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Are there any kind of, even if rough, estimates how productivity scales with the number of programmers?

(And, ideally, backed by numbers.)

Here, by productivity I mean the inverse of time that is needed to finish a project (given fixed projects, how it's time burden changes), though other measures may be as well of my interest (but other ones can be harder to be made objective).

The standard wisdom is that "Nine women can't make a baby in one month.", which in my language means that performance is sublinear. (The Brook's law goes even further - that the total performance declines, but AFAIK is related pumping people when there project is already ongoing).

Sure, there is an overhead for: communication, coordination, common technology; and very likely latter programmers may be not as skilled and motivated as the first one. But how much?

In general scenario it is called Ringelmann effect and, at least - for simple tasks like pulling rope, it is attributed to the loss of motivation.

Why the question?

Mostly, academically and out of curiosity. Scaling laws in performance are an interesting thing.

For example, for performance of scientists grow superlinearly (here: ~n^2) with team size, until the group reaches number of maximal effective collaborators. For most of cognitive tasks, group typically does better than a single person, but not n-times as good (scaling more like square root, ~n^0.5). And for institution management overhead can be directly measured, as ratio of administrative workers increase with the institution size.

See references from my talk Does 2x bigger mean 2x better? - 1st Offtopicarium.

A bit related to my question Performance of a group solving a cognitive task: How does it scale? - CogSci.SE.

Sure, it will depend on the task, and collaboration technique (e.g. two people pair-programming vs sitting at their own desks).

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closed as too broad by MichaelT, gnat, GlenH7, DougM, Robert Harvey Feb 7 '14 at 23:05

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No. There are too many variables on top of team size that affect the productivity of the team. Also, teams change over time and especially for long-term projects, it is not uncommon to finish it with completely different people than how it started. So any kind of study will inevitably be unusable to reason about real-world projects.

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The COCOMOII model has 22 variables (none of which are team size) that factor into productivity. –  MichaelT Jan 25 '14 at 17:59
The question is not what cannot be said, but what can be said (even if it means fixing some variables). Even if evan rough estimates can be for "doing a website in Django, that for a single developer takes 4 months" or so. –  Piotr Migdal Jan 25 '14 at 18:22
@PiotrMigdal Are you lucky enough to work where some form of Agile is practiced? If so, you may be able to query some team leads for complexity points accomplished per iteration and then factor in group size. For example, one of the leads at my workplace states that once he has four people on his team that he ends up doing almost entirely management work. There are so many variables here: type of work from project to project is not the exact same, level of experience can play a huge role, etc. An approach like this may give an apples-to-apples comparison if you feel like original research. –  J Trana Jan 26 '14 at 5:42
@JTrana I ask this question from academic perspective, where I am lucky enough to choose variables so something can be said :). Perhaps someone already did experiment that you described. No worries, I don't intend to use scaling for a business plan (I am very well that "it depends" and depend a lot on the project, people, whether they collaborate efficiently, etc.) –  Piotr Migdal Jan 26 '14 at 19:45
@PiotrMigdal Well thanks for the interesting question and good luck in your studies! I'm curious to see where this goes; I was aware of the Ringelmann effect but not some of the other studies around scientists etc. so I learned something even from the question being asked. –  J Trana Jan 26 '14 at 22:44

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