This paper presents a novel general expression for the bit-error rate (BER) of one-dimensional constellations, focusing on uncoded performance over an additive white Gaussian noise (AWGN) channel. The authors analyze the performance of different demodulators, including the symbol-based demodulator (SD), the approximate bit-wise demodulator (ABD), and the exact bit-wise demodulator (BD). They prove that the SD and ABD are equivalent in terms of minimizing the symbol-error rate (SER) for any constellation and labeling. The paper introduces a new BER expression valid for any one-dimensional constellation and binary labeling, and analyzes the performance of the ABD for various constellations. The analysis is performed for bit patterns that form a labeling, and the paper derives a closed-form expression for the number of patterns that give different BER for equally spaced $M$-PAM constellations. The results show that the PBER for high SNR can be predicted by the coefficient with the smallest argument in the expression. The paper also discusses the classification of patterns and compares their performance, providing a detailed analysis of the number of classes for length-$M$ patterns.This paper presents a novel general expression for the bit-error rate (BER) of one-dimensional constellations, focusing on uncoded performance over an additive white Gaussian noise (AWGN) channel. The authors analyze the performance of different demodulators, including the symbol-based demodulator (SD), the approximate bit-wise demodulator (ABD), and the exact bit-wise demodulator (BD). They prove that the SD and ABD are equivalent in terms of minimizing the symbol-error rate (SER) for any constellation and labeling. The paper introduces a new BER expression valid for any one-dimensional constellation and binary labeling, and analyzes the performance of the ABD for various constellations. The analysis is performed for bit patterns that form a labeling, and the paper derives a closed-form expression for the number of patterns that give different BER for equally spaced $M$-PAM constellations. The results show that the PBER for high SNR can be predicted by the coefficient with the smallest argument in the expression. The paper also discusses the classification of patterns and compares their performance, providing a detailed analysis of the number of classes for length-$M$ patterns.