Communication overhead is one of the major performance bottlenecks in large-scale distributed computing systems, in particular for machine learning applications. Conventionally, compression techniques are used to reduce the load of communication by combining intermediate results of the same computation task as much as possible. Recently, via the development of coded distributed computing (CDC), it has been shown that it is possible to enable coding opportunities across intermediate results of different computation tasks to further reduce the communication load. We propose a new scheme, named compressed coded distributed computing (in short, compressed CDC), which jointly exploits the above two techniques (i.e., combining the intermediate results of the same computation and coding across the intermediate results of different computations) to significantly reduce the communication load for computations with linear aggregation (reduction) of intermediate results in the final stage that are prevalent in machine learning (e.g., distributed training algorithms where partial gradients are computed distributedly and then averaged in the final stage). In particular, compressed CDC first compresses/combines several intermediate results for a single computation, and then utilizes multiple such combined packets to create a coded multicast packet that is simultaneously useful for multiple computations. We characterize the achievable communication load of compressed CDC and show that it substantially outperforms both combining methods and CDC scheme.