The youngest known powers are seven, and flukes. The numbers increase measurably with age, but slowly enough until fourteen that they are still flukes. Then the numbers increase until you reach adult levels at twenty-one. (There the constraint that takes over is that the powers can be useless flukes.)
The factor is not puberty. sigh I'm thinking about brain chemistry, but there are a number of immature though technically adult powers. It wouldn't be too bad if the adults were on average more mature and of better judgement than your average citizen, even to the impossibility of the insane getting them.
hmm. . . .