An existence theorem for a bias of the mean in the presence of data dispersion is proved. The ultimate aims are to use this theorem to explain the well-known problems of utility and decision theories, such as risk aversion, the underweighting of high and the overweighting of low probabilities, the Allais paradox, etc. The results may be used to estimate preferences, choices, decisions, (ir)rational behavior at data uncertainty, noises and experimental errors in experiments interpretation, probability theory, statistics, management, finance, investment, insurance, etc.
