I’m going to guess that about half of the overall effectiveness of common analgesics like that is due to the placebo effect; conversely the real medications are about twice as effective as placebo pills.
Here’s a meta-study that compares placebo, aspirin, acetaminophen, and caffeine to placebo controls:
There are large number of ways of looking at the effectivness of analgesics, but if you click on the button that says, “Free Full Text,” page down to page six of the PDF to see what’s up with specific measures.
Also, a note on the, “placebo effect.” I haven’t used the term very specifically here, but some of this imrpvement in placebo groups is also doubtlessly caused by natural improvement in symptoms over time. Unfortunately, a lot of these studies don’t have a no-intervention group, so we cannot really point out the actual improvment caused simply by poping a pill that people belive will make them feel better.