Interpreting the results of my Cronbach alpha test I have a survey which contains 13 variables in total. For each variable I asked 3 similar questions in order to test the internal consistency. Now I performed 13 Cronbach's alpha tests, one for each variable, to see whether people filled in the 3 questions per variable on a consistent basis. The results are presented below:
Variables ______________ Cronbachs Alpha ______ Cronbachs alpha if item deleted 
Accessibility__________________0.416_______________Not possible to get a higher CA
Accuracy____________________0.680_______________Not possible to get a higher CA
Completeness________________0.681_______________Not possible to get a higher CA
Consistency__________________0.604_________________________0.718
Information Quality_____________0.832_______________Not possible to get a higher CA
Quantity_____________________0.566_______________Not possible to get a higher CA
Relevance___________________0.327_________________________0.490
Reliability____________________0.617_________________________0.689
Satisfaction__________________0.747_________________________0.752
Timeliness___________________0.398_________________________0.487
Understandability______________0.569_______________Not possible to get a higher CA
Use_________________________0.530________________________0.556
Usefulness___________________0.655________________________0.721
My questions are the following:
If possible, do I need to delete an item in order to get a higher CA? Or should I only do this when the increase in CA is relatively high?Can I use all the variable displayed above or are some of them worthless because of the low CA score?
 A: What is your real goal/question to be answered?  If your only goal is to get high values of CA then you can just generate random noise and replicate it a couple of times, nice high alpha value, but otherwise useless.
In my experiance Cronback's alpha is usually not the end goal and sometimes more of a distraction than a help.  It could very well be that the most interesting part of the whole dataset is a variable that acts to pull down an alpha value (why is it not measuring what you thought it would?) and deleting that variable to get a high alpha would be the opposite of the best strategy.
So focus first on what you are trying to accomplish with this study (if high CA is really what is important then you need to go back and redesign the survey and possibly other data collection processes and collect new data), but it could also be that the CA values are only an interesting footnote to much more interesting results.
A: Per the comments, these are 13 results from 13 runs of Cronbach's alpha on sets of 3 questions each.
The original question is whether to delete items (from sets of 3) if alpha can be improved by much. However, I think that, considering that these are sets of 3 items that are supposed to be on the same variable, none of these alphas are great and only a few are really acceptable. I would say those with alpha under .7 need to be reworked in more ways than just deleting a question. One way to get a better sense of what's going on with the 39 items might be a factor analysis. Another, simpler, method would be to find the differences between the 3 sets of questions in each variable and see which differences are large. Such low alphas where there ought to be high alphas could be some fundamental problem like data entry or something.
However, on the original question: I would say delete items only if they raise alpha by a reasonable amount. Otherwise, you are potentially taking advantage of qualities of this particular sample. 
