This work presents a novel formulation of the ice nucleation spectrum, i.e. the function relating the ice crystal concentration to cloud formation conditions and aerosol properties. The new formulation relies on a statistical view of the ice nucleation process and explicitly accounts for the dependency of the ice crystal concentration on temperature, supersaturation, cooling rate, and particle size, and, in the case of heterogeneous ice nucleation, on the distributions of particle area and surface composition. The new formulation is used to generate ice nucleation parameterizations for the homogeneous freezing of cloud droplets and the heterogeneous deposition ice nucleation on dust and soot ice nuclei. For homogeneous freezing, it was found that by increasing the dispersion in the droplet volume distribution the fraction of supercooled droplets in the population increases. For heterogeneous ice nucleation it was found that ice nucleation on efficient ice nuclei (IN) shows features consistent with the singular hypothesis (characterized by a lack of temporal dependency of the ice nucleation spectrum) whereas less efficient IN tend to display stochastic behavior. Analysis of empirical nucleation spectra suggested that inferring the aerosol heterogeneous ice nucleation properties from measurements of the onset supersaturation and temperature may carry significant error as the variability in ice nucleation properties within the aerosol population is not accounted for. This work provides a simple and rigorous ice nucleation framework were theoretical predictions, laboratory measurements and field campaign data can be reconciled, and that is suitable for application in atmospheric modeling studies.