The two main streams in finding cluster structure from data could be to identify the number of natural clusters and, of course, to group the objects in a reasonable way. In order to achieve good results for these two, measuring goodness of clustering is required prior to beginning any related studies because it helps to establish a definition of cluster that could be ambiguous by individuals having different opinions on it. In this research we are concerned about the compactness and the connectivity of cluster as our goodness measurements. The former has been regarded as one of the most important properties that should be accomplished in a clustering task, whereas the latter that we think as a significant factor has received less attention. Since we believe that both are individually important, we employ them for better estimating the number of clusters and clustering objects. A new estimating method produces a set of promising estimates by measuring compactness and connectivity from clustered datasets which look similar to the original data but have an amount of perturbation, and then determines a single optimal number by majority voting scheme. The connectivity measure newly introduced in our research is also used as an objective to be achieved in clustering objects. We propose a new clustering algorithm, named as CNCLUST that works in a way to optimize the quantity of connectivity. The proposed clustering algorithm is a greedy heuristic that looks like a single linkage method, but it is distinguishable by the fact that it first considers local compactness of objects and later incorporates it into global connectivity. We conducted numerical experiments in order to evaluate the performances of the proposed methods based on simulated datasets and a real data. The results seem optimistic.