Use Median Points Scored in Fantasy, Not Mean (and 2022 QB Ranking Implications!)

I claim that in fantasy football, median points scored is a better metric for evaluating player quality than mean points scored is. A player with a higher mean will score more points on average but a player with a higher median will score more points more consistently. At the end of the day, valuing median over mean will decrease your week-to-week variance in points scored and maximize your probability of winning an individual matchup.

Not everyone values consistency, though. Fantasy managers that like high variance players may argue that if one of your players booms for 50 points, it doesn’t really matter what the rest of your team does – you’re nearly guaranteed to win that week. However, if that player is high variance then he might score 8 points per week the next three weeks and seriously hurt your winning chances. Is one easy win worth three nail biters?

In this article I will take you through my thought process on why fantasy football median points scored is the better metric. I’ll use an intuitive argument, some simulations, and some statistical theory to make my point in the best way possible.

The Theory of Median Points Scored for Fantasy Football

While they are related, your goal in a fantasy football matchup is to maximize your probability of winning any individual week and not to maximize the points scored over the course of the season. From a statistical perspective, medians are really measures of probability while means are measures of long-term performance. The median is defined to be the point where 50% of the data is larger and 50% of the data is smaller. From a fantasy football perspective, the median is the point where its a coin flip if the player scores more or less in a given week.

On the other hand, we can’t make any probability-base statements by just looking at the mean. It is totally possible that a boom/bust type player might have a mean points score significantly larger than their median. This would imply that in a typical week, the given player would more than likely “under-perform” relative to their average points scored.

A Simple Simulation

I’m going to run a few simulations to show why consistency is king and why mean fantasy points scored is inferior to median points scored. Let’s suppose that two teams are tied going into Monday Night Football and both have their quarterbacks remaining – Burrow v. Rodgers. Last year Rodgers was more consistent than Burrow. In order to make a fair comparison from a “mean points scored” perspective, let’s add about 0.5 points per game to Burrow’s scores. This leaves us with the following (adjusted data)

Player	Mean Points Scored	Median Points Scored
Rodgers	21.0	22.3
Burrow	21.0	20.0

I claim that you should be much happier starting Rodgers than Burrow in this game even though their average points scored is now identical! To show you how significant this is, I ran an experiment. I picked one of Burrow’s games at random (with the added extra half point) and compared it to a random game of Rodgers.

Over the course of these simulations, Aaron Rodgers outscored Joe Burrow 55% of the time. That means the team that started Rodgers, the guy with the higher median, had a much higher chance of winning the fantasy matchup than the guy who started Burrow. If you just looked at the mean points scored, you would have no idea who is the better player to start.

Let’s go more extreme. Let’s add an additional point to all of Burrow’s games. Now, Burrow has a higher mean points scored while Rodgers has the higher median points scored. Here’s the data.

Player	Mean Points Scored	Median Points Scored
Rodgers	21.0	22.3
Burrow	22.0	21.0

If you’re starting Burrow you might feel pretty confident because he averages more than Rodgers! We ran the simulations again and now…Rodgers still wins 52% of the matchups. If you look only at average points scored, this might surprise you. But, if you look at the median points scored like we suggest, you would correctly predict that Rodgers gives you the better chance at winning your matchup,

Best QBs of 2021 by Median Points Scored

We may ask now which quarterbacks in the 2021 season were the best when you use median points scored instead of mean points scored. The sortable table below shows how the rankings shook out last year for the top 31 QBs (after this point, the QBs showing up only really played a portion of the season).

	Name	Median	Mean	Difference
	Brady	25.0	22	+2
	Mahomes	22.3	21	+3
	Rodgers	22.0	21	+4
	Prescott	21.9	20	+4
	Hurts	21.8	21	+1
	Murray	21.7	21	-2
	Herbert	21.3	22	-5
	Allen	20.9	24	-7
	Stafford	19.3	19	+2
	Cousins	18.8	19	+2
	Jackson	18.6	20	-2
	Burrow	18.5	20	-2
	Russell Wilson	18.1	17	-0
	Wentz	17.4	15	+1
	Daniel Jones	16.1	15	+2
	Garoppolo	14.8	15	-0
	Tannehill	14.6	16	-3
	Bridgewater	14.5	14	+1
	Carr	14.4	15	-1
	Tagovailoa	13.7	14	-0
	Darnold	13.2	13	+5
	Roethlisberger	12.6	14	+1
	Mayfield	12.3	13	+4
	Goff	12.2	14	-3
	Ryan	12.1	13	-0
	Mac Jones	11.7	13	-2
	Heinicke	11.5	14	-5
	Zach Wilson	11.4	12	+2
	Mills	10.7	12	-1
	Lawrence	10.5	12	-1
	Fields	8.1	11	-0

There are some very interesting takeaways. Tom Brady was actually the best QB in 2021 when best is determined by “how much he helped your team win a fantasy championship”. This is different from best being determined by “most fantasy points scored”. But as I said above, we care about the first thing, not the second thing. This data point probably surprises most people.

There are some other really interesting takeaways when ranking players on median points scored instead of mean points scored. Most notably, Josh Allen may be significantly over-rated coming into the 2022 season (keep reading below to find out…). By median points scored, he only finished 8th last year. Also notably overrated was Justin Herbert. This shouldn’t be horribly surprising because these players are both quite boom/bust as QBs go.

Second, though Mahomes had a “down year” last year, even his down games tended to score in the high teens and low 20s. His median points scored last year placed him 2nd overall. Also heavily underrated were Dak and Baker.

Ranking Fantasy Best Quarterbacks 2022

I think a reasonable metric that takes into account both the median and mean is the average of the 2. If we assume that every QB has about the same performance this year as he did last year, then the top 10 QBs shake out as follows. I’m excluding Tom Brady because I don’t want to attempt to make any predictions related to his age-based decline.

Note: if you want to make projections that don’t rely only on previous year performance, you need extremely complex models that project age-based changes, personnel-based changes, and strength-of-schedule-based changes. While these are doable, we do not have good models built up yet so assume the simpler model presented here.

The value of this metric is that it doesn’t necessarily agree with the overall QB rankings. So, while I do argue for quarterback timing in other posts, at the end of the day you can’t do better than snagging a quarterback on this list who was dramatically underrated. Looking at median points scored instead of mean points scored can give you an advantage over your league-mates.

Stacking Players for Boom/Bust – Does Median Points Scored Still Win?

Things get really interesting when we talk about QB/WR or QB/TE stacks. Ask 10 different people on whether or not you should have quarterbacks and ball catchers who come from the same team and you’ll get 10 different answers.

Even from a statistical perspective, the question remains really interesting. The analysis from the last section says that the QB/WR duo with the larger median should be preferable. Let’s see if it holds up when we run the numbers comparing Rodgers + Adams versus Burrow + Chase. Again, we adjusted the pairs’ scores so that they had the same mean points scored. Here’s the data:

Stack	Mean Points Scored	Median Points Scored
Rodgers + Adams	35.9	37.2
Burrow + Chase	35.9	34

We ran 1,000 simulations and found that – as we’ve come to expect – the Rodgers + Adams stack is preferable here. The Packers stack beat the Bengals stack nearly 56% of the time.

In the next section we’ll talk a little bit about the theory surrounding when to stack or not. If you want to skip straight to the conclusions and see which QBs might make the best stacks, skip ahead to the “Should You Stack” section.

The Theory Behind Player Stacks: Central Limit Theorems and Correlation

Our central tenet is that all else held equal, players with large median points scored are preferable. The central limit theorem says that as we add together more and more independent random variables, the result gets closer and closer to a normal distribution. In fantasy football language, this means that your total team score should appear much more normally distributed than the distributions of any individual player.

However, we’ve suggested that you look for value in players who have higher median points scored than mean points scored. However, this is only possible for distributions which are right skewed – distributions which are decidedly not normal. This fact suggests that as we combine the scores of more and more players, our analysis becomes less and less valuable.

Or at least it would, if we didn’t have a way to decrease the normality of our total team score. The central limit theorem requires that the variables are independent and, therefore, uncorrelated. A quarterback’s performance is highly correlated with the performance of his best receiver. Thus, if we add together correlated random variables, we can maintain the abnormality of our total points scored distribution by stacking quarterbacks and receivers. Therefore, for quarterbacks who have larger medians than means, we actually want to prefer their receivers and tight ends, all else held equal.

Should You Stack

My conclusions are actually very straightforward:

• For a quarterback whose median is larger than their mean, a stack is good
• For a quarterback whose median is smaller than their mean, a stack is bad

How can you tell? Look back at the table in the “Best QB of 2021” section. Those quarterbacks who have better medians than means are indicated by a “+” in the change column. Those QBs are those for which a stack is good. Brady, Rodgers, Dak, and others are good to stack with their top one or two receivers. On the other hand, Allen, Herbert, Tannehill, and Lamar are players which you won’t want to stack with a receiver from the same team.

Median Fantasy Points and the Start/Sit Conundrum

There is another, simpler reason to consider median points scored over mean points scored in fantasy football analysis. A player can only contribute to your team winning if you choose to start them. If a specific player scores 30 every third week and 0 the others, you might have a hard time starting them because of the number of times they’ll put up a huge dud. If that player puts up a 30 point week while he is on your bench, it doesn’t matter. Using median points scored is a way to temper the number of points a player scores by the likelihood that you’ll actually start them on a good week.

Conclusions

We have argued that considering median points scored instead of mean points scored is a better metric for fantasy football. The rationale is that medians are inherently related to probabilities while means are averages. Because we only care about winning in any given week and not margin of victory, the median points scored is a better prediction of our fantasy success. So, when ranking fantasy football players based on their performance, we should not sort by the mean but rather by the median points scored.

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