How can I determine whether two distributions are the same?

One approach to determine if two distributions are the same is to conduct a K-S test. The K-S test compares the empirical distribution function of one sample to either that of another sample, or the cumulative distribution function of a theoretical distribution. Comparing the two distribution functions generates a D value, which represents the maximum distance between two curves, as well as a p-value, which represents the probability that the null hypothesis (stating that there is no difference between the two distributions) is true.