The Sample Covariance Matrix (SCM) is widely used in signal processing applications which require the estimation of the data covariance matrix. Indeed it exhibits good statistical properties and tractability. However its performance can become very bad in context of non-Gaussian signals, contaminated or missing data. In that case M-estimators provide a good alternative. They have been introduced within the framework of elliptical distributions which encompass a large number of well-known distributions as for instance the Gaussian, the K-distribution or the multivariate Student (or t) distribution. In this paper, we show that with an appropriate normalization, the SCM and M-estimators have the same asymptotic behavior. More precisely, they share the same asymptotic covariance up to a scale factor. Tyler (1983) obtains similar results but we propose here a simpler proof for the case of M-estimators. The important consequence is that the SCM can easily be replaced by M-estimators with minor changes in performance analysis of signal processing algorithms. This result is highlighted by simulations in Direction-Of-Arrival (DOA) estimation using a MUltiple SIgnal Classification (MUSIC) approach. In this paper, we address the case of real data. These results have also been extended to the complex case but, due to the lack of space and for clarity of the presentation, this generalization will be omitted and will be addressed later.