Get Into The Swing

By Tony Bee (a.k.a. ‘Tony’).

Here’s a quick arithmetic test.  (Don’t groan, it’s for your own good.)

Fill in the answer: 1 + 4 x 2 = ?

Quick as you can, no calculators allowed.  Got it?

(I’ll tell you the answer at the end of this article, but here’s a clue: the answer is not 10.)

Numbers.  To some they’re a source of fascination, to others a source of frustration and confusion.

As a poker player you may have hoped that arithmetic and statistical considerations could be left back in the classroom, apart from perhaps whether you have the pot odds to call with your gutshot on a rainbow flop.  And let’s face it, you can simply memorize a chart to work that out.

But you’d be wrong.  If you’re serious about the game there is one particular, very special number that you would do well to be on close terms with.

However, only a fraction of poker players have heard of it, an even smaller fraction know how to calculate it and I’d be willing to wager that only a teeny tiny percentage know what to do with it.  Yet it’s one of the most important and useful numbers a poker player (or indeed any advantage gambler) needs to know.  What is it?

Two words: STANDARD DEVIATION.

In this article I shall endeavour to explain what the big deal is about standard deviation (from here on I’ll call it SD), and I’ll give examples of how it can be used to help and inform your game.  Don’t worry if you’re not a ‘numbers person’.  There will be some arithmetic, but I’ll try to keep it light.

How is Standard Deviation (SD) Calculated?

If you play online, the simplest way to find your SD is to purchase one of a number of software packages designed to do this for you.  I personally use Poker Tracker, but there are a number of software packages with this capability.  Poker Tracker is able to automatically import your hand histories as you play and update your statistics (and those of your opponents) in close to real time.

However, if you only play live or you’re too tight to buy one of these packages, you’ll need to know how to calculate this yourself.

Here’s an example.

Let’s suppose you play $5/$10 fixed-limit Holdem. Here a big bet is $10, the larger bet used on the turn and river. I’ll write BB for big bets throughout this article; it does not mean big blinds. Check your tracking software’s definition before comparing figures.

Choose one time unit and stick to it. We’ll use blocks of 100 hands. The mean win rate M is in BB per 100 hands. The 100-hand SD is the standard deviation of the total result of a 100-hand block, measured in BB. Software often labels this “SD per 100 hands,” but it does not scale like a win rate. We’ll come back to that distinction.

Let’s suppose that your last five sessions have gone as follows. Results are net of rake and other playing costs; record those consistently.

Make a note of your net win or loss and hands played at the end of each session. To convert dollars to big bets, divide by $10. Divide the number of hands by 100 to get the number of 100-hand blocks:

There are N = 5 sessions and T = 20 blocks altogether. This method assumes independent session results with the same underlying win rate and variance per block: a session lasting t blocks has expected result M × t and variance V × t. That can be a rough model when games, stakes, opponents or your play change.

First square each session’s result, divide by its number of blocks, and add. Call this R:

$R=$$\frac{{30}^2}{3}$$+\frac{{45}^2}{8}$$+\frac{{(-35)}^2}{2}$$+\frac{{15}^2}{3}$$+\frac{{(-25)}^2}{4}$$=\mathrm{1,396.875}$

Your mean (or average) result per 100 hands is total big bets divided by total blocks. Call this M:

$M=\frac{30}{20}=1.5$BB per 100 hands.

The original calculation estimates the variance of a 100-hand result as follows.

$V=\frac{R-M^2\times T}{N}$.

$V=\frac{\mathrm{1,396.875}-({1.5}^2\times 20)}{5}$$=270.375$BB².

Your SD is the positive square root of the variance:

100-hand SD$=\sqrt{270.375}\approx 16.4$BB.

This model-based estimate has a catch. If the independent session totals are also normally distributed, dividing by N gives the maximum-likelihood variance estimate. Because we estimated the mean from the same sessions, this variance estimate tends to be too low. The usual small-sample correction divides the bracketed quantity by N − 1 instead: $\frac{\mathrm{1,351.875}}{4}$ = 337.96875 BB², giving an SD estimate of about 18.4 BB. The correction makes the variance estimate unbiased under this model; its square root is still an estimate, not an unbiased SD.

This is tedious to calculate by hand, but takes a matter of seconds in a spreadsheet. Five sessions and 2,000 hands are much too few to rely on for a win-rate or SD estimate. Thirty sessions is not a magic threshold either: session length, changing conditions and dependence between results matter. More comparable data helps, but cannot prove that your future games will behave the same way.

Now, having gone to the trouble of calculating our SD, let’s look at what we can do with it.

What is Standard Deviation and Why is it Useful?

Let’s suppose your true win rate is 1 BB per 100 hands. After 1,000 hands, your expected profit is indeed 10 BB. That is an average over many repetitions, not a prediction of what your next 1,000 hands will earn.

Even good decisions can produce losing sessions. A positive expected value after costs gives a positive average result in the model; it does not guarantee profit over any particular number of hands. Your actual results vary, and SD measures that variability.

A low SD does not prove that you play well, and a high SD does not prove that you play badly. In loose, multiway low-limit games, big pots can mean substantial swings even when you make sound value bets. Avoiding profitable plays merely to make the graph smoother can hurt your win rate.

For a normal distribution, about 99.73% of results lie within three SDs of the true mean. That is a statement about a bell-shaped model, not a guarantee about poker returns. Individual hands and short sessions can be very far from normal. A normal approximation may become more useful over many sufficiently independent hands with stable conditions and finite variance, but there is no fixed hand count that guarantees accuracy. Estimated inputs add uncertainty too.

For the examples below, assume a true win rate of 1.5 BB per 100 hands and a 100-hand SD of 16 BB. These are convenient illustrative inputs, not the small-sample-corrected estimates above. We keep them fixed so the arithmetic is easy to follow.

For 100 hands, the expected total is 1.5 BB and the SD of the total is 16 BB. A three-SD interval is:

$1.5\pm (3\times 16)$= −46.5 to +49.5 BB.

Only if the total result were normally distributed with those true inputs would this interval contain about 99.73% of repeated 100-hand results. It does not say that the endpoints are likely results, nor that those limits cannot be exceeded.

What if you play double this many hands? Under the independence and constant-variance assumptions, the expected total doubles, but the SD of that total grows only by the square root of two. More generally, for n blocks of 100 hands:

Expected total =$M\times n$BB;SD of total =$\mathrm{SD}\times \sqrt{n}$BB.

After 200 hands, the expected total is 3 BB and its SD is 16 × √2 ≈ 22.63 BB. Using the unrounded SD, the three-SD interval is:

$3\pm (3\times 16\times \sqrt{2})$≈ −64.88 to +70.88 BB in total.

Those are total big bets over 200 hands, not BB per 100 hands. Divide each total by two to express the result as a rate: the mean remains 1.5 BB per 100 hands, and the SD of the observed rate is about 11.31 BB per 100 hands. Its three-SD rate interval is about −32.44 to +35.44 BB per 100 hands.

After 300 hands the expected total is 4.5 BB and the SD of that total is 16 × √3 ≈ 27.71 BB. Total swings can grow while the average result per 100 hands becomes less variable. If results are dependent or game conditions change, the square-root rule may not describe your play well.

Swings are part of the game. Keep making good decisions, review your play honestly, and remember that a losing stretch by itself neither proves nor disproves that you have an edge. Understanding SD can help us put those swings in perspective. This leads us naturally to…

Bankroll Management

A smaller bankroll is more vulnerable to a downswing. But an estimate of SD cannot tell you a bankroll that is certain to last. “Risk of ruin” (ROR) means the chance of exhausting the playing bankroll under a specified model, not the chance of losing your next session.

The following is a simplified, continuous approximation for a player with a positive true win rate. It assumes stable stakes, a fixed win rate and variance, independent increments, and play continuing indefinitely until ruin or forever. It allows no withdrawals or added funds and no change in stakes. Real poker has discrete bets and changing conditions, so this is a teaching model, not an exact poker-bankroll calculator.

With M in BB per 100 hands and SD the 100-hand SD in BB, the approximate starting bankroll B in BB for a modeled ruin probability r is:

$B\approx -\frac{{\mathrm{SD}}^2}{2\times M}\times \ln \left(r\right)$.

“ln” is the natural logarithm, available on a scientific calculator or spreadsheet. Use a probability between 0 and 1: 0.05 for 5%, or 0.01 for 1%. A smaller target risk requires more bankroll in this model. The formula requires M > 0 and SD > 0; it cannot manufacture an edge or identify a safe bankroll for a break-even or losing player. With positive variance and a nonpositive win rate, the same indefinite-play model gives eventual ruin probability 1 for any finite bankroll.

Keep the illustrative inputs M = 1.5 and SD = 16. For a modeled 5% risk:

$B\approx -\frac{{16}^2}{2\times 1.5}\times \ln \left(0.05\right)$≈ 255.64 BB, or about 256 BB.

For a modeled 1% risk the same calculation gives about 392.97 BB, or 393 BB. The lower target risk calls for a larger bankroll in this model. Neither percentage is a recommendation or a guarantee, especially with uncertain estimates of your edge.

You can rearrange the same approximation to estimate risk for a fixed bankroll:

$r\approx \exp [-\frac{2\times M\times B}{{\mathrm{SD}}^2}]$.

“exp” is the exponential function, also written as e raised to a power. Let’s say your bankroll is $1,500 at $5/$10 fixed limit: that is 150 big bets. With the same assumed inputs:

$r\approx \exp [-\frac{2\times 1.5\times 150}{{16}^2}]$≈ 0.1724, or about 17.2%.

That is substantial modeled risk, not a green light to play. Use only money set aside for poker that you can afford to lose, and keep living expenses separate. Moving down or stopping can limit losses, but those decisions change the assumptions; this formula does not calculate their effect.

The biggest uncertainty may be your win rate: five sessions, even with an overall profit, do not establish a positive true edge. A fixed rule such as “300 big bets is enough” cannot promise safety. Choose stakes and loss limits you can afford, allow for estimation error and changing games, and do not play with “scared money.”

Summary

Hopefully this article has persuaded you of the value of tracking your standard deviation, and has given you some idea of what it’s all about.  I’ll explore the statistical aspects of the game in more detail another time, but for now just remember these three things:

1. Don’t expect to win every session. SD helps explain swings, but a short run cannot establish your true win rate.
2. Don’t chase losses. Review the decisions, take a break when needed, and stay within your limits.
3. Keep a separate, affordable playing bankroll. Treat modeled ruin probabilities as estimates with assumptions, not promises of safety.

Finally, I said I’d give you the answer to the arithmetic test in the first paragraph.  Did you get it right?  The answer is 9, not 10, because in the absence of other information we should always multiply before we add.

If you got this correct, go to the top of the class.  If you didn’t, don’t worry.  Just borrow your sister’s calculator next time.