I have a study where I'm now focusing on 4 students.

I'm trying to find the best way to interpret my data.

If I calcuate the mean using their individual scores on the test (0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4,4,4,6) I get a mean of 2.05 and a standard deviation of 1.73. I think this gives the impression that I have 20 students when there are really only 4.

If I calculate the mean using their total scores on the test (9,14,2,15) I get a mean of 10.25 and a standard deviation of 5.14.

Which one is the correct interpretation?

Please help

Signed, lost

The best way to interpret you data depends on what question you want to answer. Is it:

A: "How dispersed are the students' results"? (As in "Did most of the students do pretty much the same overall or did some do much better than others?")

Or:

B: "How dispersed were the students' performance on the individual questions?" (Did the grades on individual questions differ markedly or were they much the same?)

I think you are right not to trust your first method, because here you are confusing these two questions.

For example, you're counting each individual students performance on different answers as unrelated objects. (So if a really smart student did well on all the questions, this would contribute to a lower standard deviation; and similarly if a poor student did badly on all questions - You don't want that, if you want to measure the dispersion BETWEEN students).
Similarly you're counting results on the same question by different students as unrelated objects.

Your second approach gives a fairly good answer to question A above.
To answer question B, I think you might need to take an average over all the students of their performance on each question, and then analyse the standard deviation of these averages. Or maybe you were trying to answer a different question altogether?

I'll just make a general disclaimer: I'm not a statistician. I'm a mathematician so my knowledge of statistics is a bit rusty but I thought what I had to say might help a little.