I gather you're looking for something that analyzes loudness changes through the duration of the song, as opposed to a single number or a moment-to-moment display. Something like they used for
this article.
I use Adobe Audition for most of my own objective measurements. It has a useful histogram that gives me any idea of how squashed my song is. However, it can be misleading when there is a wide dynamic range over different parts of the song, i.e. very quiet parts and/or very loud parts. The quiet parts may indeed be too quiet, or the loud parts may indeed be too loud, but the statistics average them out and say all's well. Mostly, I rely on visual inspection of the waveform. Over the years I've developed a pretty good eye for what's too dynamic and what's not dynamic enough.
I have also used the TT Dynamic Range meter, which can report on an entire folder of songs, but which also suffers from the same problem described above. It's quick and easy, and comes in two versions: a plugin for realtime analysis and a standalone executable for batch analysis. Problem is, the developers decided they could make money with it so it's no longer free and requires a "donation" of (last time I checked) $45. But with some creative googling you should be able to locate a copy of the older free version.
Here's a tool that would probably do the job: NuGen Audio Loudness Management Batch Processor. This utility intrigues me, but at $1,000 I don't think I'll be investing in it any time soon. Your best bet may be
Toscanalyzer. It can analyze and report on multiple audio files, making it helpful for determining how similar a collection of songs, e.g. an album, is. Or comparing your own material to your favorite commercial references. It's also the only tool I know of (besides the pricey NuGen product) that graphs loudness moment-to-moment. And here's the best part: it's free.
(Edit for spelling. Yes, even I, the annoying self-appointed spelling cop, occasionally mis-spell something.)