This article is a natural follow-up to my discussion about Skill-Luck Ratio for each position (and Correlation Coefficients).

Strength in Numbers: aggregate error

All the above discussion treats fantasy positions as individual variables for betting. But fantasy football actually is a bit more predictable than I have implied so far, and this is because you use a whole team. Combining the scores of 9 players significantly reduces the amount of error relative to expectancy value of the total team score. (Not including the effect of your opponent's 9 players.)

If it's not clear to you already, this is how it works: Including more positions into your team will tend to help the outcome (sum of scores) converge to the predicted value. It's analogous to the law of large numbers, where if you flip a coin more times, the average number of heads will more reliably approach 0.5.

Let's use an example with kickers, to understand better how this works. Imagine that, instead of choosing a single kicker each week, you are allowed to choose 5 kickers and use their average score. Since we discussed above that kickers offer a predictable range of +/-0.66 points, let's stick with that number: assume you can find 5 kickers who are predicted to score a combined 3.3 points above average. A single kicker brings a random error of 4.2 points, and the ratio of 0.66/4.2 is a pathetic 0.16. What happens with 5 kickers? The predictable/controllable expected value is still 0.66 on average. (These are expectancy values, so you are allowed to sum them directly as long as you can freely choose those players, i.e. as long as they are not assigned to you randomly.) But the standard error does change for this new average 5-kicker score: you sum 5 variances to get Sqrt(5x 4.22), and you divide by 5: that's just 1.9 points of randomness! Taking the combined predictable/error ratio now, it becomes 0.35, which is the same level as D/STs and QBs.

Understanding risk/reward for the whole team

The same principle applies to your whole fantasy team, even though you're adding together different positions. Let's run a check on this and simply simulate a team of 9 positions (QB,RB1,RB1,WR1,WR1,WR1,TE,K,D/ST). I will consider a single week here-- so just realize that analyzing the statistics over a whole season is another level I will not address here. The sum of their average scores becomes 89.5 (expected score), which seems about right for standard scoring. Then, if we naively thought that positional errors kept the same level of significance and therefore could be added (which you are not allowed to do), you would get +/-55 points of randomness. Clearly this is wrong from experience: a "typical" range of scores is not 35-145 points. We only see those kinds of extreme numbers a few times per season. This "stupid example" just demonstrates that individual error contributions must get smaller when part of a team sum. And this is good news for your fantasy team.

So let's look at the expected error the correct way: we sum the square of projection errors for the 9 positions, and get Sqrt(352) = 18.8 points of randomness. Almost a 3x reduction from that wrong number of 55! More players = less relative error. (These 19 points also seem right: team scores ranging from 71-108 is more typical.) This should be compared to the summed 21.7 controllable roster points. Slightly more controllable points than points of randomness.

So let's conclude what this means for the game of fantasy football: good player selection (from reliable projection sources) lets you control within +/-21.7 points; meanwhile, combining errors gives a total team randomness of +/-18.8. The ratio is 1.16, which means Fantasy football is theoretically more predictably controllable than random (though barely).

Individual positions within the team context

Suppose you're wondering what the effect would be of taking the kicker position off your league roster. So let's understand more how grouping variables reduce risk, and ask: "how much does the error change by removing each position from the team?" This is shown on the following bar chart.

First realize that as your team size grows, each additional player contributes less and less relative risk. You can see that when a team is reduced to 8, the average contribution of the error is 2.2 points (from those 8 players). As you add the 9th player, the extra error is always smaller than this 2.2, consistent with the team analysis above. Secondly, notice how much smaller the error scale is in a team context: Kickers at 0.5 instead of 4.2 alone, QB at 1.5 instead of 7.5 alone.

As before, the next step is to ask how this marginal error affects the ratio predictable-points/random-error, nut now in the context of a whole team. The team ratio we found above was 1.16, and you will see this doesn't change appreciably by going down to 8 players.

The results are clear from the bar chart: although the first 3 positions (K/DST/QB) have a slight negative effect, their effects on the team ratio are very small. Not even kickers risk bringing the ratio under 1.0. Deleting them or any other position from you team will have only little impact.