This paper concerns the problem of estimating the noise standard deviation in different signal processing applications. The presented estimator derives from recent results in robust statistics based on sparsity hypotheses. More specifically, these theoretical results make the link between a standard problem in robust statistics (the estimation of the noise standard deviation in presence of outliers) and sparsity hypotheses. The estimator derived from these theoretical results can be applied to different signal processing applications where estimation of the noise standard deviation is crucial. In the present paper, we address speech denoising and Orthogonal Frequency Division Multiple Access (OFDMA). A relevant application should also be Communication Electronic Support (CES). For such applications, the algorithm proposed is a relevant alternative to the median absolute deviation (MAD) estimator.