It is important to know the serial number and the manufacturer. Early M1903's manufactured by Springfield Armory with serial numbers of 800,000 and under as well as M1903s made at Rock Island Arsenal serial numbered 285507 and lower have been the subject of a "do not shoot" caution for many years due to the improper heat treatment of some of the receivers made in those serial number runs. Their value is affected by this.
Is it an '03 or an '03a3? The 03s typically had a ladder sight mounted on the receiver; the 03a3s typically had an aperture sight on the bridge of the receiver. There were a number of manufacturers of the 03a3 and thus some are rarer than others.