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This paper describes a cell averaging method for the Chebyshev approximations of first order hyperbolic equations in conservation form. We present formulas for transforming between pointwise data at the collocation points and cell averaged quantities, and vice-versa. This step, trivial for the finite difference and Fourier methods, is nontrivial for the global polynomials used in Spectral methods. We then prove that the cell averaging methods presented are stable for linear scalar hyperbolic equations and present numerical simulations of shock-density wave interaction using the new cell averaging Chebyshev methods. Keywords: Pseudospectral methods; Cell averaging techniques; Shock waves; Conservation laws.