How do I know if I need a sub woofer?

First, you want at least two subs, and second there are very few systems that wouldn’t benefit significantly from subs.  Anyone who tells you differently hasn’t heard decent subs properly set up and dialed in.  Not only will your in-room bass improve but also imaging and an expanded 3D soundstage.  Once you listen with good subs you won’t wanna listen without them. 

I agree with soix.

Agree all systems benefit from an appropriate pair of subs...

you can't sleep at night!

There are all kinds of technical reasons, but the ultimate answer is, What do you hear, or miss?

Go to a live performance of chamber music and listen to an acoustic bass (instrument). Chamber, because there is more chance to hear the bass by itself. Acoustic, to get a direct feel for how an unadulterated bass instrument sounds and feels.

Then listen to similar music on your system. You will hear the bass differently. If the bass lacks depth, or floats strangely around the room, *and if you want to try to improve those shortcomings*, then you might try subs.

Cheers.