I don't think there's a mistake.
On page 51: It says it takes a week training to "Improve" a skill. not open a new skill. AFAIK, you don't OPEN common skills, you're going to get your stats at base anyway.
On page 52: If you want to OPEN a NEW ADVANCED skill, it takes a month to do so.
This infers to IMPROVE an advanced skill via training (one you already have), it takes a week (being the same amount of time it would take to improve a common skill via training).
Or at least that's how it looks to me.