The classical Pearson’s procedure for testing whether a random sample has been drawn from a continuous distribution is based on the ‘difference’ of the observed cell counts and their model based counterpart. The test statistics known as the chi-square statistics can be very sensitive to the choice of cell origin. Different choice of cell origin may lead to different result of goodness of fits test. Worse still is that test based on grouping of data is often expected to be less powerful. To cope with the above two problems, Wu and Deng (2010) proposed to repeatedly partition the sample space into k cells for L times to obtain L respective chi-square statistics, and use their average to serve as the test statistics. Call the resultant test statistics the averaged shifted chi-square statistics (ASCS). The purpose of this thesis is to derive the exact distributions of ASCS for some small values of (L,k) by exhaustive permutation of all possible values ASCS. By comparing the exact distributions with the limit distribution of ASCS, we find that the limit distribution approximates well its target exact counterpart. A simple method is proposed to improve the approximation of the exact distribution by the limit distribution. Simulation study reveals that the power of ASCS is less sensitive to the choice cell origin and that ASCS can be more powerful as the values of k increases.