Umbral dots (UDs) are transient, bright features observed in the umbral region of a sunspot. We study the physical properties of UDs observed in sunspots of different sizes. The aim of our study is to relate the physical properties of UDs with the large-scale properties of sunspots. For this purpose, we analyze high-resolution G-band images of 42 sunspots observed by Hinode/SOT, located close to disk center. The images were corrected for instrumental stray light and restored with the modeled point-spread function. An automated multilevel tracking algorithm was employed to identify the UDs located in selected G-band images. Furthermore, we employed Solar Dynamics Observatory/HMI, limb-darkening-corrected, full-disk continuum images to estimate the sunspot phase and epoch for the selected sunspots. The number of UDs identified in different umbrae exhibits a linear relation to the umbral size. The observed filling factor ranges from 3% to 7% and increases with the mean umbral intensity. Moreover, the filling factor shows a decreasing trend with the umbral size. We also found that the observed mean and maximum intensities of UDs are correlated with the mean umbral intensity. However, we do not find any significant relationship between the mean (and maximum) intensity and effective diameter of UDs and the sunspot area, epoch, and decay rate. We suggest that this lack of relation could be due to either the distinct transition of spatial scales associated with overturning convection in the umbra or the shallow depth associated with UDs, or both.