Yes in ideal conditions a faction would be more productive then a single player. But everything with players is case relative. Like lets assume each player is on for 2 hours a day. If a faction has 10 people they have 20 hours of work. Where as a player has 2 hours. But that often not the case. And it seems to me that most factions contain 3-4 diehards while everyone else on the roster is barely on once a day or even week. Where as a die hard player is often on and can accomplish much as they don't have to check with base-mates and discuss what to do. Their base may not produce as much overall to faction in terms of pure amounts,but a player with a good setup could best a faction in terms of each player to goods.

Factions also seem to get raided alot by either internal or external forces. Where as a player with an underground, hidden setup will probably never be raided. Like the "Sky Kin" a relativly new faction was recently raided.

A faction of well working members with a hidden setup where every member is loyal 100% would be ideal, but near impossible to achieve. But it is possible.

So in conculsion i feel that the question "Is a player better than a faction?"has 2 answers, yes and no. Yes they could easly produce more goods and get more profit as they don't have to split it. Yes they are alot safer from a raid. No in that they may not produce as much raw goods overall. It is a balancing act every player much choose. I have played both as a loner and as a faction member and they both are fun, but there is an undeniable sense of accomplishment when one can freely build and help others in a faction.

@R4iscool1 True it was just an example number.
Manpower is somewhat irrelvant with auto farms that only require the chunk to be loaded.
During which time that person can improve their base etc.
One player can undoubtly match a faction with dedication and skill.

anything one player has a good faction has x 20 or more
pretty simple concept

THEORETICALLY: What if a player had x20 more than a faction, what about then?

As i stated previously i think power should be judged on a player:item ratio. Meaning even though there may be more people in a faction do they really have more items(player:item). That makes sense to me im not sure about you guys.

@TheDunmerRaven Well, Who do you think is more powerful,The faction with 10 people yet are almost never heard from, or the faction with 3 people who are regonised by nearly everyone?

Ahh an interesting question. I did not include a "fame" variable in my thoughts. This is very thought provoking. Thanks for asking! May I ask for more details on the players, if you want me to answer with just the given I can, but I would prefer to know the following.
For referance group A is the group of 10 unknowns, while Group B is the 3 regs.

1. Are group A and group B players on the same amount of time, ie each player is on for 3 hours a day.

@TheDunmerRaven Lets say that everything about the two groups are the same, except for how well they are known on the server.

And for a third option as a control, lets say group C who have 5 members and are moderately well known but still the same in all other aspects.

Ok as BoneChi11er said recruitment is a challenge if you are unknown. They have more people and as you said they have the same effiency rate and time served. So yea obviously Group A is more powerful in terms of gear. With Group C next. Then Group B. The fame factor is next. Group B is easier to recruit. While Group C and A have a harder time respectivly. But they start with more people to make up for the unknown feature. So in a way you could consider them equal. But I'd probably have to give it to Group A just due to output.

@TheDunmerRaven Fair enough, How about we change the conditions.

So that group A is very well organised, group B are really badly organised and group C are in between.
All other conditions remain the same, and they are all known the same on the server.

Which would you think would be the most powerful?

Group A is organized so yea obviously the more organized the better, or so we think. If it is organized to the point of militant status it's members will grow discontent and an inevitable war would begin. If all members are on the same page then I would say that the faction would be unstopable. A common goal and everyone would work in unity. The faction would be near ultimate in terms of output. Raiding isnt as big of a thing due to unity and probably several bases.

Group B on the other hand is poorly organized. If by poorly organized you mean loosly organized then it may appeal to the loner crowd who prefer a loose relation of people with no definite teams and common storage. This type of a loose group would be debatable as a "faction"in the sense of the word. But if by poorly organized you truly mean poorly organized then it would most likely be leveled quickly. Unable to keep up with the documentation of players and not caring about who joins they recruit a few bad players and are quickly destroyed. But depending on the situation their downfall may be their saving grace. Unorganized they have bases everywhere and quickly relocate and rebuild. But as a general rule a poorly organized faction is not as fun as a decently organized one. It often falls prey to anarchy and the leaders are often to not found on. The faction doesnt go out with a bang but a whisper as the remaining members try to gather their forces.

Group C is the decently organized faction. It is a nice mix of both. Allowing both freedom and stability for its members. Its upsides are that it doesn't maintain a strict policy and is easier on the players. But at the same time raiding is more likely then Group A. It is a bit heptic at times and some players may feel they are pulling all the weight leading to discontent and eventual break off factions. It is probably lead by a few leaders. It probably has a strong possibilty of being one raid away from collapse as they havent organized a full defense/backup plan.

As far as who I think is the best. Well it is hard to say. Group B is at a bit of a disadvantage for a player like myself but some more trolly players would jump on board. Group A is fun until the player's ideas clash with that of the leader insuring a stressful war of drama and back stabbing. Group C is pretty good except the lack of a certain system may confuse and annoy hardcore players.
So I say a tie between Group A and Group C. But it is all case pertainate. It is all revelavant to the man in charge. Sorry for dragging on I just like this sort of discussion.

@R4iscool1 Not if the player is better than the faction.
Its a pretty simple concept quality over quantity.
They are useful but they aren't necessarily the most powerful entity on the server.

Fish has a rather good outlook on it.

a player again won't be trading items to villagers for 10-20 hours a day
a very good player can't compete monetarily with a good faction

@Mabzino a player again won't be trading items to villagers for 10-20 hours a day
a very good player can't compete monetarily with a good faction

Yea i would say that is true almost all the time. But if a player produces 10 diamonds a day, and a faction of 5 produces 40 then our ratio are 1:10 player and 1:8 faction. So the player is really producing more. See saying more people, more profit is true as yes raw materials is higher, but more people more mouths to feed meaning one must split it more ways and outfit more people. So while a faction may acquire a lot of stuff, and true it may be able to level up its members quicker, a faction that hasn't had time to make a large setup would be inferior to a player in my opinion. But yes you could say "well having 40 diamonds is better than 10 and the faction functions as one so player to item ratio doesnt exist" and that is true to some degree.
But here look at it like this. A player is selling elytra for 30 diamonds. Our lone player quickly acquires this and buys the elytra. A faction of 10 gets the 30 diamonds alot quicker then the player and buys the elytra. But the faction must repeat this 10 times for every player(assuming all players are equal).