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The "Law of Averages" (1 Viewer)

There are about three or four different discussions going on here, some of which are useful (correlation of week-to-week performance), and some of which are not (jurb desperately trying to change the subject from his original argument to something more sensical).

The bottom line is that Maurice and others are right about the Gambler's Fallacy. How much it really applies to fantasy football is up for debate, of course. I could even see the argument that a Smith owner who had his week one matchup in the bag would want him to do poorly -- fewer touches means less chance of injury, and if Smith plays badly he's less likely to be the focus for defensive co-ordinators the rest of the month.

But to argue that you should root against him so he doesn't "use up" his projections is just wrong. That's where the fallacy comes in, which is all people were trying to point out.
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I have not changed my stance on this one bit. There are just too many different topics and subtopics going on in this thread right now. I contest only that every player regardless of who he is is bound to have both good and bad weeks through the course of a 16 game season. I can only hope that that bad or subpar week comes this week as I have sured up a win anyway. I simply would not like to see it happen when the performance is more of a need to my specific team needs. What is so difficult about this? Just because you make a projection does that suddenly blind you of the reality that there are ups and downs to every palyer and team through out the course of a year? If Smith blows up tonight for 150/2, what good did that do my team this week? None what so ever. I think the man is talented enough to approach numbers of around 1300/10. I make this projection knowing full well taht good games and bad games are LIKELY though the year. If Smith has to have a bad game, why then would I given my current situation, not want for it to be during a week that I have already won anyway?
 
Let me try this and see if it helps people to better understand my position. Out of a 16 week season. Of all of those 16 games, I hope that this is the lowest scoring week of the 16 for S.Smith. In other words, if Smith ends up having at the end of the year his worst game being:1rec 10 yds, 0 tdsor if it is: 5 rec, 100 yds, 1 tdSo long as this ends up being his worst performance of the year I will be relatively happy.

 
...it is only after an infinite number of flips that the even split will occur.
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This is part of the same fallacy. As the number of coin flips goes to infinity, the margin of error goes to zero on a percentage basis. However, the expected absolute error increases the more you flip the coin--all the way out to infinity. In fact, the probability of the expected number of heads (m) equalling exactly half the total number of flips (n/2) approaches zero as n approaches infinity.
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note: this post is kind of off topicok, this just totally blew my mind....so i spent the last hour reading about probability--something I haven't done in about 8 years.

Here is how I thought the coin flip probability worked:

If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio NH/N.

As N gets larger and larger, we expect that in our example the ratio NH/N will get closer and closer to 1/2. This allows us to define the probability Pr(H) of flipping heads as the mathematical limit, as N approaches infinity, of this sequence of ratios:

Pr(H)=lim NH/N

i wasn't able to find anything about the existance of errors in "fair" coin tosses, but what you said make intuitive sense to me. I think my brain is going to melt though, i haven't looked at this stuff in ages.

someone was asking earlier about something new in coin-toss probability:

Here's what I found.

 
i wasn't able to find anything about the existance of errors in "fair" coin tosses, but what you said make intuitive sense to me. I think my brain is going to melt though, i haven't looked at this stuff in ages.
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Infinities are really hard, even for people who are good at math.It's easier to use N=1, N=5, N=10, and N=20 and then extrapolate from the trend.

If you flip a coin just once, you will expect to get 0.5 heads. But you will also expect your error to be 0.5 (or 100% of your expectation). That's because you know you're either going to get 0 heads or 1 head; you can't possibly get 0.5 heads, so you're going to be off by 0.5.

If you flip a coin five times, you will expect 2.5 heads. You will be off by either 0.5, 1.5, or 2.5 (all equally likely) for an expected error of 1.5 (or 60% of your expectation). Edit: "all equally likely" is wrong, and therefore so is 60% -- it's lower than that.

So as we flipped more coins, the absolute value of the expected error went up, but the error as a percentage of expectation went down.

If you flip a coin 10 times, you will expect 5 heads . . . and again the expected error will go up in absolute terms but down in percentage terms. That keeps happening as we raise N. The bigger N gets, the bigger our expected error gets.

 
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Man, if this thread had an I.Q. score it would be over 130!Personally in regards to fantasy football I stick to Newton's first law of motion is often stated as:An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

 
i wasn't able to find anything about the existance of errors in "fair" coin tosses, but what you said make intuitive sense to me.  I think my brain is going to melt though, i haven't looked at this stuff in ages.
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Infinities are really hard, even for people who are good at math.It's easier to use N=1, N=5, N=10, and N=20 and then extrapolate from the trend.If you flip a coin just once, you will expect to get 0.5 heads. But you will also expect your error to be 0.5 (or 100% of your expectation). That's because you know you're either going to get 0 heads or 1 head; you can't possibly get 0.5 heads, so you're going to be off by 0.5.If you flip a coin five times, you will expect 2.5 heads. You will be off by either 0.5, 1.5, or 2.5 (all equally likely) for an expected error of 1.5 (or 60% of your expectation).So as we flipped more coins, the absolute value of the expected error went up, but the error as a percentage of expectation went down.If you flip a coin 10 times, you will expect 5 heads . . . and again the expected error will go up in absolute terms but down in percentage terms. That keeps happening as we raise N. The bigger N gets, the bigger our expected error gets.
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I finally get it.It's not that the the 100 initially tossed heads will ever be "made-up" in the absolute sense, but only that as the number of total tosses increases the 100 initially tossed heads becomes a smaller percentage of the total tosses. So at 100 million tosses the percent of heads and tails tossed may be almost 50% each but the actual difference in real tosses may be over some 500 tosses for each.
 
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I tend to agree with Jurb.Though I have absolutely no science, math, greek or art history to back it up, when I have a game in hand, I tend to "root" for my remaining players to have subpar performances.

 
I have not changed my stance on this one bit. There are just too many different topics and subtopics going on in this thread right now. I contest only that every player regardless of who he is is bound to have both good and bad weeks through the course of a 16 game season. I can only hope that that bad or subpar week comes this week as I have sured up a win anyway. I simply would not like to see it happen when the performance is more of a need to my specific team needs. What is so difficult about this? Just because you make a projection does that suddenly blind you of the reality that there are ups and downs to every palyer and team through out the course of a year? If Smith blows up tonight for 150/2, what good did that do my team this week? None what so ever. I think the man is talented enough to approach numbers of around 1300/10. I make this projection knowing full well taht good games and bad games are LIKELY though the year. If Smith has to have a bad game, why then would I given my current situation, not want for it to be during a week that I have already won anyway?
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Rightly or wrongly, you make it sound like you think that Smith will be less likely to have good games in the future if he has a big game tonight. You also make it seem like you believe that Smith having a bad game tonight will help you in some way. Both seem like high-level examples of the gambler's fallacy.It's quite possible that that's not what you mean at all -- after all, this whole thread started with someone else representing what they thought you meant. In any case, it was the springboard for an interesting debate about probabilities.
 
I have not changed my stance on this one bit.  There are just too many different topics and subtopics going on in this thread right now.  I contest only that every player regardless of who he is is bound to have both good and bad weeks through the course of a 16 game season.  I can only hope that that bad or subpar week comes this week as I have sured up a win anyway.  I simply would not like to see it happen when the performance is more of a need to my specific team needs.  What is so difficult about this?  Just because you make a projection does that suddenly blind you of the reality that there are ups and downs to every palyer and team through out the course of a year?  If Smith blows up tonight for 150/2, what good did that do my team this week?  None what so ever.  I think the man is talented enough to approach numbers of around 1300/10.  I make this projection knowing full well taht good games and bad games are LIKELY though the year.  If Smith has to have a bad game, why then would I given my current situation, not want for it to be during a week that I have already won anyway?
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Rightly or wrongly, you make it sound like you think that Smith will be less likely to have good games in the future if he has a big game tonight. You also make it seem like you believe that Smith having a bad game tonight will help you in some way. Both seem like high-level examples of the gambler's fallacy.It's quite possible that that's not what you mean at all -- after all, this whole thread started with someone else representing what they thought you meant. In any case, it was the springboard for an interesting debate about probabilities.
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Agreed. The only thing I have been trying to contest this whole time though is this: Players will have high weeks and low weeks. Given my current situation I can only hope that for Smith that low week is today rather than down the road when I MAY or MAY NOT need the points for a 'W'. I don't like the gambler fallacy in its use with FF, which I have stated. I don't in any way though think that the proven fact that players will have good days and bad days falls into that subset though. Maybe that is what you guys mean by this fallacy though and I'm just missing your point(s), I don't know. Either way it deals with events with little to no variables and no human elements. Football does. For that major reason I find its use in FF to be flawed.
 
I don't in any way though think that the proven fact that players will have good days and bad days falls into that subset though. Maybe that is what you guys mean by this fallacy though and I'm just missing your point(s), I don't know.
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The fallacy is in thinking that the number of good days and the number of bad days are fixed ahead of time, such that a good day today makes a bad day tomorrow more likely, or vice versa.
 
I don't in any way though think that the proven fact that players will have good days and bad days falls into that subset though. Maybe that is what you guys mean by this fallacy though and I'm just missing your point(s), I don't know.
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The fallacy is in thinking that the number of good days and the number of bad days are fixed ahead of time, such that a good day today makes a bad day tomorrow more likely, or vice versa.
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I'm not thinking that it is fixed though. Only that Smith is good enough and talented enough to not let that happen very often. Hence the human element and were this fallacy has no place in FF. Of course that assumption on my part is subjective though. However it does not fall into the gamblers fallacy IMO.
 
...and the correlation is positive, not negative.
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This is true, but isn't it so partially due to the fact that nearly all if not all of the data points are positive? If you started with a benchmark other than 0, say 10 points as the floor for RB scoring or something, we could see some negative correlations don't you think?
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I don't know what benchmark you used, but it shouldn't be zero points.The predicting formula will bey = mx + bwhere y = "weeks 2-15 PPG", x = "week 1 points", and m and b are whatever optimize the accuracy of that equation. (Non-linear equations might produce better results, but lets keep it simple.)When I claim that week 1 and weeks 2-15 are positively correlated, I'm just saying that m will be positive for the best values of m and b. I would certainly not expect the best value of b to be zero. (b will almost certainly also be positive, and m will be between 0 and 1.)(Excel uses regression analysis to find the best-fitting line; I assumed that's how you did it, but maybe not.)
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I fully understand the regression equation, but that was not my point.I was addressing your negative correlation remark. In order for something to have a negative correlation, it would have to move in the opposite direction as predicted. In the case we have been working with, that typically isn't possible since fantay players always score from 0 up.For example,I predict Steve Smith to score 1,000 points this season. If he scores 1, 10, 100, 500 or 1,000, you'd have somewhere between 0 correlation and 1. In other words, not negative.That's not saying much though IMO to say there is no negative correlation as ANY positive score from 1 - 1000 would fit that model. If you set the benchmark at 10, and reset all scores to X - 10, you could then start to see some negative correlations.I understand what you are saying, I was just going to take this a step further. Without negative fantasy scores, you simply won't have - correlation.
 
Without negative fantasy scores, you simply won't have - correlation.
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Not true at all.Consider:X[1] = 10, X[2] = 20, X[3] = 30, X[4] = 40Y[1] = 120, Y[2] = 100, Y[3] = 80, Y[4] = 60There are no negative values there, but X and Y are negatively correlated. As X gets bigger, Y gets smaller. (Specifically, Y = -2x + 140.)
 
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Without negative fantasy scores, you simply won't have - correlation.
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Not true at all.Consider:X[1] = 10, X[2] = 20, X[3] = 30, X[4] = 40Y[1] = 120, Y[2] = 100, Y[3] = 80, Y[4] = 60There are no negative values there, but X and Y are negatively correlated. As X gets bigger, Y gets smaller. (Specifically, Y = -2x + 140.)
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You are right, but only partly so. There is only 1 prediction for the player's annual points, not 4 as you state. There aren't 16 Y's, just one. That's were the difference lies.Y = X + b1 + b2 + bn + eAny positive prediction with positive points scored is a positive correlation.Your illustration isn't applicable as far as I can tell for this purpose. If I had a weekly prediction and a weekly result, you are right, but that's not what I thought we were talking about.
 
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You are right, but only partly so. There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
 
You are right, but only partly so. There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
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Run them all and find me one with a negative correlation, it's not going to happen.
 
You are right, but only partly so.  There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
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Run them all and find me one with a negative correlation, it's not going to happen.
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:confused: ?If you plot all the Xs and Ys on a graph, it will be a scattering of data points that looks like a big blob, but the blob won't be a disc. It'll be a bit oblong, kind of smushed from the upper left and bottom right. In other words, the trendline will slope upwards and to the right.There will, however, be some points that appear in the upper left and lower right quadrants -- i.e., there will be some players who did better than average in week 1 and worse than average in weeks 2-15 or vice versa. Is that what you mean? Name one? Okay, LaDainain Tomlinson.
 
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You are right, but only partly so. There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
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Run them all and find me one with a negative correlation, it's not going to happen.
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The correlation calculation goes like this; take the first game performance of all starting NFL WRs in 2003; that is your X. Then take stats for week 2-17 for the same group; that is your Y. The hypothesis is that X is positively correlated with Y; that players who performed well in week 1 were more likely to perform well in weeks 2-17, and players who performed poorly in week 1 were more likely to perform poorly in weeks 2-17. I agree that this seems so obvious as to not be worth putting in the effort to do the calculation. There will not be a negative correlation, because these quantities are positively correlated. Whether the correlation is negative or not has nothing to do with whether the baseline is zero or not; if week 1 top performers tended to do more poorly than week 1 duds over the rest of the season, this calculation would result in a negative number.
 
You are right, but only partly so.  There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
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Run them all and find me one with a negative correlation, it's not going to happen.
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:confused: ?If you plot all the Xs and Ys on a graph, it will be a scattering of data points that looks like a big blob, but the blob won't be a disc. It'll be a bit oblong, kind of smushed from the upper left and bottom right. In other words, the trendline will slope upwards and to the right.There will, however, be some points that appear in the upper left and lower right quadrants -- i.e., there will be some players who did better than average in week 1 and worse than average in weeks 2-15 or vice versa. Is that what you mean? Name one? Okay, LaDainain Tomlinson.
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Show me the equation that works for LT.
 
You are right, but only partly so.  There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
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Run them all and find me one with a negative correlation, it's not going to happen.
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:confused: ?If you plot all the Xs and Ys on a graph, it will be a scattering of data points that looks like a big blob, but the blob won't be a disc. It'll be a bit oblong, kind of smushed from the upper left and bottom right. In other words, the trendline will slope upwards and to the right.There will, however, be some points that appear in the upper left and lower right quadrants -- i.e., there will be some players who did better than average in week 1 and worse than average in weeks 2-15 or vice versa. Is that what you mean? Name one? Okay, LaDainain Tomlinson.
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Show me the equation that works for LT.
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The best-fitting equation is the one that produces the smallest overall error (for all players considered). But the error will be on the large side for LT because he had a poor week 1 and a good week 2-17.
 
You are right, but only partly so.  There is only 1 prediction for the player's annual points, not 4 as you state.
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There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
Click to expand...
Run them all and find me one with a negative correlation, it's not going to happen.
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:confused: ?If you plot all the Xs and Ys on a graph, it will be a scattering of data points that looks like a big blob, but the blob won't be a disc. It'll be a bit oblong, kind of smushed from the upper left and bottom right. In other words, the trendline will slope upwards and to the right.There will, however, be some points that appear in the upper left and lower right quadrants -- i.e., there will be some players who did better than average in week 1 and worse than average in weeks 2-15 or vice versa. Is that what you mean? Name one? Okay, LaDainain Tomlinson.
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Show me the equation that works for LT.
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Do you think each player gets his own equation? That's not how it works.
 
You are right, but only partly so.  There is only 1 prediction for the player's annual points, not 4 as you state.
Click to expand...
There are a zillion of them.If you take the top 50 QBs, RBs, WRs, TEs, PKs, and DTs from 2003, there are 300 sets of Xs and Ys to look at. If you go back more seasons, there are even more.
Click to expand...
Run them all and find me one with a negative correlation, it's not going to happen.
Click to expand...
:confused: ?If you plot all the Xs and Ys on a graph, it will be a scattering of data points that looks like a big blob, but the blob won't be a disc. It'll be a bit oblong, kind of smushed from the upper left and bottom right. In other words, the trendline will slope upwards and to the right.There will, however, be some points that appear in the upper left and lower right quadrants -- i.e., there will be some players who did better than average in week 1 and worse than average in weeks 2-15 or vice versa. Is that what you mean? Name one? Okay, LaDainain Tomlinson.
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Show me the equation that works for LT.
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Do you think each player gets his own equation? That's not how it works.
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No kidding, you are the one that said LT not me.Let's do it this way.Show my a negative correlation as I all can discover are positives.
 
No kidding, you are the one that said LT not me.
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I named LT as an example of a player who did worse than average in week 1 and better than average in weeks 2-17. He is, in fact, an example of that.
Show my a negative correlation as I all can discover are positives.
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In single males aged 24-40, body fat percentage is negatively correlated with number of female sex partners.
 
No kidding, you are the one that said LT not me.
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I named LT as an example of a player who did worse than average in week 1 and better than average in weeks 2-17. He is, in fact, an example of that.
Show my a negative correlation as I all can discover are positives.
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In single males aged 24-40, body fat percentage is negatively correlated with number of female sex partners.
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:brush: I'll take that.Obviously I can find endless cases of - correlation, just don't see it here.Thanks for the debate :D
 
A single player is a point, not a line. One X (Week 1 yards), and One Y (Avg Yards Weeks 2-16). So, there is no way to show a "negative correlation" when there is nothing to which to correlate this input. So, you need at least 2 players to make a line.Obviously, the correlation is positive, taking all players into account. All else equal, average yards per game for weeks 2-16 increase as week 1 yards increase. This is certainly not a perfect correlation, but positive nonetheless.But you can take a limited sample of players and create a negative correlation. Plug in the numbers of LaDainian Tomlinson and Trung Canidate (using the season average in place of avg yards weeks 2-16), and you'll get a correlation of -4.

 
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OK, so if we expect Steve Smith to break one leg during the season, and he breaks it in week one, does that have a positive or a negative impact on his projections the rest of the way?

(Ouch... apparently the fantasy gods still believe in karma.)

 
My, my, this really did get silly at some point.Let me say that when a good fantasy player has a bad week 1:(a) It is likely that they will improve, regressing toward their mean of past performances. On average. No reason to panic.(b) Statistically one still has alter future projections downward even if only a small amount. Many of the reasons why a player did worse week 1 (a poorer O-line this year, etc.) are still there. To ignore the game 1 data poitn woudl be just as illogical as ignoring all of the past games from previous seasons.© Hoping a fantasy player on my team does poorly is really nonsenesical. I want every single one to do balls-out great. It improves their value and increases my ability to improve my team through trading. Even if I have already won that week's game, even if they are on my bench. (d) There are lots and lots of negative correlations in fantasy football (age of starting RB and fantasy points per game). But the one between week 1 and week 2 is postive and fairly robust. I once ran a bunch of these and was like .63 or something. This goes down as you correlate with later weeks more distally removed but almost as big for the mean of all later weeks (because the composite is more reliable than an individual game). This pattern is known as a simplex and is found for many measures of performance, such as job performance (Deadrick & Madigan, 1990; Hofmann, Jacobs, & Baratta, 1993), academic performance (Humphreys, 1968), and sport performance (Hofmann, Jacobs, & Gerras, 1992). So week 1 helps you predict week 2 and such things as WDIS.

 
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