The average and the spread
Two numbers describe a repeated bet. The first is the average result per hand, which in blackjack is slightly negative for the player under standard rules because of the house edge. The second is how far real results typically wander from that average. The second number is what statisticians call variance, and it is why one evening at a blackjack table can look nothing like the long-run average.
This article explains variance in plain terms and works through illustrations with simple numbers. It does not describe a way to win; it explains why outcomes are so uncertain.
What variance and standard deviation measure
Variance measures how spread out outcomes are around their average. Its square root is the standard deviation, which is easier to use because it is measured in the same units as the bets. Roughly speaking, a standard deviation tells you the typical distance of a result from the average.
For a single flat-bet hand of blackjack, the standard deviation is commonly cited as being a little over one bet, about 1.1 to 1.2 times the amount wagered. It is larger than for a simple even-money coin flip, which has a standard deviation of exactly one bet, because doubling and splitting can put more than one bet at stake and a natural pays more than even money. A fuller treatment of the per-hand figure is under standard deviation.
For the illustrations below, this article uses 1.15 as a round, assumed value.
How spread grows with the number of hands
Here is the important statistical fact. If hands are independent, the standard deviation over n hands equals the per-hand figure multiplied by the square root of n. The expected loss, in contrast, grows in direct proportion to n.
That difference in growth rates is the whole story. The expected loss rises steadily, while the spread rises more slowly, so over a very large number of hands the average begins to dominate. Over short stretches, the spread dominates.
A worked illustration
Assume, purely for illustration, a flat bet of 1 unit per hand, an average loss of 0.5% of the amount wagered (0.005 units per hand), and a standard deviation of 1.15 units per hand. These are simple stand-in numbers, not the figures for any particular game.
| Hands | Expected result | Standard deviation | Rough range (average plus or minus 2 standard deviations) |
|---|---|---|---|
| 100 | −0.5 units | 1.15 × √100 = 11.5 | about −23.5 to +22.5 |
| 1,000 | −5 units | 1.15 × √1,000 ≈ 36.4 | about −77.8 to +67.8 |
| 10,000 | −50 units | 1.15 × √10,000 = 115 | about −280 to +180 |
Check the first row: 2 × 11.5 = 23, so −0.5 − 23 = −23.5 and −0.5 + 23 = +22.5. For the second, 2 × 36.4 = 72.8, giving −77.8 and +67.8. For the third, 2 × 115 = 230, giving −280 and +180.
Now compare the spread with the expected loss:
- At 100 hands, the standard deviation is 11.5 against an expected loss of 0.5, which is 23 times larger.
- At 1,000 hands, 36.4 against 5, about 7 times larger.
- At 10,000 hands, 115 against 50, about 2.3 times larger.
The ratio shrinks as hands accumulate. Even at 10,000 hands the range is still wide, and a result above zero remains possible. Real outcomes are not perfectly bell-shaped, so the plus-or-minus two standard deviations range is a rough guide, often described as covering about 95% of outcomes, rather than a guarantee. The idea of putting a range around an estimate is developed under confidence intervals.
What this means for a short session
A player who plays 100 hands could finish well ahead or well behind, and in the illustration above the range of plausible results is roughly equal on both sides of zero. That explains why some people leave a casino ahead and conclude that the game is beatable. A single session says very little about the underlying average.
The same swings also run the other way. A bad session does not mean a player made mistakes, and a good one does not mean skill or a special method. The same logic applies to runs of consecutive results, which can be long even when each hand is independent, as in losing streaks.
Why the long run looks different
As the number of hands increases, the average result per hand settles closer to the theoretical value. This is the law of large numbers, set out under large numbers. Note what it says and does not say. It says the average per hand converges. It does not say that the total difference from expectation shrinks; in absolute units the typical gap actually grows with the square root of the hands played, as the table shows.
This is also why a loss does not get “made up”. Past results stay in the total, and future hands do not compensate for them. Believing otherwise is the mistake discussed under the gamblers fallacy.
Variance and money management
Variance is the reason a limited budget can run out even when the average cost per hand is small. With a bankroll of, say, 50 units in the illustration, a swing of a few standard deviations over a few hundred hands could exhaust it. The probability that a stake is wiped out before any recovery is studied under risk of ruin.
The practical lesson is a protective one. Because short-term swings are large relative to the average, it is wise to decide in advance how much you can afford to lose, treat that as the cost of entertainment, and stop when you reach it. Never borrow to play or try to win back losses.
Summary
- The house edge describes the average per hand; variance describes how far real results stray from it.
- The standard deviation over n hands is roughly the per-hand figure times the square root of n, while the expected loss grows in proportion to n.
- Over short sessions, spread is far larger than the expected loss, so outcomes are mostly down to chance.
- Over many hands the average dominates more and more, but the total can still be on either side of zero.
- None of this makes blackjack a way to make money. If gambling is causing you harm, a local support service can help.