We develop and investigate methods for assessing significance in obtained clustering solutions. The developed procedures compare two  solutions and declares the more complicated one to be a better candidate if there is significant improvement in  the goodness-of-fit criterion. Methodology is developed separately for both the model-based and nonparametric distance-based clustering algorithms. This leads to a comprehensive tool called the quantitation map which displays significance and quantitatively summarizes all pairwise clustering comparisons. This map can be used for, among other applications, deciding on the best among a set of candidate clustering solutions. We illustrate and evaluate performance on some standard classification datasets and simulation experiments. The distribution-free methodology is applied to the problem of color quantization of images, and demonstrated to be a viable approach for determining the minimal and optimal numbers of colors needed to adequately display an image without significant loss in resolution, while the model-based methodology is applied to a study of the voting preferences of senators in the 109th US Congress.

 (This talk is based on joint work with Soumendra Lahiri of Texas A & M University, Volodymyr Melnykov of North Dakota State University and Wei-Chen Chen of Iowa State University, and supported in part by the National Science Foundation under its Grant No. CAREER-DMS 0437555.)