Online social media such as Twitter are widely used for mining public opinions and sentiments on various issues and topics. The sheer volume of the data generated and the eager adoption by the online-savvy public are helping to raise the profile of online media as a convenient source of news and public opinions on social and political issues as well. Due to the uncontrollable biases in the population who heavily use the media, however, it is often difficult to measure how accurately the online sphere reflects the offline world at large, undermining the usefulness of online media. One way of identifying and overcoming the online-offline discrepancies is to apply a common analytical and modeling framework to comparable data sets from online and offline sources and cross-analyzing the patterns found therein. In this paper we study the political spectra constructed from Twitter and from legislators' voting records as an example to demonstrate the potential limits of online media as the source for accurate public opinion mining, and how to overcome the limits by using offline data simultaneously.