A paper was submitted to a journal, and now I am thinking of submitting a preprint on ArXiv. However, I wouldn't want to release publicly one of the examples in the paper yet, and would submit the preprint without that particular example (it's probably 5% of the paper). Could this be a problem given the reviewers might check the preprint? Is there any rule on the level of similarity between the preprint and the journal paper, or they are fully independent?
I don't see any problem with this. The submitted version and the arXiv version certainly do not need to be exactly the same, although in my experience it is more common the other way around (something in the arXiv version that was omitted from the submission).
A few possible issues this might cause:
- A referee might see that there is an arXiv version and assume it is the same, therefore that everything in the paper is publicly available, not realising you are trying to keep that example under wraps.
- Once the article is accepted, if you want arXiv to count for "green open access" you would need to update the arXiv version to match the accepted version, including the example.
- Seeing the arXiv version might prompt someone else to come up with the same example and write it up (either publicly or just sent to you), assuming you don't know about it.
I think that most preprints are papers still a bit "in process". Therefore it is natural that they aren't in the same form as the published version, which is likely to be changed on advice of the reviewers in any case.
So, assuming that the journal permits preprints that they don't control themselves, it should be fine. A preprint isn't intended as an alternate publication method (though used that way often enough) but a PRE print: an early version.
Your usage is a bit non-standard, but no one, including reviewers, will likely notice nor care.
In math, for example, preprints are favored so as to give earliest notice of results that might be useful to others, given the somewhat long publication process.