How do I show if two paired sets of quantitative values vary in relation to each other?
First I always — as in ALWAYS remind people — correlation is NOT causation. Consider the synonyms for the word correlation:
Nowhere on this list is the work “causation.” Now it doesn’t mean that there is never a causation factor between variables BUT the rigor, study (controlled) and statistical analysis that are required to show causation are EXTENSIVE (think tobacco). So, remember, as a general rule, correlation is NOT causation.
Okay, now that I have gotten that on the record one more time…
If you need to show how two paired sets of quantitative data values vary in relation to each other and in which direction, i.e., positively or negatively use of a scatter plot (this is not your only choice, but a most common and good choice).