For a USGA Handicap, you need a minimum of 5 rounds. While it's true that the best 10 rounds out of the last 20 are used in the calculation, that's only after you've played 20 rounds. You can officially start after only 5 rounds.
Here's the breakdown:
# of Rounds Played / # of Rounds Used
5 or 6: Lowest 1
7 or 8: Lowest 2
9 or 10: Lowest 3
11 or 12: Lowest 4
13 or 14: Lowest 5
15 or 16: Lowest 6
17: Lowest 7
18: Lowest 8
19: Lowest 9
20 (or more): Lowest 10 of the last 20
It's not necessarily the lowest scores that are used, it's the lowest Differentials, which take into account the course's slope and rating.
If you want to see all of the details explained in a pretty straightforward way, check out http://www.imagolfer.com (which is free for individuals) and create an individual account. Click on the Sign-Up button and be sure to select INDIVIDUAL, which is free and doesn't time-out or anything.
You can enter your scores and it will calculate your handicap index and course handicaps. You can click on your handicap for any round for a detailed explanation of which scores were used, the resulting differentials, and all of the calculations that are used to determine your handicap.
It's not an official USGA Handicap Index since you need to be a member of a sanctioned club to have a recognized handicap, but it will calculate exactly the same.
Here are the formulas for the calculation:
Handicap Differentials:
(Adjusted Gross Score - Course Rating) x 113 / Slope
Round to nearest tenth
Handicap Index:
(Average Handicap Differential) x 0.96
Truncate to tenth (do not round)
Course Handicap:
Course Handicap = (Handicap Index) x (Slope) / 113
Once you have a Handicap Index, it will be converted to a Course Handicap based on the slope of the course.
I know this post is pretty long, but it's correct. A few of the other responses are way off.
- Rich