Closeness measures between distributions, between vectors, and between matrices abound. Many practical problems, however, call for measures of closeness between the first two moments of two unspecified distributions given only a sample of one of them or of a third distribution without other information. We present several metrics for such problems that demand special, distribution-dependent solutions, and show their good qualities. We demonstrate their rich applications in various areas, such as estimation performance analysis, efficiency of distributed fusion, metrized Kullback-Leibler divergence, decision performance evaluation, credibility of estimators, filter initialization, and empirical distribution function problems.