I know that some people find it impossible to read through the songs/poems in LOTR text and just skip them.
Are there any sort of statistics on what percentage of readers do this?
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Okay, so I am answering this like five years late, but I did find something that might have what you want:
This is an archive of past surveys on the site, and the last one on this page is "Which part of the LotR books do you skip most often?" (emphasis mine)
Which part of the LotR books do you skip most often?
Nothing, I read every last letter (44%, 1,856 Votes)
All those poems and songs (21%, 907 Votes)
Everything with Tom Bombadil in it (9%, 387 Votes)
The Hobbit stuff in FotR (8%, 318 Votes)
Other (6%, 250 Votes)
The Council of Elrond, of course (3%, 140 Votes)
Treebeard, because the Ents are so slow and boring (3%, 137 Votes)
I haven't read the books at all (3%, 106 Votes)
Shelob, she freaks me out (1%, 62 Votes)
All the battles in Return of the King (2%, 57 Votes)
Total Voters: 4,220, Start date: March 30, 2005
So far as I am aware, there are no stats for this. Indeed, it's unlikely that there could be stats for this, given that the books have been read for several decades by uncounted people. Any statistical survey would have to be done today (or already underway) and would represent only the modern society, not the prior generation of readers.
I would expect that those who DO skip the songs and poems have a harder time with the books - if you skim books as dense and intricate as the LotR series, you WILL miss things, and the songs and poems give some relatively important background information.
Surveys based on anecdotal accounts are probably not going to be that reliable (and I don't know of any that have been done).
However, this sort of data could in principle be collected by e-book readers like the Kindle.
I am not entirely sure that I am comfortable with giving Amazon and Sony reams of data about how long people spend in which part of the text, however. I would rather remain ignorant about how much people skip such passages. But I suspect that the commercial pressures to gather data about reading habits will eventually be too great. So you should, potentially, have your answer in a decade or two.