Quantum computation in noisy, near-term implementations holds promise across multiple domains ranging from chemistry and many-body physics to machine learning, optimization, and finance. However, experimental and algorithmic shortcomings such as noise and decoherence lead to the preparation of imperfect states which differ from the ideal state and hence lead to faulty measurement outcomes, ultimately undermining the reliability of near-term quantum algorithms. It is thus crucial to accurately quantify and bound these errors via intervals which are guaranteed to contain the ideal measurement outcomes. To address this desire, based on the formulation as a semidefinite program, we derive such robustness intervals for the expectation values of quantum observables using only their first moment and the fidelity to the ideal state. In addition, to get tighter bounds, we take into account second moments and extend bounds for pure states based on the non-negativity of Gram matrices to mixed states, thus enabling their applicability in the NISQ era where noisy scenarios are prevalent. Finally, we demonstrate our results in the context of the variational quantum eigensolver (VQE) on noisy and noiseless simulations.