9 Variance, Standard Deviation & the Long Run

Poker is a game of skill played over a fog of luck. Every decision you make has a correct answer in the long run, the play that wins the most chips on average, but on any single hand the universe is free to ignore your skill entirely. You can get all-in with aces against a single opponent holding deuces, an 81% favorite, and lose. You can lose three times in a row in that exact spot, watch three full buy-ins evaporate, and still have done nothing wrong.

This chapter is about that gap: the space between how well you play and how your results actually look. Understanding variance is not optional knowledge for a serious player. It is the difference between a professional who calmly reloads after a brutal session and an amateur who blows up, chases losses, and quits the game convinced it is rigged. The math here is genuinely intuitive once you see it, and seeing it is one of the most emotionally protective things you can do for your poker career.

This chapter will let you put honest error bars on your own winrate, recognize a routine downswing for what it is, and read an all-in EV graph without being fooled by it.

9.1 What variance actually is

In everyday speech, “variance” just means “swings.” In statistics it has a precise meaning worth holding onto, because it makes everything downstream easier.

When you play a hand of poker, the outcome (how many big blinds you win or lose) is a random variable. Over many hands, that random variable has two numbers that describe it:

The mean (your expected value, or EV): the average result per hand, or more usefully per 100 hands. This is your winrate. A good small-stakes online cash player might run at, say, 5 bb/100: five big blinds of profit for every hundred hands dealt.
The standard deviation (SD): a measure of how far individual results typically stray from that mean. This is the size of your swings.

Variance is technically the square of the standard deviation, but in practice players use the words loosely and you should care about the standard deviation, because it is in the same units as your winrate (big blinds) and so you can compare them directly.

Key idea

Your winrate tells you how much you make. Your standard deviation tells you how much that winrate hides under noise. The whole problem of “the long run” is that in poker the noise is enormous relative to the signal. The SD dwarfs the winrate, so it takes a shocking amount of data before your real skill becomes visible in your results.

9.2 Typical standard deviations: knowing the numbers

These figures come from tracking-software databases across millions of tracked hands and years of solver-era population study. Treat them as typical ranges rather than laws of physics; your personal SD depends on your style (loose-aggressive players swing more), the format, and the player pool.

Cash games (6-max No-Limit Hold’em), per 100 hands:

A tight, straightforward player might run around 75–85 bb/100 of standard deviation.
A typical solid regular sits around 85–100 bb/100.
A loose, aggressive, three-bet-happy player can push 100–120 bb/100 or higher.

So the common rule of thumb, a standard deviation of roughly 80–100 bb/100 in 6-max cash, is a good working number. Full-ring (9-handed) games run lower, often 65–85 bb/100 (and aggressive modern full-ring tables can reach ~90 bb/100), because you see fewer flops and play fewer big pots, though the gap to 6-max is narrower than it used to be. Heads-up runs much higher because you play every hand.

Tournaments (MTTs) are a completely different animal. Variance there is far higher and is usually measured in ROI (return on investment) rather than bb/100, because the prize structure is top-heavy. Most of your profit comes from a small number of deep runs and final tables, which by their nature happen rarely. A winning MTT player with a healthy 30% ROI can routinely go hundreds of buy-ins without a meaningful score. We will not put a single SD number on MTTs because it would mislead; the practical takeaway is that tournament downswings are measured in months and thousands of games, not sessions. Chapter 41 returns to why this forces dramatically more conservative bankroll rules for tournament players.

Common mistake

“I’m a losing player. I haven’t won in three weeks.” Three weeks is not a sample; it is a coin flip. Many genuinely winning players have month-long losing stretches, and many genuinely losing players have month-long heaters that convince them they’ve “figured it out.” You cannot diagnose your winrate from how the last few sessions felt. You need data, and a lot of it.

9.3 Why a winrate needs a gigantic sample

Here is the single most important quantitative fact in this chapter.

Suppose you are a real, true 5 bb/100 winner with a standard deviation of 100 bb/100. After playing N hundred-hand blocks, two things happen:

Your expected total profit grows in proportion to N. Over N blocks you expect to win 5 × N big blinds.
Your uncertainty (the standard deviation of your total result) grows in proportion to the square root of N. It is 100 × √N.

That asymmetry is the entire story. Early on, the square-root term is monstrously bigger than the linear term, and your results are basically noise. Only as N gets large does the linear signal finally outrun the square-root noise.

Let us make it concrete by looking at the standard deviation of your measured winrate (your profit divided by the number of blocks). That uncertainty shrinks as:

\[\text{SD of measured winrate} = \frac{100}{\sqrt{N}} \text{ bb/100}\]

where N is the number of 100-hand blocks. Plug in some sample sizes:

Hands played	Blocks (N)	SD of your measured winrate
10,000	100	10.0 bb/100
100,000	1,000	3.2 bb/100
500,000	5,000	1.4 bb/100
1,000,000	10,000	1.0 bb/100

Now read that table with the eyes of our true 5 bb/100 winner. The ranges in this section are all one standard deviation wide, roughly a 68% band around the true winrate (the spread you’ll land inside about two times in three). The next section switches to the stricter two-SD, ~95% convention.

At 10,000 hands the noise around your measured winrate is ±10 bb/100 (one SD). Your true winrate is 5. So after ten thousand hands (a couple of weeks of serious online grinding) your observed winrate could plausibly be anywhere from −5 to +15 bb/100. You genuinely cannot tell whether you are a solid winner or an actual loser. The data simply does not contain the answer yet.

At 100,000 hands the one-SD noise is ±3.2 bb/100. Now your true 5 produces an observed result usually somewhere around 2 to 8. You are probably a winner, but you still can’t pin the number down well.

At 1,000,000 hands the one-SD noise is ±1.0 bb/100. Finally your 5 bb/100 edge is clearly visible above the noise. This is why serious players talk about a million hands as the neighborhood where a cash winrate becomes trustworthy.

Key idea

A useful slogan: edge is linear, luck is square-root. Doubling your sample does not halve your uncertainty; you have to quadruple the sample to halve it. That is why confirming a winrate is so brutally data-hungry, and why your gut feeling about “how I’m running” is almost always premature.

9.4 Confidence intervals: putting error bars on yourself

A confidence interval turns the table above into an honest statement about what you know. The standard (roughly 95%) interval is your observed winrate plus or minus about two standard deviations of the measurement. As flagged above, we now switch from the one-SD bands to this wider two-SD convention; a 95% interval is exactly twice as wide as the 68% one, so make sure you compare like with like.

Worked example. You have played 50,000 hands of 6-max cash. Your tracker says you are winning at 6 bb/100. Your SD is 100 bb/100. Should you be confident you’re a 6 bb/100 crusher?

Step 1: count the blocks. 50,000 hands = 500 blocks of 100. So N = 500.

Step 2: SD of the measured winrate: 100 / √500 = 100 / 22.4 ≈ 4.5 bb/100.

Step 3: build the interval at both confidence levels so the contrast with the previous section is explicit:

One SD (~68%): 6 ± 4.5, i.e. roughly +1.5 to +10.5 bb/100. This is the same flavor of band as the 10k/100k/1M ranges above.
Two SD (~95%): 6 ± 2 × 4.5 = 6 ± 9.0, i.e. roughly −3 to +15 bb/100. This is the one we report.

Read that 95% interval out loud. After fifty thousand hands of winning poker, the honest statistical statement is “I am 95% confident my true winrate is somewhere between losing 3 bb/100 and winning 15 bb/100.” The interval still straddles zero: you cannot yet rule out that you are a small loser running hot. That is simply the arithmetic of a high-variance game.

One more dose of humility, because the clean arithmetic above is actually optimistic. It treats your standard deviation as a known constant (100 bb/100) and your true winrate as a fixed target sitting still while you collect data. Neither is quite true. Your SD is itself an estimate squeezed out of the same, often small, sample, and your real winrate drifts over time as the games change, the player pool adjusts, and your own skill improves or rusts. Both effects make the honest, real-world error bars wider and softer than the idealized 100/√N suggests. If anything, the table understates how little you know early on.

This should change how you behave:

You should not radically overhaul a winning strategy because of a bad week, nor crown yourself a genius after a good month.
You should treat your tracker’s winrate as a moving estimate with error bars, not a verdict.
You should keep studying regardless of short-term results, because the results are too noisy to be your teacher on their own.

Drill

Open your own database (or imagine plausible numbers). Take your current sample size in hands, divide by 100 to get N, and compute 100/√N. Multiply by 2. That is your 95% error bar in bb/100. Write down the sentence: “My true winrate is probably between [observed − error] and [observed + error].” Sit with how wide that range is. Most players are stunned the first time they do this honestly.

9.5 Downswings: how deep and how long

A downswing is just a stretch where the square-root noise happens to run against you. Because the noise is so large, downswings that feel catastrophic are completely routine.

A rough way to picture the depth: early on, and permanently for a break-even player, the size of a “normal” worst-case dip from a peak scales with your SD and with the square root of how many hands you play. Read literally, that square-root growth would imply downswings deepen forever, but for a genuine winner they do not. Once your winrate is reliably positive, the expected depth of your worst drawdown stops growing with sample size and levels off; that plateau is what people mean when they say your winrate slowly fills the hole back in. A standard result puts the central bound at roughly SD² ⁄ (2 × winrate). For our 100 bb/100, 5 bb/100 winner that is 100² ⁄ (2 × 5) = 1,000 bb, about 10 buy-ins, as the typical worst-case depth, with the unlucky tail of the distribution stretching to the 20–30 buy-ins people dread. So large downswings of 20+ buy-ins (one buy-in = 100bb) are not rare events; they are expected to happen multiple times across a career, though they live in the tail rather than the center. The genuinely dangerous case is the break-even-or-slightly-winning player: with almost no winrate to anchor the bound, that SD² ⁄ (2 × winrate) ceiling blows up, and they can be stuck in a downswing lasting hundreds of thousands of hands, simply because their thin edge takes forever to climb out of a hole the variance dug.

Two consequences follow, and they are the practical heart of this chapter:

The depth of a downswing you should plan for is driven by your SD, not by how well you play. A great player with high variance can have deeper dollar swings than a mediocre player with low variance. Skill reduces the frequency and eventual recovery of downswings; it does not exempt you from them.
You will spend a large fraction of your playing life below your all-time peak. This is mathematically guaranteed for everyone, winners included. The graph in your head where good players’ bankrolls march smoothly upward is a fantasy. Real winning graphs are jagged staircases with long, ugly plateaus and sickening drops, trending up only when you zoom all the way out.

9.6 All-in EV: separating decisions from outcomes

Modern tracking software (Hold’em Manager, PokerTracker, and similar) shows two lines on your results graph: your actual winnings (the green line, your real money) and your all-in EV (sometimes the blue or red line). The all-in adjusted line answers a specific, useful question: what would my results look like if every all-in had resolved exactly according to its probability?

Here is how it works. You get all-in on the flop with a flush draw and a pair, holding 45% equity for a 200bb pot. In reality you either win the whole 200 or win nothing. But the EV line credits you with your fair share: 45% of 200 = 90bb, every single time, regardless of what the river actually brings. Do this across thousands of all-ins and the EV line smooths out the river luck.

If your green (actual) line is below your EV line, you have been running below expectation on all-ins: “running bad,” losing flips and getting drawn out on more than your share. The gap is, to a first approximation, luck rather than bad play.
If your green line is above your EV line, you have been running good, winning more all-ins than your equity deserved.

Common mistake

Two errors cluster around the EV line, and they are opposite mistakes.

Error one: treating the EV line as your “real” winrate. It isn’t. The EV line removes river variance only at the moment of the all-in: once the cards are tabled, it computes your true equity against the revealed hand (it does not assume your equity was correct, it reads it straight off the cards both players turned over). What it leaves untouched is the entire world of non-all-in pots, where most hands are decided and most of your money is quietly won or lost. A player can have a pristine all-in EV line and still bleed chips through weak postflop play in pots that never reach a shove. Nor does it flatter a bad get-in: a player who repeatedly stacks off as a 30% underdog, putting in roughly half the pot to realize only 30% of it, is booked at about −0.20 × pot every single time, so their EV line steadily declines. The EV line adjusts for luck; it does not grade decisions.

Error two: using “I’m running below EV” as a permanent excuse. Over a big enough sample the green and EV lines converge. If after 300,000 hands your green line is still dramatically below your EV line, the likely explanation is no longer bad luck: perhaps your equity is systematically worse than you think because you’re getting it in against ranges that crush you, or your reads at the moment of stacking off are off. Variance hides skill problems; it does not erase them.

The healthy use of the EV line is as a tilt-control instrument. After a brutal session, before you conclude you played badly, check whether the green line cratered while the EV line held steady. If so, you ran bad; your decisions were probably fine and the river was unkind. That knowledge won’t refund the money, but it can stop you from “fixing” a strategy that was never broken.

9.7 A fully worked hand: right play, wrong result

Let us walk one all-in through both lenses, decision and outcome, so the distinction is concrete.

Setup. 100bb cash, 6-max. You are in the cutoff with A♠K♠. UTG (a tight, solid regular) opens to 3bb. It folds to you; you 3-bet to 10bb. UTG 4-bets to 24bb. You shove 100bb. UTG snap-calls and tables Q♥Q♦.

The board runs out J♣8♦3♠9♥2♣. You miss everything. You lose 100bb.

Decision analysis. Against a tight UTG’s 4-betting range (typically QQ+, AK, and a few bluffs at most pools) AK is doing fine. Against this specific range, call it QQ, KK, AA, AK with a bluff or two, your AKs has somewhere in the neighborhood of 40–45% equity; take ~42% as a working number. Be honest about what that buys you on the called branch: 42% of a roughly 200bb pot is only about 84bb, less than the 100bb you put in. At showdown alone, this get-in is slightly underwater. What drags it back to profitable is the money already dead in the middle plus your fold equity, the value of the times UTG folds. Note that when you jam you are putting in the last bet: UTG can only call or fold, so there are no further streets on which you can be bluffed off your equity. Against a near-pure-value range, though, that fold equity is thin, perhaps one combo in ten folds, and the arithmetic is unforgiving. Put the whole jam together. When UTG folds (≈10%) you scoop the 35.5bb already in the middle for a profit of about +25.5bb (you contributed only 10 of it). When he calls (≈90%) you net 0.42 × 201.5 − 100 ≈ −15.4bb. Blend the branches: 0.10 × 25.5 + 0.90 × (−15.4) ≈ −11bb, and even if you talk yourself into the optimistic 45%, the shove is still around −6bb. So against the literal near-pure-value range you wrote down, the jam is slightly negative rather than break-even-to-profitable. Two things rescue the play in practice. First, real tight regs almost never 4-bet a pure-nut range: add the bluffs and thinner value they actually carry and both your equity and your fold equity climb until the jam is clearly +EV. Second, against a genuinely value-heavy 4-bettor you are not obliged to jam at all. Flatting (just calling) the 4-bet at 100bb deep is a standard, often higher-EV continue, because shoving blasts out exactly the bluffs AK dominates and gets called mainly by the hands that already beat or flip with you. The one line that is simply a leak is folding AKs to a 4-bet at 100bb; that is what costs you long-run money.

Outcome analysis. Here is the subtle part that trips up almost everyone reading their own graph. The ~42% from the decision analysis was your equity against UTG’s entire 4-betting range, exactly the right number for deciding whether to jam, because at the moment you shove you don’t yet know which hand you face. But your tracker’s all-in EV line does not use that range equity. Once the money is in and the cards are tabled, the software computes your equity against UTG’s actual, revealed hand, QQ, and AKs against QQ is a hair under a coin flip at about 46% (A♠K♠ vs Q♥Q♦ runs ≈ 46.3%), a few points higher than your equity against the whole range. So the EV line credits you 0.46 × (final pot), regardless of the board. With about 201.5bb in the middle (your 100 + UTG’s 100 + 1.5 in blinds), that fair share is roughly 93bb, and your actual result was zero. You ran about 93bb below expectation (roughly nine-tenths of a stack, nearly a full buy-in) on this single pot. Keep the two equities firmly distinct: ~42% is equity vs the range (the get-in decision), and ~46% is equity vs the revealed QQ (what the EV line books). Mix them up and you will misread your own all-in-adjusted graph, expecting it to reflect your range reads when it only ever scores you against the cards your opponents turned over. Multiply spots like this over a session and you get the gap between a green line and an EV line.

The lesson. If you replay this hand and feel sick, if you start thinking “I should fold AK to tight regs,” you are letting a single high-variance outcome rewrite a sound strategy. (The other thought, “I should have just called the 4-bet,” is not a tilt error at all: as we just saw, flatting is a perfectly standard, sometimes superior, line against a value-heavy 4-bettor. What you must never conclude is that AK belongs in the muck.) The fix is to internalize that you can lose a hand you played well, lose it again the next orbit, and still be the favorite to profit by continuing with AK in that spot over a career. The board J♣8♦3♠9♥2♣ owes you nothing this hand; the math owes you over ten thousand hands.

Drill

For your next five losing all-ins, write two columns: “Was the get-in correct? (equity + range read)” and “What was the result?” Force yourself to grade the decision without looking at the outcome. The goal is to build the mental habit of evaluating choices independently of results, the single most important psychological skill variance demands. (We go much deeper on this in the tilt and mental-game chapters.)

9.8 The emotional cost, and why it demands bankroll discipline

Everything above is math, but the reason the math matters is human. Variance is not just a statistical property of poker; it is a sustained assault on your emotional stability, and it is built, almost by accident, to break exactly the cognitive machinery you need to play well.

Here is the trap. Humans are wired to learn from outcomes: do a thing, get a reward, repeat; get punished, stop. That instinct is brilliant for most of life and poison for poker, because poker deliberately decouples the reward from the decision in the short run. If you let outcomes train you, variance will teach you to fear correct plays that recently lost and to love reckless plays that recently won, reprogramming you by noise into a worse player. The defense is to anchor your sense of “did I play well?” to your decisions and your study, and explicitly not to the session’s result.

But emotional discipline alone is not enough, because even a perfectly stoic player can go broke through bad luck if they are underfunded. This is the bridge to bankroll management (the full treatment is Chapter 41). The logic is simple and ironclad:

Downswings of many buy-ins are not tail risks; they are expected events for everyone.
If your entire bankroll is only a handful of buy-ins, an ordinary, statistically routine downswing can bust you before your edge ever gets the sample size it needs to show up.
Therefore the size of your bankroll is not really about how good you are; it is about how big your standard deviation is. High-variance formats and styles demand fatter bankrolls. Cash games, with their ~80–100 bb/100 SD, are commonly played on something like 20–40 buy-ins; tournaments, with their wildly higher variance, demand hundreds of buy-ins. Those numbers are downstream of the variance math in this chapter.

Key idea

Bankroll management is variance insurance. You are not buying it because you doubt your skill; you are buying it so that your skill gets to play out over a long enough sample to matter. An undercapitalized genius is gambling on the short run. A well-capitalized solid winner is investing in the long run.

9.9 The mindset that survives the long run

Pulling it together, the winning relationship with variance rests on a few load-bearing beliefs. Hold these and the swings become tolerable; lose them and even a real edge won’t save you.

Results in the short run are evidence about luck, not skill. Your skill is measured by the quality of your decisions, which you can evaluate hand by hand. Your luck is measured by the gap between your green line and your EV line, and by your distance from your error-barred true winrate.
The long run is longer than you think. Tens of thousands of hands is barely a teaser. Hundreds of thousands begins to mean something. A million is where a cash winrate stands up. For tournaments, multiply all of that.
Downswings are weather, not climate. They are guaranteed, they are deep, they are no one’s fault, and they end. Your job during one is to keep making good decisions, keep your stakes appropriate to your shrinking bankroll, and refuse to let the noise rewrite your strategy.
Detachment from outcomes is a trainable skill, and it is the master skill of the high-variance game. Everything in the psychology chapters (tilt control, focus, confidence) ultimately reduces to this: can you keep playing your best when the cards are actively lying to you about how well you’re playing?

Get the math into your bones, fund yourself properly, judge your decisions and not your dollars, and the variance that destroys most players becomes the very thing that pays you, because it keeps the weaker players, the ones who can’t see past the short run, coming back to the table again and again.

Summary

Edge grows with N while luck grows with √N, so confirming a cash winrate takes on the order of a million hands.
A 95% error bar on your winrate is roughly 2 × (SD ⁄ √N) bb/100, and early samples routinely straddle zero.
Downswings of 20+ buy-ins are expected events, and the depth you should plan for is set by your SD, not your skill.
The all-in EV line adjusts for river luck and scores you against the cards shown; it neither grades your range decisions nor rescues a bad get-in.
Bankroll requirements track your variance, buying your edge enough hands to show up.