Optical tweezers are a versatile tool in biophysics and have matured from a tool of manipulation to a tool of precise measurements. We argue here that the data analysis with advantage can be developed to a level of sophistication that matches that of the instrument. We review methods of analysis of optical tweezers data, primarily based on the power spectra of time series of positions for trapped spherical objects. The majority of precise studies in the literature are performed on in vitro systems, whereas in the present work, an example of an in vivo system is presented for which precise power spectral analysis is both useful and necessary. The biological system is the cytoplasm of fission yeast, Schizosaccharomyces pombe in which we observe subdiffusion of lipid granules. In a search for the cause of subdiffusion, we chemically disrupt the actin network in the cytoplasm and further consider in vitro networks of filamenteous actin undergoing similar chemical disruption.