The current document is directed to methods and systems that collect metric data within computing facilities, including large data centers and cloud-computing facilities. In a described implementation, two or more metric-data sets are combined to generate a multidimensional metric-data set. The multidimensional metric-data set is compressed for efficient storage by clustering the multidimensional data points within the multidimensional metric-data set to produce a covering subset of multidimensional data points and by then representing the multidimensional-data-point members of each cluster by a cluster identifier rather than by a set of floating-point values, integer values, or other types of data representations. The covering set is constructed to ensure that the compression does not result in greater than a specified level of distortion of the original data.