.40171937 -0.40171937
-2 71380000 0.0418772 -0.0837544
-3 3559202 0.00208811 -0.00626434
-4 828010 0.00048578 -0.00194311
-5 152687 0.00008958 -0.00044789
-6 30536 0.00001791 -0.00010749
-7 3972 0.00000233 -0.00001631
-8 305 0.00000018 -0.00000143
Total 1704507420 1 -0.00291455

This table reflects a standard deviation of 1.1418.

Chart provided by WizardofOdds

By applying this number directly to a ‘normal distribution’ – or bell curve - we find that over an infinite sample, in a single hand of blackjack you will win or lose 1.1418 betting units or less 68% of the time, win or lose 2 standard deviations or 2.2836 betting units or less 95% of the time and win or lose 3 standard deviations or 3.4254 units or less 99.7% of the time.

While this score is interesting, the application of it benefits from adding a second variable – the number of hands played.

Standard Deviation In Blackjack – Using The Information To Predict Win / Loss Runs

Finally we get to the key practical application of working out standard deviations in blackjack games – assessing the likelihood of winning or losing certain amounts of units over specified numbers of hands. Here is the formula to work this out based on your hand sample size:

(Square Root Of The Number Of Hands Played)*1.1418

Here are some working examples:

100 hands played, square root = 10 * 1.1418 = 11.418

This shows that 68% of the time you will win or lose 11.418 units or less over the course of 100 blackjack hands, 95% of the time you will fall within 2 standard distributions and win or lose less than 22.836 units – while 99.7% of the time your outcome over 100 hands will be within 3 standard deviations, or + / - 34.2 units.

300 hands played, square root = 17.32 * 1.1418 = 19.77

Here the distribution of blackjack outcomes predicts you will win or lose 19.77 units 68% of the time you play 300 hands, 95% of the time you will fall within 2 standard deviations and win or lose 39.54 units and 99.7% of the time you will fall within 3 standard distributions and win or lose < 59.31 units.

As you can see, the higher the number of hands played the smaller the relative impact of chance. Of course you also need to take into account the house edge of 0.05% or so when making these calculations!

Do not worry if you do not have a pocket calculator with you at the casino, it is straight forward to work out the blackjack standard deviations for different sized sessions in advance and gain an insight into how the average distribution of outcomes affects your chances of either winning or losing certain amounts of cash. Once you get an idea of the types of swings which are normal in the game you will feel more comfortable at the blackjack tables, whether live or online!