In protein molecules each residue has a different ability to form contacts. In this paper, we calculated the number of contacts per residue and investigated the distribution of residue-residue contacts from 495 globular protein molecules using Contacts of Structural Units (CSU) software. It was found that the probability P(n) of amino acid residues having n pairs of contacts in all contacts fits Gaussian distribution very well. The distribution function of residue-residue contacts can be expressed as: P(n) = P0 + aexp[-b(n - nc)2]. In our calculation, P0 = -0.06, a = 11.4, b = -0.04 and nc = 9.0. According to distribution function, we found that those hydrophobic (H) residues including Leu, Val, Ile, Met, Phe, Tyr, Cys, and Trp residues have large values of the most probable number of contact nc, and hydrophilic (P) residues including Ala, Gly, Thr, His, Glu, Gln, Asp, Asn, Lys, Ser, Arg, and Pro residues have the small ones. We also compare with Fauchere-Pliska hydrophobicity scale (FPH) and the most probable number of contact nc for 20 amino acid residues, and find that there exists a linear relationship between Fauchere-Pliska hydrophobicity scale (FPH) and the most probable number of contact nc, and it is expressed as: nc = a + b × FPH, here a=8.87, and b=1.15. It is important to further explain protein folding and its stability from residue-residue contacts.