Suppose I have a webpage located at http://www.website.com with an in it. Like this:
...
As of August 2015 all the major browsers (Safari, Chrome, Firefox) should set the Referer HTTP header to http://www.washington.edu in this case.
But I encourage you to test how specific browsers versions behave in specific cases if your application depends on Referer value heavily.
I write this because for example I have found out that Safari 8 currently makes requests with incorrect Referer for JavaScript loaded in iframes, probably if this JavaScript have previously been requested with a different Referer. This is a bug that's similar to this particular one that was closed in Chromium only in 2014. (I have already reported it to Apple, btw.)
Historically Internet Explorer 6 showed different behaviour than described above - it would set the Referer HTTP header to http://www.website.com.