The instability of passive atomic frequency standards caused by shot noise introduced by the atomic reference is calculated. This instability can be characterized by a figure of merit for the atomic reference, and the asymptotic functions for the rms frequency fluctuation for long and short averaging times expressed in terms of this quantity. Measurements of the rms frequency fluctuation of cesium atomic beam and rubidium vapor frequency standards are compared with theory, and the predicted performances of various existing and proposed atomic standards are tabulated. It is feasible to build atomic beam frequency standards whose stability for averaging times less than one second is limited by noise in the crystal oscillator rather than by shot noise, a criterion generally met by rubidium vapor frequency standards. For long averaging times, shot noise will always be the fundamental limiting factor in atomic frequency standard stability.