Correlations between sequence separation (in residues) and distance (in Angstrom) of any pair of amino acids in polypeptide chains are investigated. For each sequence separation we define a distance threshold. For pairs of amino acids where the distance between Ca atoms is smaller than the threshold, a characteristic sequence (logo) motif, is found. The motifs change as the sequence separation increases: for small separations they consist of one peak located in between the two residues, then additional peaks at these residues appear, and finally the center peak smears out for very large separations. We also find correlations between the residues in the center of the motif. This and other statistical analyses are used to design neural networks with enhanced performance compared to earlier work. Importantly, the statistical analysis explains why neural networks perform better than simple statistical data-driven approaches such as pair probability density functions. The statistical results also explain characteristics of the network performance for increasing sequence separation. The improvement of the new network design is significant in the sequence separation range 10-30 residues. Finally, we find that the performance curve for increasing sequence separation is directly correlated to the corresponding information content.