Low Hand Poker Definition

Poker Stats:VPIP : PFR : Std Dev : The Red Line

Standard deviation ('Std Dev' or sometimes just 'SD') is one of those terms you've bumped in to a few times, but you've never fully got to grips to with it. Or, you've heard about it, but it just sounds far too mathsy for you to understand.

In low games, like razz, the lowest-ranking hands win. In high-low split games, both the highest-ranking and lowest-ranking hands win, though different rules are used to rank the high and low hands. Each hand belongs to a category determined by the patterns formed by its cards. A hand in a higher-ranking category always ranks higher than a hand. The rules of Omaha hi-lo is usually played with a 'qualifier' for the low hand, meaning all of the cards making up a low hand have to be ranked eight or lower. That's where the 'split-8-or-better'. Straightforward, 'by the book' poker meaning you raise when you think you have the best hand, fold when you don't and rarely bluff. There are a number of terms and phrases that are used often in poker. In this glossary we have provided definitions and explanations of the most common ones, as well as abbreviations, acronyms, and slang terms used for poker hands.

So to help you out, I've decided to explain what standard deviation in poker is all about and what it means to you as a poker player. You don't need a background in statistics to enjoy the pleasures of the SD stat, I promise.

What is standard deviation in poker?

Your standard deviation stat gives you an indication of how “swingy” your game is.

• The higher your standard deviation, the higher your variance.
• The lower your standard deviation, the lower your variance.

Standard deviation can also give you an indication of how far you can expect to veer from your current winrate over 100 hands (hence why Std Dev stats are shown in BB/100 ['Big Bets' or 'Big Blinds' per 100 hands]).

That's simple enough, but it doesn't give us much to work with. For a better understanding of the poker std dev stat, we'll need to find out what this standard deviation stuff is all about.

An introduction to standard deviation.

Standard deviation is a measure of how spread out numbers are from the average.

To give an example, lets say you go to your local supermarket and measure the width of 5 loose oranges. The widths of these 5 loose oranges are:

• Widths of 5 loose oranges.
• 6cm
• 7cm
• 13cm
• 5cm
• 10cm
• Average width of these 5 oranges = 8.2cm

Now, instead of measuring 5 loose oranges, you decide to measure another 5 oranges from a pack of “medium oranges”. The widths of the 5 oranges from the pack are:

• Widths of 5 'medium' oranges from a packet.
• 7cm
• 8cm
• 9cm
• 8cm
• 9cm
• Average width of these 5 oranges = 8.2cm

You'll notice that the average widths of these two sets of 5 oranges are the same, despite the width of each individual orange in each set being quite different.

• In the first set of loose oranges, the results are varied and spread out.
• In the second set of 'medium' oranges, the results are very close together.

Now, because standard deviation is a measure of the spread of results from the average, we can say that:

The standard deviation of the loose oranges is higher than the standard deviation of the 'medium' oranges.

I could run through an equation and give you an exact number for the 'standard deviation' of each of these two sets of oranges, but I don't want to get too mathsy (and you don't necessarily need to know how to work it out). If you are interested in the equation though, check out this very straightforward article: basic standard deviation example and equation.

This standard deviation video on youtube is also very helpful.

Note: Okay, I can't leave any stone unturned. The SD of the loose oranges is 3.3, and the SD of the 'medium' oranges is 0.8. These numbers tell us the average spread.

Standard deviation and poker winrates.

Okay, that's enough about oranges. How does standard deviation in poker work and what does it tell you about your winrate?

Well, instead of measuring oranges, what if you split up your poker career in to 100-hand mini-sessions (or 'chunks'), and measured your winrate for each of these sessions?

From a set of results like this, you could work out your mean average winrate (this will give you the average winrate stat that you're used to seeing) and also your standard deviation. The more your winrate varies in each chunk of 100 hands you play, the bigger your standard deviation will be.

Additionally, if you plotted the frequency of the winrates (how often each winrate occurs) over each of these 100 hand chunks in a graph, it would look something like this:

This is called a “bell curve” or a “normal distribution”. There are 2 important things to note about this graph of your winrates:

1. Your average winrate is in the very centre of the curve (the peak).
2. The further you move from the centre of the curve, the lower the chances of these sort of winrates occurring over 100 hands.

This makes sense, because more often than not your winrate will remain close to your average, but there will be rare occasions where it ends up being quite far from average.

So for example, if you have an average winrate of 4BB/100, achieving a winrate of 4.5BB/100 over 10,000 hands is very likely. However, achieving a winrate of 18BB/100 over 10,000 hands is highly unlikely.

High and low standard deviation graphs.

So, we know that if we plotted a graph of our individual winrates from many small 100-hand sessions it would form a bell curve, with our most frequent winrates collecting around the centre.

But what do high and low standard deviation graphs look like?

I'm hoping that you can make sense of these graphs by just looking at them. If not though:

• If you have a low standard deviation, your winrates are generally going to remain close to your average winrate over each 100 hand chunk.
• If you have a high standard deviation, you winrates are going to wildly vary over each 100 hand chunk.

The lower your std dev, the thinner the bell curve. The higher your std dev, the fatter it will be to account for the wide variety of winrates.

What does standard deviation tell us about our winrate?

Three useful things:

1. There is a 68% chance that you will be within 1 SD of your winrate over 100 hands.
2. There is a 95% chance that you will be within 2 SDs of your winrate over 100 hands.
3. There is a 99% chance that you will be within 3 SDs of your winrate over 100 hands.

Why these percentages? Because this is just how bell curves work. Here's another bell curve graph with some notes to help you understand what's going on.

For more information, here's a short wikipedia article that actually isn't overly complicated: 68-95-99.7 rule wiki.

Using standard deviation example.

Lets say that you're a $50NL player with a winrate of 5BB/100. You also have a std dev of 30BB/100.

Note: In this example we're using poker tracker BBs (Big Bets), which are equal to 2x the size of the big blind (i.e. 1BB at $50NL is equal to $1 instead of $0.50).

68% of the time.

There is a 68% chance that your winrate will fall between 1 standard deviation of your current winrate.

• 1SD = 5BB +/- 30BB
• 1SD = -25BB <=> 35BB
• 1SD = -$25 <=> $35

So there is a 68% chance that you will win (or lose) between -$25 and $35 over the next 100 hands given your current winrate and standard deviation.

95% of the time.

There is a 95% chance that your winrate will fall between 2 standard deviations of your current winrate (2 x 30BB = 60BB).

• 2SD = 5BB +/- 60BB
• 2SD = -55BB <=> 65BB
• 2SD = -$55 <=> $65

99% of the time.

There is a 99% chance that your winrate will fall between 3 standard deviations of your current winrate (3 x 30BB = 90BB).

3SD = 5BB +/- 90BB
3SD = -85BB <=> 95BB
3SD = -$85 <=> $95

What affects your standard deviation stat?

Your playing style.

If you have a LAG (loose-aggressive) playing style you will have a high standard deviation. This makes sense, as you should already know that maniacs and LAGs experience a lot of variance.

On the other hand, if you are a nit that only ever plays premium hands, your standard deviation will be low.

If you're a TAG, you'll be somewhere in the middle.

What's are low and high standard deviations in poker?

The majority of poker players' standard deviations lie between 35BB/100 and 50BB/100 (70bb/100 and 100bb/100).

If your SD is far outside of this bracket, you're probably playing too tight or too loose to achieve an optimum winrate.

Which is better, a high or low SD?

It doesn't matter. Your winrate is the most important thing.

It's like traveling to the next town in a car compared to traveling to the next town on horseback. You get the same destination in the same amount of time, you just take different routes (with one being bumpier than the other).

Most people would prefer the less 'swingy' route with a low SD, but don't change your game just because you don't like the look of your SD. Unless your std dev stat is huge and you're due for a myocardial infarction, I wouldn't worry about it.

See how standard deviation affects your results for yourself.

Play with this awesome poker variance simulator.

Keep your winrate the same and enter varying standard deviations and see how the affect your results over different sample sizes.

it just goes to show how you shouldn't worry (or get too cocky) about your winrate over small sample sizes. The differences in the results is surprising.

Poker standard deviation conclusion.

Standard deviation is a measure of how much you are likely to swing up and down during the course of small sample sizes of hands. The higher your SD is, the larger the swings you will face and the longer it will take for your results to converge to your actual win rate.

The higher your Std Dev stat is, the more (and the bigger the) swings you will experience throughout your poker career.

I'm hoping that this big article with all the graphs has helped you to understand standard deviation in poker a little better. It can be a little tricky at first (especially with some of those equations), but it's not so bad when you start to get your head around it.

Here are some more links that I found really helpful:

Go back to the interesting Texas Hold'em Articles.

Can You Afford Not To Use
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“I wouldn’t play another session of online poker without it”

“I play $25NL, and in under 1 week PT4 had paid for itself”

Comments

This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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Preliminary Calculation

Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

These are the same hand. Order is not important.

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.

Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is

This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.

Poker Hands Chart

The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.

If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.

Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.

Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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The Poker Hands

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

Definitions of Poker Hands

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Counting Poker Hands

Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

Two Pair and One Pair
These two are left as exercises.

High Card
The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

Hands Of Poker In Order

Probabilities of Poker Hands

Poker Low Hand Rankings

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2017 – Dan Ma