Understanding chromatin function requires knowing the precise location of nucleosomes. MNase-seq methods have been widely applied to characterize nucleosome organization in vivo, but generally lack the accuracy to determine the precise nucleosome positions. Here we develop a computational approach leveraging digestion variability to determine nucleosome positions at a base-pair resolution from MNase-seq data. We generate a variability template as a simple error model for how MNase digestion affects the mapping of individual nucleosomes. Applied to both yeast and human cells, this analysis reveals that alternatively positioned nucleosomes are prevalent and create significant heterogeneity in a cell population. We show that the periodic occurrences of dinucleotide sequences relative to nucleosome dyads can be directly determined from genome-wide nucleosome positions from MNase-seq. Alternatively positioned nucleosomes near transcription start sites likely represent different states of promoter nucleosomes during transcription initiation. Our method can be applied to map nucleosome positions in diverse organisms at base-pair resolution. DOI: http://dx.doi.org/10.7554/eLife.16970.001