The Bell Curve and Fantasy Football

Note: this article was written prior to the 1998 season (while Brad Johnson was playing for the Vikings). It was posted at the FanEx site for awhile so you may have seen it. A few of the names may have changed, but you'll still find the information useful, I hope. In addition, I've added some lists of the most and least consistent performers of the last five years.

Most fantasy football veterans have a strong opinion on the merits of the "hookup." If you've got Brad Johnson as your QB, would you rather have Cris Carter or an equally talented non-Viking (say Tim Brown, maybe) as one of your WRs? The pro-hookup argument is simple, and appeals to the inherent greed in all of us: every Johnson-to-Carter TD gets you double points. Who doesn't want that? Those two will connect for at least 9 or 10 scores in a given year, and it's tempting to want to capitalize on that.

The anti-hookup crowd will at this point note that depending on a combo tends to put a lot of emphasis on one team. Should that team sputter, or decide to get serious about the running game, or run into a snowstorm or a really tough defense, you've got not one but two key contributers contributing little.

The Switzerland position in this conflict, and the one I had always taken until the middle of last season, is that combos should be neither sought nor avoided. If Carter and Johnson hook up for 2 TDs, then having Carter and Johnson is better than having Tim Brown and Johnson if Brown scores only one TD, but it's not as good as having Brown and Johnson if Brown scores 3. I want to maximize my total amount of TDs. I don't care whether they happen on the same play or several thousand miles apart.

Last season, it was pointed out to me that it's not my total points that I really want to maximize, it's my probability of victory. It would seem that these two goals are synonymous, but that is not always the case. A combo improves your chances of hitting the jackpot, but it also improves your chances of crashing and burning. It's a high-risk, high-reward strategy. Underdogs should use these kinds of strategies. Favorites should not.

For the rest of this article, I will mathematically formalize the ideas described in the paragraph above. Is this just theoretical mumbo-jumbo, or can it make a big difference in a real live fantasy football game? How can you tell how high the risk is, and how high the reward might be? How much of an underdog do you really need to be in order for it to make sense to play a worse WR for the sake of capitalizing on a hookup?

The method used is what statisticians would call a correlated Gaussian. (For the record, I can't take credit for applying a correlated Gaussian to sports. I stole this idea from Dean Oliver and his terrific basketball analysis page.) Let's take this slow. We'll get to "correlated" in a minute. What the heck is "Gaussian?" Gaussian (which is a noun, not an adjective) is just another word for a standard normal distribution, and a standard normal distribution is more commonly known as a bell curve. "Correlated" means that we'll be looking at two quantities which may or may not be related.

In this case, the two quantities we'll be looking at are your fantasy football team's score and your opponent's score. OK, I've got two quantities that may or may not be related, so how does the bell curve enter into the picture? Well, it's a reasonable assumption that, if two fantasy football squads were to square off 100 or 1000 or 10,000 times (theoretically, of course) that the distribution of the difference between the two teams' scores would be shaped like a bell curve. Suppose one of the two teams was, in actuality, 10 points per week better than the other. Does that mean that they'll win by 10 every time? Of course not. Sometimes they'll win by 25. Sometimes they'll lose by 5. They'll win by 12 a lot more often than they'll win by 60, and they'll win by 5 about as often as they win by 15 (remember, they're 10 points better).

If you were to plot how often they win or lose by each margin, you'll see a bell curve with the hump centered right at +10. That's the bell curve we're interested in. In particular, we're interested in getting as much of that curve as possible on the positive side of 0. Remember, in plain English, "the difference between my score and my opponent's score is greater than zero" means "I win!"

I mentioned above that our two quantities (the two teams' scores) may or may not be related. How's that? This is the essence of the "anti-hookup," the strategy of choosing a WR from the same team as your opponent's QB in hopes of cancelling him out. I realized that the correlated Gaussian, because of the "correlated" aspect, was perfect for analyzing this strategy as well. Another issue on which this method can shed some light is the issue of consistency. Many fantasy footballers prefer to have consistent-but-unspectacular performers (like Emmit Smith was last year) rather than streaky players (like Corey Dillon last year) who could win you a game single-handedly or give you a goose-egg. We'll look at that as well.

Before I can go further, I need to tell you exactly how the correlated Gaussian works. Essentially, what it does is to look at all your scores, look at all your opponents scores, and calculate the probability that you're going to win. That's the basic idea. If you trust me, and you don't want to slog through some math, you might want to skip this paragraph. For the skeptics and mathematical zealots that are still with us, I'll go through the details. Say X is a random variable denoting my score, and Y is a random variable denoting your score. Then the probability that X is greater than Y (i.e. the probability that I win) is given by:

               _                                                         _
              |               (average of X) - (average of Y)             |
P(X>Y) = NORM | ----------------------------------------------------------|
              |_SQRT[(variance of X) + (var of Y) - 2*(covar of X and Y)]_|

where NORM(Z) means the probability that a normally distributed variable with mean 0 and standard deviation 1 is less than Z. If you don't remember how to compute variance, covariance, etc, dust off the old stat books (or learn how to make a spreadsheet do it for you). The thing to remember is that the result is greater than 50 0f the quantity in brackets (which I'll call z) is positive and less than 50 0f z is negative. The bigger z is, the closer the result is to 100%. The bigger z is in the negative direction, the closer the result is to 0%.

A few quick things to notice about z. First, the denominator is always positive, so z will be positive if and only if the numerator is. In fantasy football terms, this means that your chance of winning is greater than 50 0f your team has a higher scoring average than your opponent's team. Don't need a fancy formula to tell us that.

Next, note that a bigger variance for either team makes the denominator bigger, which makes z closer to zero, which in turn pushes the probability closer to 50%. The lesson here is that inconsistency is a levelling force. If you have a good team, you want them to be rock-like. If you have a bad team, you want them to be wild and crazy. This is mathematical justification of something I alluded to earlier: underdogs should pursue high-risk, high-reward strategies, favorites should be conservative.

Finally, note that a high (positive) covariance decreases the denominator, which forces z away from zero. In other words, if your team and your opponent's team are connected in some way (such as if you have Scott Mitchell and your opponent has Herman Moore and Johnny Morton), then this favors the better team. That is, the "anti-hookup" is a wise move for a favorite, and a strategical blunder for an underdog.

Again let me stress that this is all theoretical. It is not at all clear at these considerations should cause us to deviate from a "play the guy you think is going to score the most points" strategy. In order to see, we need to look at some actual data, so that's just what we'll do.


Now let's take a look at what kind of variances we can expect out of an actual fantasy football squad. Using 1997 stats and the scoring system from one of my leagues (all TDs = 6, 10 yds rush/rec = 25 yds pass = 1 pt, INT = -3 pt.), I identified four teams with different profiles. First, two good teams, one consistent, and one inconsistent. The numbers in parentheses are the variance of that particular individual's fantasy scoring. Remember, higher variance means more inconsistent.

      Consistent                        Inconsistent
Player         Avg  (var.)          Player         Avg  (var.)
J George      18.4  (69)            J Elway       16.6  (94)
J Bettis      13.9  (56)            N Kaufman     12.9  (68)
M Faulk       11.8  (51)            C Dillon      11.9 (139)
B Coates       7.1  (13)            R Dudley       7.1  (41)
C Carter      11.0  (46)            Y Thigpen     11.0  (78)
D Alexander    9.1  (30)            J Reed         8.9  (64)

The consistent team's total fantasy scoring has a variance of 153, and the inconsistent team's variance is 544. Both teams average around 70 points per week.

Now let's build two bad teams. Again, one consistent and one inconsistent:

      Consistent                        Inconsistent
Player         Avg  (var.)          Player         Avg  (var.)
D Marino      12.9   (56)           S Young       13.3   (97)
E Smith        9.0   (19)           J Stewart      8.2   (89) 
J Anderson    11.0   (37)           Rob Smith     11.1   (63)
T McGee        4.4   (17)           M Chmura       4.4   (25)
M Jackson      6.9   (23)           I Bruce        6.7   (97)
R Brooks       8.6   (24)           J Jett         8.7   (47)

Both teams averaged about 53 points per week. The consistent team had a variance of 69 and the inconsistent team had a variance of 372.

The purpose of this was to build teams which had extreme variance (or extreme lack of variance). You're not likely to see a more consistent actual fantasy football team than the two we've built here, nor are you likely to run across a flakier team than those you'll find above. So at this point I'm going to abandon these teams and build some theoretical teams, which I'll call Team GI (good, inconsistent), Team GC (good, consistent), Team BI (bad inconsistent), and Team BC (not Boston College -- bad, consistent, but you probably already figured that out).

We're moving from the real world to a theoretical one to eliminate some covariance that we don't want right now (for example, Jett is bad and inconsistent, while Jeff George is good and consistent, that could've skewed the estimates that we'll get shortly). Also, we can now make sure that our teams have exactly the same ability level. Here are the precise specs of our four squads:

Team      Avg.  Var.
GI         70   550
GC         70   150 

BI         53   372
BC         53    65

The question now is, what are the chances that one of the bad teams can beat one of the good teams (assuming no covariance)? Glad you asked.

BI will beat GI  28.80f the time
BC will beat GI  24.70f the time

BI will beat GC  22.90f the time
BC will beat GC  12.30f the time

Lesson 1: either of these teams is significantly more likely to beat the good inconsistent team than they are to beat the good consistent team despite the fact that the two good teams are of identical quality. In other words, if you're good, you want to be consistent as well.

Lesson 2: the bad inconsistent team has a significantly greater chance of beating either good team than does the bad consistent team. Playing around with the numbers a bit, we find that, with a variance of 65, you would need an average of over 59 points per game to have a 22.9hance of beating the good consistent team. So because of their inconsistency, when playing Team GI, Team BI has a better chance than a consistent team which is 6 points better on average!

If your team stunk last year, they could've been helped a lot more by Isaac Bruce and James Stewart than Michael Jackson and Emmitt Smith, despite the fact that the overall stats of the above duos are nearly identical. Though the difference is not huge, it's not insignificant either.

The Hookup

Feast your eyes on these two teams:

Team 1A:  S Mitchell, J Bettis, J Anderson, H Moore, J Morton, F Wycheck
Team 1B:  S Mithcell, J Bettis, J Anderson, Y Thigpen, T Owens, F Wycheck

Team 1A averages 62.4 points per game, and Team 1B averages 62.8 (Let's do some creative rounding and call them both an even 63). The difference, of course, should be in the variance. Team 1A features the all-or-nothing Mitchell and not one but two hookups, Moore and Morton. OK, so how much difference is there in the variances? Team 1A has a variance of 385.6. Team 1B is at 289.7. How much difference does that make? Let's have these two teams square off against Team G and Team B, a good and a bad team that are neither particularly consistent nor particularly inconsistent.

Team 1A will beat Team G  39.80f the time
Team 1B will beat Team G  39.10f the time

Team 1A will beat Team B  65.80f the time
Team 1B will beat Team B  67.10f the time

The hookup makes you more vulnerable to an upset but gives you a better chance of pulling off an upset of your own. However, the difference is tiny. Let's try it again.

Team 2A:  J George, K Ab-Jabbar, E Smith, R Dudley, T Brown, J Jett
Team 2B:  J George, K Ab-Jabbar, E Smith, B Coates, M Irvin, K McCardell

Both teams average 66 points a week. The variances are 255 for Team 2A and 182 for Team 2B. Again, not nearly enough to make any difference.

Team 2A will beat Team G  43.50f the time
Team 2B will beat Team G  43.10f the time

Team 2A will beat Team B  72.50f the time
Team 2B will beat Team B  74.20f the time

This next one is very surprising:

Team 3A:  B Favre, R Watters, E George, R Brooks, A Freeman, M Chmura
Team 3B:  B Favre, R Watters, E George, Rod Smith, J Jett, T Drayton

In this case, both teams have the same average *and* the same variance. This is a testament to Favre's ability to spread the ball around. The point is that stocking up on Packers is no better or worse than stocking up on equally good non-Packers.

I tried various other things, like switching out the randomly-selected RBs for different ones and using only one of a team's WRs instead of both (and the TE, in 2 of the cases), but none of these changes altered the variance by enough to make any appreciable difference.

Moral of the story: the hookup is completely irrelevant. Don't seek it out and don't avoid it.

The Anti-Hookup

This one is a bit stickier, because we can no longer avoid the covariance term. Suppose your opponent has Elway and not much else. His lineup looks like this

Team BE (bad, Elway): Elway, W Dunn, Murrell, Drayton, Emmanuel, Alexander   

You have the following guys pencilled into your lineup: Bledsoe, Bettis, Kaufman, and Cris Carter. You have to choose one more WR and have Rod Smith and Antonio Freeman to choose from. At the TE slot, you can opt for Rickey Dudley or Shannon Sharpe. In both spots, you can expect the same average production. Do you want to go with the Bronco pair or not?

I mentioned above that, if you think you have the better team (which in this case you obviously do), you should go for the anti-hookup. Play Smith and Sharpe. Let's call that team Team AH (anti-hookup), and the Dudley-Freeman team Team DF (for lack of a better acronym).

As expected, the covariance between Team BE and Team AH is greater than the covariance between Team BE and Team DF. The bottom line is as follows:

Team AH will beat Team BE 81.80f the time
Team DF will beat Team BE 76.40f the time

Unfortunately, Team AH has a lower variance than team DF, which makes the difference look bigger than it is. It's tough to find examples which don't have some sort of bias. It does appear that a small but not negligible advantage can be gained by opting for the anti-hookup if you're the favorite.

Now suppose that your buddy who has Elway makes some trades. He somehow manages to pawn off his entire starting lineup (except Elway) and acquires Barry Sanders, Dorsey Levens, Wesley Walls, Rob Moore, and Joey Galloway in return. His team can now change its name to Team GE (good, Elway). Quicker than you can say "collusion," have gone from being a favorite to being an underdog. Hence the Gaussian says you should go for Team DF. Sure enough....

Team AH will beat Team GE 39.50f the time
Team DF will beat Team GE 40.80f the time

Again, there may be some bias here (some kind of relationship between Levens and Freeman, for example), so this shouldn't be viewed as conclusive evidence of anything.


First, a warm welcome to those of you that skimmed (or just outright skipped) the majority of the article and decided to join us here for the good part. Here's the very short version: using some properties of normally distributed variables (the bell curve), we determined that, in theory, three things are true:

  1. Consistent players help good teams more than equally good but inconsistent players do. Likewise, Inconsistent players help bad teams more than consistent players of the same caliber do.

  2. The Hookup (choosing a WR and/or TE from the same NFL team as your QB) is a good strategy for underdogs, and a bad strategy for favorites.

  3. The Anti-hookup (choosing a WR and/or TE from the same NFL team as you opponent's QB) is a wise strategy for favorites and a poor strategy for underdogs.

However, we don't play fantasy football in theory (well, now that I think about it, fantasy football is 100% theoretical, but you know what I mean). How much (if any) actual difference can these factors make in terms of your actual win-loss record? Let's take them one at a time:

  1. This is actually the one that appears to have the most impact on your chances of winning a game. However, there's a big problem: we don't know before the season starts who the consistent and inconsistent players are going to be, so we really can't make use of this information. Check out some of the guys who appeared on the Consistent list: Jeff George (??!?!), Marshall Faulk, Derrick Alexander. Likewise, we see names like Elway and Jake Reed on the Inconsistent list. Anyone know whether George is going to be more consistent than Elway next year? Unless you can tell me, the conclusion of this section is useless to you.

    Update (7/7/2000): I ran some numbers on this, and it turns out that my guess was on the mark. That is, consistency in year N is a very weak indicator of consistency in year N+1. To be specific, I looked at all players from 1995-1998 who had over 50 fantasy points, and I measured their game-by-game standard deviation for that year and the next year. For RBs, I got a correlation coefficient of .09. For WRs, it was .29, and for QBs, it was -.11. Players who were consistent one year showed no strong tendency to be consistent the following year.

  2. This was the one with the least effect. The Hookup is completely irrelevant. If you think Tim Brown is going to score 1 more point over the course of a fantasy season than Cris Carter is, draft Tim Brown, regardless of who your QB is. If you have to choose between the two of them for a given week and you honestly have no idea which of them is going to score more, go for the hookup if you're the underdog, and don't if you're the fav (I have a feeling if you're choosing between these 2 for your 2nd WR slot, you're probably the fav), just don't expect it to make too much difference.

  3. I'm going to have to call this inconclusive. I just don't know how to do away with all the biases that polluted the study of the anti-hookup. All we have here is the theoretical knowledge.

Bottom-line time. Are you any better off having read this article than if you hadn't read it? Honestly, probably not. The advantages you can gain here are so minimal that you may win an extra one or two fantasy games per decade at most. But hey, what else were you going to do with the time you spent reading this?