Paired t-test for skewed data I have a cohort of subjects (n=262) with a single-time intervention. Subjects have been asked to answer on a questionnaire before and after the intervention. The questionnaire comprises five items where each item is in the range [0, 1, 2, 3] and the total score is the mean value of these 5 items. Thus, the total score of each subjects is given by a rational number in the range 0-3.
Here is the histogram of their responses:

I applied a standard paired t-test to find an affect of the intervention on the cohort and found (with python stats):
>> scipy.stats.ttest_rel(data_before, data_after)    
>>Ttest_relResult(statistic=3.4864105747384686, pvalue=0.00074914757737233801)

So, it looks interesting and supports nicely my hypothesis that the intervention has affected the cohort. But I am wondering if the application of paired t-test on these skewed data with n=262 is justified in this case? 
Any suggestion how to check my findings with other tests?
UPDATE
I shared the data through dropbox
 A: The original values aren't assumed to be normal, the differences are, so the skewness in the first two histograms is not an issue.
While your differences aren't normal, they're bounded, relatively symmetric and not very heavy-tailed (somewhat fat, with peaky center, but the boundedness helps), so this may not affect the t-test much. 
The main concern would be that it looks like there might be a lot of 0 values, but a quick simulation with numbers very similar to yours seems to indicate very little problem with the distribution of the usual one sample t-statistic -- i.e. the significance level should be very close to the chosen level.
Power might be mildly affected by the heavier tails, but I wouldn't have much concern in this case.
I really don't see that there would be much problem here. 
If you are concerned about challenges on the t-test, you could always consider a permutation test of the mean differences. [An alternative might be a Wilcoxon signed rank test but the high proportion of ties could be a concern.]
