Can non-paired data have dependency? I have two vectors of data of different sample size. Both are the same experiment, but executed in a different environment. Since the sample sizes are different and they don't share any subject, is it safe to assume there is no dependency possible?  From what I gathered, there can be no correlation nor can I create a scatterplot, which automatically means there can be no linear dependency.
I know it must be a simple question, but it's surprisingly hard to find something about it. I might not be using the correct terms.
 A: Your use of "paired" could be leading to confusion. In a statistical context, "paired" data often refer to paired observations (moms and kids; twins; cells from the same culture; etc) where you would expect information about 1 half of the pair to be informative about the other half of the pair. This means that the observations are "dependent" and this often requires special treatment. Your use of "paired" instead seems to refer to variables being paired within an observation. 
That bit of semantics aside, you are correct that you can't assess relationships between 2 variables when each is measured in a different set of observations, for the exact reason you state. 
A: To counter D.L. slightly, there certainly can be dependency in the data (either within the same environment or between environments). More details on the experiment and how the groups are separated are necessary to say if such dependence is unlikely.
For one example, lets say that you had two different classes of students, and you conducted an experiment on one class. If it is possible for students in the classes to interact, there could be feedback (which causes dependency) from one condition to another. (For a specific example, I knew of an individual who conducted surveys on deviant behavior in adolescents, and one of the students by the end of the day asked "When are you going to ask me about the sex questions?")
Other common examples are time series or spatial data. If for some reason the outcome in either condition has a relationship with time, there could be a dependency in the experimental or control condition (e.g. observations nearer in time are more correlated than those farther away).
To assess those dependencies are more difficult though than graphing paired data in a scatterplot. One way to do it is to examine all pair-wise connections that might be theoretically correlated (such as for network or spatial or time series data). 
Examples of this are a Moran scatterplot in which you graph values against the average of the spatial neighbors. Other examples for data indexed in space or time are the variogram or correlogram.
