  Symbolic Data Analysis (SDA) is a relatively new field of statistics that
extends conventional data analysis by taking into account intrinsic data
variability and structure. Unlike conventional data analysis, in SDA the
features characterizing the data can be multi-valued, such as intervals or
histograms. SDA has been mainly approached from a sampling perspective. In this
work, we propose a model that links the micro-data and macro-data of
interval-valued symbolic variables, which takes a populational perspective.
Using this model, we derive the micro-data assumptions underlying the various
definitions of symbolic covariance matrices proposed in the literature, and
show that these assumptions can be too restrictive, raising applicability
concerns. We analyze the various definitions using worked examples and four
datasets. Our results show that the existence/absence of correlations in the
macro-data may not be correctly captured by the definitions of symbolic
covariance matrices and that, in real data, there can be a strong divergence
between these definitions. Thus, in order to select the most appropriate
definition, one must have some knowledge about the micro-data structure.
