Q:

How to measure and quantify concentration or focus?

How can we measure and quantify concentration or focus? For example, if I were to say:

"If you do Activity X while using Y, you will be more concentrated
  than if you were using Z"

How can I quantify that? And what tests should I do to actually prove that statement? Examples are appreciated.
Background
I am trying to build some different technology products and I want them to help the user stay more focused than they would using a competitor product of the same kind. The only research I've done so far was to build some interfaces and ask the user how they feel.

A:

Short answer
Attention can be quantified with a sustained attention to response task.
Background
I think with focus or concentration you mean sustained attention to a certain task. A sustained attention to response task (SART) (Silverstein & Palumbo, 1998) could be helpful to you (here is a free PsychoPy script). SART seems to be a reliable measure for attention (Smilek et al., 2010). 
Basically SART consists of a set of stimuli (e.g., simple shapes) and the subject is asked to repeatedly give a response (e.g. a button press) when a certain event occurs (e.g. a shape changes color). Reductions in correct rates, increased lapse rates, and/or decreased reaction times all may signal reduced attention to the task. 
Admittedly I am not too familiar with these tests, but I do have a lot of experience with people loosing attention during tedious psychophysical tasks :) I hope the links and references provided here may be of further help.
References
- Silverstein & Palumbo, Computers in Human Behavior (1998); 14(3): 463-75
- Smilek et al., Neuropsychologia (2010); 48(9): 2564-70

