There have been many studies on the spatial configuration of cities, but few attempts to quantify the difference in building patterns between the old and new parts of cities. This may be partly attributable to lack of suitable study methods. This paper presents a new application of statistical methods for quantifying the geometric difference between different parts of a city using, as a case study, the old (historical) and new parts of the city of Yazd in Iran. We measured 341 edge lengths of 4 bazaars, 302 edge lengths of 5 mosques and tombs, and 239 edge lengths of 3 schools. We also measured 6,804 edge lengths and the areas of 1,243 well-preserved courtyard houses in the old part and 4,948 edge lengths and the areas of 1,237 houses in the new part of the city. In the old part, all edge-length and house-area frequency distributions, to a first approximation, follow power laws, indicating that there are many small and very few large buildings. By contrast, in the new part the edge-length and house-area frequency distributions follow bimodal (two-peak) distributions. The calculated entropies (measures of dispersion) of the house edge lengths and areas in the old part are much higher than of those in the new part and provide a clear, quantitative measure of the geometric differences between the built-up structures of old and the new parts of the cities. The power-law distributions in the old part indicate a gradual and continuous variation in sizes of edge lengths and house areas, whereas the bimodal distributions in the new part indicate abrupt (discontinuous) changes in the edge lengths and house areas. The entropy results imply that the size distributions of houses in the old part are more dispersed than those in the new part, indicating more interconnected structures in the old part of the city. The results also demonstrate quantitatively that there is a lack of coherence between the structures of old and new parts of city.