(shamelessly stolen from Smugzoid so we can test it):
Relevant attribute = chance
17-18 = 3/6 chance of success
13-16 = 2/6 chance of success
5-12 = 1/6 chance of success
1-4 = no chance
Examples of skills
STR: Athletics, Lifting, Rough wall climbing (not smooth - thief skill only)
DEX: Acrobatics, Sleight of Hand, Stealth, Walking on ice or an edge, Tying a rope quickly, Lighting a torch quickly
CON: Holding breath, Endurance, Nausea
INT: Arcana, History, Investigation, Nature, Religion, Scribing, Reading
WIS: Animal Handling, Insight, Medicine, Perception, Survival, Nature, Tracking
CHR: Deception, Intimidation, Performance, Persuasion
At 3rd, 6th, 9th and 12th level you can increase the odds of any one stat skill by 1. For example if your DEX skill started at 1/6, at 3rd level you can make it 2/6. At 6th level you can increase it again or, increase a different stat skill by 1.
These increases do not change your six character stats, it only increases your skill odds.
No STAT skill can be greater than 5/6
Note: Thief and Assassin skills remain RAW