Thanks guys. I found out that I have many more files where the rtf is bigger than the doc. The only thing is that with these files the rtf files are just a few KB bigger so no big deal here.
About my problem where a .doc format that has a size of 1.313KB becomes 1.971KB when I save it as a .rtf.
I managed to get it from 1.971KB to 1.131KB! How? I saved it as a Word6.0/95(*.doc). And then saved it back to a rtf. The result is exactly the same formatting etc. but a much smaller filesize. It must have dumped a lot of garbage when saved as Word6.0/95(*.doc). The rtf file is still a few KB bigger than the doc but it's more balanced like my other files. It's a mystery to me how a file with so many garbage can still be displayed correctly!
I tried it also on my other files but they stayed the same. Still it is weird that a rtf is bigger(even if it's a few KB) than a doc.