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I have five groups of students and 40 in each. And we counted the number of references spelling mistakes in their homework. We would like to compare to see if there exists any groups differences in this count dependent variable? What is the best suited test to use? We have a lot of zeros, because students never made any spelling mistake at all.

Groups      Count Data
Group1      0
Group1      0
Group1      15
Group1      20
…until 40       
Group2      1
Group2      0
Group2      0
Group2      25
…until 40       
Group3      0
Group3      0
Group3      0
Group3      1
…until 40       
Group4      0
Group4      15
Group4      16
Group4      24
…until 40       
Group5      0
Group5      1
Group5      1
Group5      16
…until 40       

the question is: does the sum of the count of mistakes differ from one group to another? if so, between which groups?

update

each group actually belongs to a specific language. Let's say, group1 is English, meaning that students are English speaking and submitting their homework in English, and the second group is German, third French , fourth Spanish, and fifth Italian. Students write up some kind of homework and we are interested in knowing which language group students make more mistakes in this particular kind of homework?

update2

actually, it is a special kind of phrasing or special kind of linguistic feature, that I call "mistake", for the sake of explaining clearly. In fact, the more of these, the better! The more the students uses this special kind of "mistake", the better the score! So we want to know if one language is more favourable to the use of this special kind of "mistake". The length of the context in which they appear is not limited, no time, no number of words! It is theoretically endless until the student stops and finishes him/herself. The interviewer doesn't stop the participant. It is possible that in a span of one page, one student includes 30 instances of this kind 'mistake' and another includes only 1 or nothing at all. The students aren't aware of the fact that they are producing such special phrases. We want to test the effect of language on the production of such "mistakes". I hope this is now clear.

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  • $\begingroup$ Please edit your question to address the following: What is the larger goal of this study? What do you mean by randomly selecting groups from cities? (That sounds like you had one group in each of many more than 5 cities and then randomly chose 5 of those groups, but I doubt you did that.) Do you have a per-student measure of how much the student spoke overall, regardless of topic? How many students in total never mentioned the topic of interest? $\endgroup$ – Kodiologist Jun 27 '16 at 20:07
  • $\begingroup$ Thank you, that helps, although this new description is not entirely compatible with the old one. Also, I still can't tell what the larger goal of the study is (i.e., what scientific question you're trying to answer or applied problem you're trying to solve). Perhaps it has to do with what distinguishes the groups. What distinguishes the groups? $\endgroup$ – Kodiologist Jun 27 '16 at 21:43
  • $\begingroup$ @Kodiologist i updated the question to get the overall idea. $\endgroup$ – cplus Jun 27 '16 at 21:46
  • $\begingroup$ I guess I still haven't been clear. You say that you want to find differences between groups, but why? Why are you doing a data analysis in the first place? Are you trying to test a scientific theory, and if so, what theory? Are you trying to solve a practical problem, such as reducing children's spelling mistakes, and if so, what is the problem and how do you propose to solve it? This is what I mean by asking what the larger goal of the study is. $\endgroup$ – Kodiologist Jun 27 '16 at 22:05
  • $\begingroup$ @Kodiologist i updated the question, am I clear now? by the way, thanks for your comments and trying to help me clarify the whole thing. $\endgroup$ – cplus Jun 27 '16 at 22:32
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You're in a really bad situation because you're missing a key contextual piece of information, which is how long each student's essay is. That means you can't tell how much a difference in counts reflects a real difference in a tendency to make mistakes, and how much it results from just a difference in essay length. The only way we can proceed is to assume that all students' essays were of the same length, which we already know is wrong.

You can try Poisson regression (or OLS with a suitably transformed DV) with a dummy variable for each language, then compare the coefficients for the dummies to each other (and to the intercept).

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