In this paper, we focus on binary sequences obtained from piecewise logistic map over integers, and show the relation between the Lyapunov exponents of the maps and results of the NIST tests for the sequences. When the Lyapunov exponent is a negative value, we confirm that the branch diagram is sparse and the bit occurrence rate is also greatly biased. We also give a sample of that the positive/negative signs of the Lyapunov exponents mostly coincide on the evaluations of the NIST tests. From the branching diagram of the piecewise logistic map, we can moreover find that there are many good pseudorandom sequences generated by the individual control parameters of the map.