Every Pokemon has what are called Individual Values or IVs and these gene-like structures determine the stats of each Pokemon and explain why there's some wide variation within a single species.
Stats Judges evaluate the IVs of a Pokemon and determine the range of values the IVs fall under. While Stats Judges are normally found at the Battle Facility, Pokemon X and Y's Stats Judge is located in the Kilcoude City Pokecenter once you've defeated the Elite Four.
IVs are important for those interested in selective breeding and training of Pokemon so you be the best there ever was. If you've only recently picked up X or Y or haven't reallly looked at IV and how important they can be before, you might be interested in learning what the Stats Judge has to say.
IV stats checks and what they mean can be a tad confusing at first but thankfully, Reddit's MistaWhiska has produced a guide to help.
The Stats Judge will assess your Pokemon and give a rating based on his evaluation as well as your Pokemon's best stat (or stats if two are equally high). The best way to improve IVs is to breed and that'll mean a lot of trips to the Day Care center but for now, here's the guide to the Stats Judge and IVs in Pokemon X and Y and we've got plenty more walkthrough for X and Y looking at Mega Stone locations, O-Powers Locations, Friend Safari, fast money, shiny fishing, and more.
