Yes, AFIK, all drill chucks have a taper where they mate to the driveshaft.
The most common taper for chucks is the Jacobs Taper (JT). JT are numbered. Midway down this wiki is the dimension of the different JT.
http://en.wikipedia.org/wiki/Machine_taper
The two usual causes of out-of-round conditions in chucks is a raised burr on a chuck jaw or debris inside the ways of jaws (i.e. the groove where the chuck jaws live). Jacobs chucks are rebuildable, and you can purchase rebuild kits rather cheaply assuming the chuck body is not damaged. Hit up youtube for instructions on rebuilding a chuck.
If your chuck is beat, or not a Jacobs or other reputable brand, they are surprisingly inexpensive (given the materials and precision involved) to purchase and upgrade your machine. If I were made of money, I'd put an Allbrecht keyless chuck on every drill I owned.
To buy the correct chuck, you need to remove the old one and determine which taper your drill press has, and order accordingly.
Another aspect of removing frustration from hole drilling is sharp drill bits. I bought a Drill Doctor a number of years ago, and have been extremely pleased with it. A dull drill is the work of the devil. It makes you lean real hard on your drill press, causing both the drill bit and drill press to flex, causing oversize holes. Seriously, at $140, a Drill Doctor 750x is a very reasonable long-term investment -- you never have to replace drills, and you can even sharpen carbide masonry bits to cut hardened steel!