We investigate the upper and lower bounds of the quantization distortion for identically and independently distributed source in the finite-block length regime. Based on the convex optimization framework of the rate-distortion theory we derive a lower bound for the quantization distortion for finite block length, which can be proved to be larger than the asymptotical distortion results of the rate-distortion theory. We also derive two upper bounds for the quantization distortion based on random quantization codebook, which can achieve any distortion above the asymptotical one. Moreover, we apply the proposed upper and lower bounds for two types of sources, the discrete binary symmetric source and the continuous Gaussian source.