This paper presents an approach to geographical hypothesis testing based on the concept of expected information. The expected information formula and its relationship to other entropy formulas is first introduced and this concept is then used to test various hypotheses concerning the distribution of population and its density, in the New York, London, and Los Angeles regions. A related method of analysis based on the idea of deriving equivalent forms of system in which entropy is maximized and expected information minimized, is then presented and this provides alternative ways in which the various hypotheses can be tested. Finally, the use of the spatial entropy formula in fitting continuous population density functions to cities is explored and some comparative tests with other methods of estimation are presented. The ease with which the entropy estimator method can be used in this manner is then offset against its disadvantages, and in conclusion, these techniques are drawn together and evaluated.