Is a pre-post intervention study with a control group without random assignment quasi-experimental? I am currently rather confused about the design type for my study.
I am looking at a number of measures pre and post intervention with control group in a community sample. The sample collected was not through randomization. Is the design type quasi-experimental? What kind of statistics can i use for this design. 
Note: Sample size is 20 in each group before and after. 
 A: If participants will be assigned to the control or treatment groups randomly, this is a run-of-the-mill experiment (not a quasi-experimental design) and you can use all of the usual statistics. On the other hand, if the control group is really a bunch of people who did not qualify for the intervention or were excluded by you or someone else based on some other criteria, then what you have is indeed a quasi-experiment.
Of course, if people registered voluntarily to participate in the study or were screened beforehand, its generalization to a broader population is open to discussion (as are all clinical trials in fact) but the “randomization” part in introductory psychology research method texts usually refers to treatment assignment, not to the fact that the participants form a random sample of anything. The difference therefore does not lie in the type of sample (random or community sample, treatment sample, convenience sample) but in treatment assignment (who goes to the control and who gets the intervention).
If you want more specifics on the analysis you will need to post a lot more details about the study (perhaps in a follow-up question).
A: I'm not sure I understood your phrase "not through randomisation".
This answer assumes that you did not randomly assign participants to groups.
If there was no random assignment of participants to group you could call it quasi-experimental.
Thus, control and treatment groups may differ anyway. The fact that you have pre and post measurement hopefully reduces this issue because to some extent you are adjusting for baseline differences. That said, it doesn't resolve the issue completely.
Assuming that the dependent variable is numeric, you could look at the suggestions on this question about how to analyse such a design.
