# Variance and Poker Pt. 1: How Good Cash-Game Players Outrun Luck

07-09-2014, 16:09

In this two-part series PokerOlymp's Arved Klöhn explains what variance is and how it dictates your poker results.

First he'll take a look at variance in live poker and then he'll compare live poker variance to online poker.

By Arved Klöhn

Variance isn't a dirty word per se, but for most of us it’s a very theoretical term and only few of us understand what it actually means.

Variance has a lot to do with mathematics, which – again – is not a dirty word but a science that makes sure our cars drive, our planes fly and our smart phones are smart (to some extent at least).

To understand what variance means one does not have to study probability theory or know that the abbreviation CLT means “Central Limit Theorem."

It’s actually a lot simpler than that. Basically in poker, variance is a term (or more specifically a number) which describes how far your results spread around the mean.

To put it simpler – a lot of variance means many swings; less variance means fewer swings.

But let’s look at an example to get a better grasp at the concept of variance in poker and put some numbers on it.

### Variance for Live Poker Players

Let's take a look at a decent low-stakes live poker player. We'll call him John.

John plays $1/$2 No-Limit Hold’em in his local casino. He plays 20 days per month and averages six hours of play per day.

He usually plays between 30 and 35 hands per hour. Thus in total he plays about 4,000 hands per month.

Now since John is a decent player he usually wins at his game. Per month he averages $2,000 profit after rake and tips. So his win rate is $50 per 100 hands.

Of course John doesn’t win uniformly every day. He has good days where he wins a ton and bad days where he loses quite a bit. “Good days and bad days” – that’s his variance.

Now let’s try to put a number on John’s variance. Therefore we ask him for one month to accurately write down his results every 100 hands.

So John produces this chart for us:

$50

$180

$0

$160

-$190

$140

$440

-$200

-$300

$230

$230

$320

$130

$390

$330

$340

$200

-$220

$130

$640

-$170

$90

-$50

-$120

-$270

-$200

-$260

$20

$90

-$160

$0

-$340

$190

-$250

$120

-$300

$400

$90

$200

-$80

Those are John’s winnings for each 100-hand-stretch. If you add up those 40 numbers it’s exactly $2,000. But for individual stretches his results range from $340 losses to $640 wins.

On average we expect John to win $50 per 100 hands, but let’s take a closer look at his variance.

Therefore we calculate the average difference from his expected winnings ($50 per 100 hands) for all forty 100-hand-stretches (a program like excel comes in quite handy here because it has built in formulas to do those calculations).

We did this calculation and it turns out the average difference is $238. This number is called standard deviation.

This means John will miss his expected winnings of $50 per 100 hands by $238 on average – either by winning way more than $50 or by losing over a single 100-hand-stretch.

### The 3 Magic Numbers for Variance

Now we've gathered the three important numbers to fully understand variance:

Win Rate: In the case of John: $50 per 100 hands Standard Deviation: In the case of John: $238 per 100 hands Number of Hands Played: In the case of John: 4,000 (for one month)

Those three numbers are all we need to run a bunch of tests, explain John’s results over long periods and even make educated guesses about his future winnings.

We take those three parameters (win rate, standard deviation and number of hands) and head to over to any variance calculator to get some more insight into the variance John can expect.

Pokerdope.com probably hosts the best free available variance calculator for poker players. So let’s go there, enter those three parameters and take a look at the results (this calculator works with big blinds instead of dollars so we translated John’s numbers to big blinds):

Now what do we see here? Firstly the calculator shows 20 simulated samples how John might perform over the course of one month (the thin colored lines).

It also shows the best and worst run out of 1,000 trials (the bold blue and red lines).

The bold green lines show the confidence intervals. The two dark green lines are the most important ones – they indicate the 95-per cent confidence interval.

This term means that in 95 per cent of all cases the actual results will be somewhere between those lines. Meaning: at any given time there’s only a 5 per cent chance that his winnings will be above the upper dark green graph or below the lower dark green graph.

Let’s take a closer look at some numbers the variance calculator produces:

The probability that John does not make any profit over 4,000 hands is 9.2 per cent. 5 per cent of the time John will be stuck in a downswing over more than $2,000 (1,000 big blinds). 3 to 4 per cent of the time John will be stuck in a downswing over more than 10,000 hands (that’s two-and-a-half months if he plays 4,000 hands per month). On average John can expect to win $2,000 per month but there’s a 5 per cent chance that he will either win more than $5,000 in one month or lose at least $1,000 in one month.

To put those numbers into a single, simple sentence: John may expect some wild rides and should not be too surprised if he has a losing month and should not boast excessively if he has a very good month.

Now what happens to his variance if John plays more than just one month?

Naturally one would expect the variance to have a lesser impact when playing more. So let’s assume John plays for 10 months (40,000 hands) with the same win rate ($50 or 25 big blinds per 100 hands) and the same standard deviation ($238 or 119 big blinds per 100 hands).

Here's what the variance calculator shows:

Now those lines look much friendlier – all of them go up. And even the worst run out of 1,000 trials is way above the zero line. In fact, let’s take a closer look at some numbers:

The probability of John losing money over 40,000 hands (10 months worth of live poker) is miniscule: 0.001 per cent. In 95 per cent of all trials John will win between $10,400 and $29,500 over 10 months.

Those numbers lead to one conclusion:

Very good live cash game players have no variance problems

The example we just went through shows that after a few months of play a good live cash-game player can expect to simply outlast any negative variance he might encounter along the way.

If you’re good enough and your win rate is high enough, you will come off well despite any downswing you might stumble into.

But the example also shows that over short periods of time you must take variance into consideration. In our example there’s a very realistic chance of having a losing month (9.2 per cent).

So you should always make sure your bankroll can handle those short-term swings.

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