Good IMPS Average

[Trinidad]

Rik just to be clear:

- To me what this looks like is 1.1 is the standard error of the average of the IMPs, which would make sense if you were playing 20-40 boards. (As you said, 26 boards.)

Yep.

 

Rik just to be clear:

- And if I multiply this by the square root of 26 I get 5.6, which would be the standard deviation of the IMPs score on any individual board. This is very consistent with my math.

 

Again: Yep.

- Just to be clear -- 1.1 is the standard deviation of the 7 nightly IMPs averages? Can you type in the data so I can see it? :)

Of course, the standard deviation varied a little bit. I calculated the results for 7 evenings (just because they fit on one web page). You can find the raw data below. As you can see, the amount of tables in play varied.

 

Rik

 

1.62   1.71   2.35   1.53   1.54   1.75   2.52

1.28   1.51   1.27   1.31   1.42   1.63   1.46

0.9   1.13   1.21   1.03   1.3   1.44   1.16

0.75   1.19   1.24   0.86   0.91   1.44   0.93

0.74   1.06   0.92   0.79   0.55   0.5   0.87

0.61   0.48   0.53   0.58   0.41   0.45   0.68

0.17   0.31   0.5   0.5   0.4   0.34   0.59

0.17   0.26   -0.15   0.3   0.21   0.23   0.53

0.1   -0.02   -0.2   0.32   0.2   0.23   0.27

0.04   -0.02   -0.28   -0.04   0.18   0.17   0

-0.04   -0.21   -0.45   -0.18   -0.2   0.06   -0.08

-0.07   -0.59   -0.51   -0.53   -0.27   -0.14   -0.15

-0.17   -0.69   -0.6   -0.94   -0.38   -0.18   -0.35

-0.62   -1.14   -0.66   -0.87   -0.61   -0.58   -0.37

-0.71   -1.35   -0.82   -1.22   -1.14   -0.61   -0.99

-1.21   -1.55   -0.85   -1.27   -1.68   -0.64   -1.06

-1.25   -2.15   -0.92   -1.95   -3.04   -0.7   -1.25

-2.44           -1.18                   -0.75   -1.98

                -1.42                   -0.76   -2.97

                                        -0.92

                                        -1.47

                                        -1.49

[HighLow21]

These are the nightly average IMPs at each table yes?

[HighLow21]

OK so there were between 17 and 22 tables each night, and the standard deviation across all tables varied between 0.94 IMPs and 1.28 IMPs. The average standard deviation per night is 1.07 IMPs and the standard deviation of all the results pooled together is 1.05.

 

Assuming of course that all pairs (teams?) are of equal ability (they aren't) and each hand is independent of one another (they aren't, but they are probably close enough), this would mean that on a per-hand basis, the standard deviation per hand is between 4.80 IMPs and 6.51 IMPs, with the best estimate being 5.36 IMPs. This may be slightly high if we were talking about the expectation for a single partnership, but I can't imagine the haircut would be more than 10% of this.

 

Besides, my own recorded results show a standard deviation of 5.28 IMPs per hand, which would work out to 1.04 IMPs for my results on a random 26-board night. I'd expect it to be less than this for a partnership with a single individual, and indeed, if I look at my 3 most common partnerships, the standard deviations are 4.36, 5.03, and 5.21. Pooled, I get 5.02.

 

So there you have it. On a single board, the standard deviation is around 5 IMPs. On 26 boards, this means a standard deviation of around 1 IMP; on 8 it's around 1.8 IMPs, and on 32 it's around 0.9 IMPs.

[helene_t]

To go from 26 boards to 168 boards you would have to divide everything in the left-hand column by the square root of 7, which is the square root of the ratio of hands played. The variation decreases as a function of the square root of the number of observations.

Now you are assuming that the variance is entirely random. To see why this can't be correct, imagine that you went from 100,000 boards to a million boards. Surely you wouldn't expect the SD of the mean crossimps per pair to decrease by a factor sqrt(10).

 

I am afraid there is no simple way out of this. The more the field is dominated by "stupid idiots", the more the random variance will be, but the same is probably true for the skill variance.

[Trinidad]

These are the nightly average IMPs at each table yes?

No. Sorry, I should have been clearer.

 

There are 7 columns. Each column represents one night. In each columns is the final result for each evening: the ranking. So, on the first night, the winning pair had an average of 1.62 IMPs per board, number 2 scored 1.28 IMPs per board, etc.

 

Every pair played 26 boards, except when there were an odd number of pairs when there was obviously a sit out. In those cases 13 of the pairs played 24 boards only. Obviously, their average scores are the sum of their IMP scores divided by 24.

 

Rik

[HighLow21]

No. Sorry, I should have been clearer.

 

There are 7 columns. Each column represents one night. In each columns is the final result for each evening: the ranking. So, on the first night, the winning pair had an average of 1.62 IMPs per board, number 2 scored 1.28 IMPs per board, etc.

 

Every pair played 26 boards, except when there were an odd number of pairs when there was obviously a sit out. In those cases 13 of the pairs played 24 boards only. Obviously, their average scores are the sum of their IMP scores divided by 24.

 

Rik

Yes that's the way I interpreted it.

[HighLow21]

Now you are assuming that the variance is entirely random. To see why this can't be correct, imagine that you went from 100,000 boards to a million boards. Surely you wouldn't expect the SD of the mean crossimps per pair to decrease by a factor sqrt(10).

 

I am afraid there is no simple way out of this. The more the field is dominated by "stupid idiots", the more the random variance will be, but the same is probably true for the skill variance.

No, that's correct -- I was assuming that you were drawing from the same pool of results and that all results were random and independent. It helps to assume all pairs are actually of equal skill in this calculation; clearly this isn't the case, but unless the pool contains pairs that are wildly better or worse than the others, it shouldn't matter terribly much.

 

And agreed -- the more rabbits/fools in the pool, the more the standard deviation should increase.

 

The main point of the conversation -- figuring out a rough estimate of how much variation there is, hand-by-hand, seems to stand though. I come to an estimate of 5 IMPs per hand under normal conditions and am fully prepared for the correct answer to be 4 or 6, but not 2 or 10. What do you think?

[HighLow21]

No. Sorry, I should have been clearer.

 

There are 7 columns. Each column represents one night. In each columns is the final result for each evening: the ranking. So, on the first night, the winning pair had an average of 1.62 IMPs per board, number 2 scored 1.28 IMPs per board, etc.

 

Every pair played 26 boards, except when there were an odd number of pairs when there was obviously a sit out. In those cases 13 of the pairs played 24 boards only. Obviously, their average scores are the sum of their IMP scores divided by 24.

 

Rik

This was my response to the earlier average of 1.1 IMPs you gave me. The spreadsheet and table I produced, along with discussion, are based on an understanding of your data that matches what you just said.

[Zelandakh]

And agreed -- the more rabbits/fools in the pool, the more the standard deviation should increase.

Is a pair playing Precision rabbits or fools? What about Polish Club? Playing a different system from others in the room will increase variation. That does not mean that the standard has been decreased.

[HighLow21]

Is a pair playing Precision rabbits or fools? What about Polish Club? Playing a different system from others in the room will increase variation. That does not mean that the standard has been decreased.

That's true too -- anybody playing an unusual system, particularly one against which the opposing partnership doesn't have agreements, will likely increase the volatility of results.

 

Basically, the standard deviation is caused by mistakes--anything that increases the frequency or severity of those mistakes (poor players, unusual systems, very wild distributions, etc.) will increase standard deviation. Similarly, a group of expert/world class players would probaby demonstrate a much lower typical swing per board.

[Zelandakh]

No, when different systems are in play you can get swings without any mistakes having been made at all. Even just changing the NT range from 15-17 to 14-16 will create swings without a mistake being required. A swing is not the same as a mistake.

[Trinidad]

It is important to make a distinction between the IMPs per board average for a given pair in a field and the IMPs per board standard deviation on the board set.

 

When we have a field of pairs, there will be pairs that are stronger than others. There is a range of strengths, and this strength can be quantified, e.g. in the form of an "expected IMPs per board score" for each pair. Each pair has only one given strength, compared to the rest of the field. The whole field is made up of several pairs and this whole field has a "strength distribution". One can model the strength of this field. One can assume, as I did, that the strength of the players is "normally distributed". A normal distribution can be described by two parameters: the average and the standard deviation of the distribution. It is important to realize that this standard deviation is not an error. It is a parameter of the distribution, just like apples on a tree will have a weight distribution. If you weight two apples and you get a different result, it is likely that it is because of the difference in weight of the apples, not because of a random error in the scale.

 

Unfortunately, errors are also expressed as standard deviations. It is a bit like the difference between pound (mass) and pound (force): They have the same name, and there is a relation between the two, but they are fundamentally different things.

 

The errors only come into the discussion once we want to measure the strength of a pair, e.g. in a tournament. And we want to measure them, because we don't know the strength of the pairs. We know that each pair must have a strength, but we just don't know its value. So, we estimate their strength, based on the results of the tournament. And when we estimate, we make errors.

 

In a bridge tournament, there are large random errors in the measurement. This is one of the charms in bridge: There is a realistic probability that Aunt Millie - Uncle Bob beat Meckwell on a single board. It would be premature to conclude that Aunt Millie - Uncle Bob are stronger that Meckwell based on that one board (though I would never deny Aunt Millie and Uncle Bob their moment of glory). The probability that Aunt Millie and Uncle Bob manage that on more boards gets less and less realistic.

 

So, if we play more and more boards, our tournament result will converge towards the true "strength distribution". The error in the estimate (whether due to the variation in the swinginess of boards or random differences caused by kings sitting over or under aces) will get closer to 0. In contrast, the standard deviation in the pair strength distribution will remain unchanged, because the true value of all the pairs' strength doesnot depend on the measurement.

 

So, what we are doing in a bridge tournament is the same as determining the weight distribution of the apples on a windy day with a scale that is jumping up and down in the wind: we have a large error in the measurement of the weight of the individual apples. But if we keep repeating the measurement over and over again, we will get a better estimated value for the weight of each apple. And we can use these to determine the weight distribution of the apples in the tree.

 

In this thread, the question was more or less: the mass of my apple (my bridge strength) from my tree (played in BBO) is x% (x IMPs/bd) larger than the average. What does that mean? I took another tree (bridge club), measured the weight distribution of its apples (measured the strength distribution of the players) and said: "If your tree (BBO) is comparable to mine (my bridge club), it would mean that so many % of the apples (pairs) are lighter (worse) than yours (you)."

 

All this shows that it is perfectly possible to compare apples and pairs... ;)

 

Rik

[awm]

It may be important to keep in mind that IMP expectations are not transitive (i.e. if pair A is +1 IMP/bd against B and B is +1 IMP/bd against C it does not follow that A is +2 IMP/bd against C). In fact due to methods and style you cannot cleanly rank pairs like this. Further, it is difficult to compare scores against a field when the field is very far off from the pairs being compared.

[HighLow21]

No, when different systems are in play you can get swings without any mistakes having been made at all. Even just changing the NT range from 15-17 to 14-16 will create swings without a mistake being required. A swing is not the same as a mistake.

It depends on how you define mistake, doesn't it? Any time a pair fails to get its optimal result on the board, it could be argued that it's a mistake. Many such mistakes will be made routinely by even the best players in the world, but if you define it that way, swing = mistake.

[HighLow21]

It is important to make a distinction between the IMPs per board average for a given pair in a field and the IMPs per board standard deviation on the board set.

 

When we have a field of pairs, there will be pairs that are stronger than others. There is a range of strengths, and this strength can be quantified, e.g. in the form of an "expected IMPs per board score" for each pair. Each pair has only one given strength, compared to the rest of the field. The whole field is made up of several pairs and this whole field has a "strength distribution". One can model the strength of this field. One can assume, as I did, that the strength of the players is "normally distributed". A normal distribution can be described by two parameters: the average and the standard deviation of the distribution. It is important to realize that this standard deviation is not an error. It is a parameter of the distribution, just like apples on a tree will have a weight distribution. If you weight two apples and you get a different result, it is likely that it is because of the difference in weight of the apples, not because of a random error in the scale.

 

Unfortunately, errors are also expressed as standard deviations. It is a bit like the difference between pound (mass) and pound (force): They have the same name, and there is a relation between the two, but they are fundamentally different things.

 

The errors only come into the discussion once we want to measure the strength of a pair, e.g. in a tournament. And we want to measure them, because we don't know the strength of the pairs. We know that each pair must have a strength, but we just don't know its value. So, we estimate their strength, based on the results of the tournament. And when we estimate, we make errors.

 

In a bridge tournament, there are large random errors in the measurement. This is one of the charms in bridge: There is a realistic probability that Aunt Millie - Uncle Bob beat Meckwell on a single board. It would be premature to conclude that Aunt Millie - Uncle Bob are stronger that Meckwell based on that one board (though I would never deny Aunt Millie and Uncle Bob their moment of glory). The probability that Aunt Millie and Uncle Bob manage that on more boards gets less and less realistic.

 

So, if we play more and more boards, our tournament result will converge towards the true "strength distribution". The error in the estimate (whether due to the variation in the swinginess of boards or random differences caused by kings sitting over or under aces) will get closer to 0. In contrast, the standard deviation in the pair strength distribution will remain unchanged, because the true value of all the pairs' strength doesnot depend on the measurement.

 

So, what we are doing in a bridge tournament is the same as determining the weight distribution of the apples on a windy day with a scale that is jumping up and down in the wind: we have a large error in the measurement of the weight of the individual apples. But if we keep repeating the measurement over and over again, we will get a better estimated value for the weight of each apple. And we can use these to determine the weight distribution of the apples in the tree.

 

In this thread, the question was more or less: the mass of my apple (my bridge strength) from my tree (played in BBO) is x% (x IMPs/bd) larger than the average. What does that mean? I took another tree (bridge club), measured the weight distribution of its apples (measured the strength distribution of the players) and said: "If your tree (BBO) is comparable to mine (my bridge club), it would mean that so many % of the apples (pairs) are lighter (worse) than yours (you)."

 

All this shows that it is perfectly possible to compare apples and pairs... ;)

 

Rik

 

Good explanation. I think the most important point that follows is that some of the variation you'll see in the pooled results will be caused by randomness, and some will be caused by the relative strength/weakness of individual pairs. The more variation in talent across the pairs, the higher the standard deviation of the observed results. The randomness component will, of course, always be there irrespective of the relative quality of the pairs.

[Zelandakh]

It depends on how you define mistake, doesn't it? Any time a pair fails to get its optimal result on the board, it could be argued that it's a mistake. Many such mistakes will be made routinely by even the best players in the world, but if you define it that way, swing = mistake.

OK, take a classic. Slam is 50%. One pair bids it and the other stays in game. A swing is guaranteed. Which pair has made the mistake? If it depends on the cards of the opponents then you are on fairly slippery ice with your definition.

[HighLow21]

OK, take a classic. Slam is 50%. One pair bids it and the other stays in game. A swing is guaranteed. Which pair has made the mistake? If it depends on the cards of the opponents then you are on fairly slippery ice with your definition.

I see your point, and I think you see mine. One is a mistake regardless of the result, and the other is a mistake given the result. The argument is starting to become semantic, and remember -- the original question revolved around empirical observations of IMP score variation per hand (or per set).

 

My point is that any factor which leads to more swings--different bidding systems in play, different levels of ability among the pairs, trickier hands, etc.--will obviously correlate to a higher standard deviation of IMP results per hand. Whether you define mistake as "egregious error" or "deviation from par result", at least a large part of the variation will be due to mistakes. And more mistakes will be made when you're playing against a bidding system you aren't familiar with or don't have an effective defense against.

[Trinidad]

It may be important to keep in mind that IMP expectations are not transitive (i.e. if pair A is +1 IMP/bd against B and B is +1 IMP/bd against C it does not follow that A is +2 IMP/bd against C). In fact due to methods and style you cannot cleanly rank pairs like this. Further, it is difficult to compare scores against a field when the field is very far off from the pairs being compared.

That is true.

 

However, within a field you can rank them and assign a strength: It would be the result if everybody would play the same, infinite amount of boards against everybody.

 

When you then take the top 10 pairs of that field (or any other subset) and do the same exercise on that smaller field, it is entirely possible that their ranking will be different from the original one.

 

But that doesn't mean that you can't rank within a given field.

 

Rik

[barmar]

OK, take a classic. Slam is 50%. One pair bids it and the other stays in game. A swing is guaranteed. Which pair has made the mistake? If it depends on the cards of the opponents then you are on fairly slippery ice with your definition.

Here's a different way to think about it.

 

At IMPs, you're supposed to bid 50% games. If game is 50%, but you don't bid it, you've probably made a mistake (perhaps your bidding methods are poor and don't allow you to determine the odds, or you had the information but used poor judgement in applying it). About half the time you won't be punished, because of the lie of the cards, but if you bid like this in general, your long term results will suffer because the scoring table awards bidding the games more than staying out of them.

 

The converse of this is that a bidding misunderstanding lands you in a horrible contract, but you find a lucky lie of the cards, or the opponents misdefend, so you make it. Regardless of the results, you still made a mistake during the bidding, although you may have compensated for this with your strong declarer play, or demeanor that didn't give away the show to the defenders (perhaps that's why they misdefended).

[Zelandakh]

In truth this is not a good example for this thread for a completely different reason, namely that the incidence of such hands is constant in the long run and can therefore be modelled. The times when a small system difference leads to a different end contract without any mistake having been made are a better example because that depends purely on how many pairs are playing alternate systems. What I am trying to show is that making a decision that does not lead to the "optimsl" contract does not mean a mistake has been made. Often enough better bidding is rewarded with a negative swing because of the distribution od defenders' cards. That is not a mistake, although it is often seen as such. Often enough that there is a word for it on these forums - resulting. Trying to give resulting a statistical basis and thereby some quasi-mathematical endorsement is something I think should be shot down quickly. There are some benefits to comparing long-term results against par but no benefit or justification whatsoever in calling good actions that happened to produce a negative swing mistakes.

[WellSpyder]

Good explanation.

Yes indeed! I upvoted trinidad's post for the clarity of the statistical explanation rather than for the light-hearted touch at the end - though I enjoyed that, too.

[Fluffy]

My friends averaged +3.9 IMps/board between the 2 tables at a tournament this weekend, they scored more than 160 VPs from a maximum of 180 in the 20 VP escale 9 rounds (there was a bye, so actually the maximum was less)

 

http://www.aebridge....dos/buttler.pdf

 

The average hand was to win 4 IMPs or so lol.

[Zelandakh]

Sign them up to your team for a run at the next Bermuda Bowl, Fluffy!