Ron Guth: 1803 Half Dollars are found with either a Large or a Small 3 in the date. The most noticeable difference between the two varieties is the thickness and shape of the top bar and serif of the 3's. The Large 3 is a fairly common variety and appears to be roughly five times more common than the Small 3 variety.
The 1803 Large 3 Half Dollar is extremely rare in Mint State, with only five examples certified thus far by PCGS (as of February 2012). The finest example certified by PCGS is a single MS-64.
Hain Family Collection - Stack's, sold privately in 1/1990 - Stack's 1/2002:1220, $25,300 - D. Brent Pogue Collection - Stack’s/Bowers & Sotheby’s 5/2015:1107, $76,375