Urbanization is an index of transformation from traditional rural economies to modern industrial one. It is a progressive concentration of population in cities/towns. To describe the size distribution of cities, we use Zipf’s law, which states that the size distribution of cities follows a simple Pareto distribution with shape parameter equal to 1. In this paper we have performed a test for Zipf's law in respect for Andhra Pradesh city and town sizes distribution for the period 1951-2001. The analysis of data reveals that class I cities dominate Andhra Pradesh urban scene in terms of their share in urban population. We have performed sensitivity analysis and observed the resulting impact on the Pareto coefficient. An indepth study of the City Size Distribution demonstrates that the value of the Pareto coefficient increases when the threshold population increases for 2001 census data.