The circadian clock, responsible for coordinating organism function with daily and seasonal changes in the day-night cycle, is controlled by a complex protein network that constitutes a robust biochemical oscillator. Deterministic ordinary differential equation models have been used extensively to model the behavior of these central clocks. However, due to the small number of proteins involved in the circadian oscillations, mathematical models that track stochastic variations in the numbers of clock proteins may reveal more complex and biologically relevant behaviors. In this paper, we compare the response of a robust yet detailed deterministic model for the mammalian circadian clock with its corresponding stochastic version that takes into account low protein number noise. We then use signal analysis techniques in order to examine differences in behavior among components of the stochastic system oscillator. This approach reveals differences in the system response between the stochastic and deterministic model and also allows us to extend bifurcation analysis into the stochastic domain. From our analysis of the unfitted stochastic model, we propose novel explanations of some previous experimental results.