What exactly does 'representative sample' refer to? When reading passages like the following: 


Based on a representative sample of 88 recent raids, we show that the Turkana sustain costly cooperation in combat at a remarkably large scale, at least in part, through punishment of free-riders. 


I wonder what 'representative sample' might refer to. 
Is it related to power calculations (for example) in statistical inference or is there some way to assess the number of samples required from the total population for it to be considered representative?
 A: There seem to be three things going on in this discussion. John is correct in the statistical parlance that "representative sample means at random with no bias in the selection methodology. That needs to be distinguished from what Jose said about sample size. Sample size is not a measure of how reliable a sample is, but it more a measure of how accurate it is in the context of variance. This is what is usually referred to in the polling data for example as "accurate withing plus or minus five percentage points."
That does not mean that the reference you quoted meant to use the phrase in the way John and I have described it. To figure if that is so you would have to read more of what they told you about how the sample was selected - and perhaps ask some rather pointed questions.  The term is frequently misused to mean in " Using my judgment I purposely selected cases that I felt were representative." 
A: The representative sample is that how preciously you collect sample from the target population. There is no quantitative method for this purpose, and the person who make sampling frame can assess how much sample is representative of population. Nevertheless, one can also reduce the bias of estimates to claim that sample is representative of population.
Definition with Example
A subset of a statistical population that accurately reflects the members of the entire population. A representative sample should be an unbiased indication of what the population is like. In a classroom of 30 students in which half the students are male and half are female, a representative sample might include six students: three males and three females.
When a sample is not representative, the result is known as a sampling error. Using the classroom example again, a sample that included six students, all of whom were male, would not be a representative sample. Whatever conclusions were drawn from studying the six male students would not be likely to translate to the entire group since no female students were studied.
For more information check this URL.
A: A representative sample is one which is drawn without bias from the population of interest.
For example, suppose I want to find out how many people drink milk with breakfast. If I am a vegan, and I ask a random sample of my friends and associates (many of whom are also vegans), then the sample I have taken is not representative of the population as a whole: I will of course find that a low proportion of people drink milk with breakfast, but this is an artifact of my choice of sample, not because so few people do in reality.
Basically, if there's any factor which causes us to select our sample in a non-random way, the inference is questionable. In the example above, a representative sample is one drawn randomly from all raids. We could also draw samples only during winter, in which case we might get a skewed result (maybe the cold makes people less prone to cooperation).
Hope that clears things up a bit...
A: There are many different meanings of a term "representativity" in different fields. To give an answer I will post a quote from Bethlehem, Cobben, Schouten (2009) Indicators for the Representativeness of Survey Response.

The concept of representativity is often used in survey research, but
  usually it is not clear what it means. Kruskal and Mosteller (1979a,
  1979b and 1979c) present an extensive overview of what representative
  is supposed to mean in non-scientific literature, scientific
  literature excluding statistics and in the statistical literature.
  They found the following meanings for ‘representative sampling’: (1)
  general acclaim for data, (2) absence of selective forces, (3)
  miniature of the population, (4) typical or ideal case(s), (5)
  coverage of the population, (6) a vague term, to be made precise, (7)
  representative sampling as a specific sampling method, (8) as
  permitting good estimation, or (9) good enough for a particular
  purpose. They recommended not using the word representative, but
  instead to specify what one means.

These are the references to Kruskal, and Mosteller:


*

*Kruskal, W. and Mosteller, F. (1979a). Representative sampling, I: Nonscientific literature. International Statistical Review, 47, 13–24.

*Kruskal, W. and Mosteller, F. (1979b). Representative sampling, II: Scientific literature, excluding statistics. International Statistical Review, 47, 113–127.

*Kruskal, W. and Mosteller, F. (1979c). Representative sampling, III: the current statistical literature. International Statistical Review, 47, 245–265.

*Kruskal, W. and Mosteller, F. (1980). Representative sampling, IV: the history of the concept in statistics, 1895 - 1939. International Statistical Review, 48, 169–195.

A: Let me disagree with previous responses. A "representative sample" is that sample that have enough data to represent the natural behavior of the variable. Enough data seems to be difficult to establish, and depends on the kind of variable, however, sample of more than 30 experimental units are often considered as enough.
