Using the standard deviation of the real samples μ n ≥ … ≥ μ 1 and λ n ≥ … ≥ λ 1 , we refine the Chebyshev's inequality (refer to [5]), As a consequence, we obtain a new proof of the Benedetti's inequality (refer to [1], [2] and [4]) where Cov [μ, λ], s (μ) and s (λ) denote the covariance, and the standard deviations (≠ 0) of the sample vectors μ = (μ 1 , …, μ n ) and λ = (λ 1 , …, λ n ), respectively. We can also get very interesting applications to eigenvalues and singular values perturbation theory. For some kinds of matrices, the result that we present improves the well known Homand-Weiland's inequality.