Symbolic data analysis (SDA) is an emerging area of statistics based on aggregating individual level data into group-based distributional summaries (symbols), and then developing statistical methods to analyse them. It is ideal for analysing large and complex datasets, and has immense potential to become a standard inferential technique in the near future. However, existing SDA techniques are either non-inferential, do not easily permit meaningful statistical models, are unable to distinguish between competing models, and are based on simplifying assumptions that are known to be false. Further, the procedure for constructing symbols from the underlying data is erroneously not considered relevant to the resulting statistical analysis. In this paper we introduce a new general method for constructing likelihood functions for symbolic data based on a desired probability model for the underlying classical data, while only observing the distributional summaries. This approach resolves many of the conceptual and practical issues with current SDA methods, opens the door for new classes of symbol design and construction, in addition to developing SDA as a viable tool to enable and improve upon classical data analyses, particularly for very large and complex datasets. This work creates a new direction for SDA research, which we illustrate through several real and simulated data analyses.