This method of clustering is very different from the Joining (Tree Clustering) and Two-way Joining methods. Suppose that you already have hypotheses concerning the number of clusters in your cases or variables. You may want to "tell" the computer to form exactly 3 clusters that are to be as distinct as possible. This is the type of research question that can be addressed by the k-means clustering algorithm. In general, the k-means method will produce exactly k different clusters of greatest possible distinction.

Example. In the physical fitness example (see Two-Way Joining), the medical researcher may have a "hunch" from clinical experience that her heart patients fall basically into three different categories with regard to physical fitness. She might wonder whether this intuition can be quantified, that is, whether a k-means cluster analysis of the physical fitness measures would indeed produce the three clusters of patients as expected. If so, the means on the different measures of physical fitness for each cluster would represent a quantitative way of expressing the researcher's hypothesis or intuition (i.e., patients in cluster 1 are high on measure 1, low on measure 2, etc.).

Computations. Computationally, you can think of this method as analysis of variance (ANOVA) "in reverse." The program will start with k random clusters, and then move objects between those clusters with the goal to 1) minimize variability within clusters and 2) maximize variability between clusters (see also Differences in k-Means Algorithms in Generalized EM & k-Means Cluster Analysis vs. Cluster Analysis). This is analogous to "ANOVA in reverse" in the sense that the significance test in ANOVA evaluates the between group variability against the within-group variability when computing the significance test for the hypothesis that the means in the groups are different from each other. In k-means clustering, the program tries to move objects (e.g., cases) in and out of groups (clusters) to get the most significant ANOVA results. (Because, among other results, the ANOVA results are part of the standard output from a k-means clustering analysis, you may want to refer to ANOVA/MANOVA to learn more about that method.)

Interpretation of results. Usually, as the result of a k-means clustering analysis, we would examine the means for each cluster on each dimension to assess how distinct our k clusters are. Ideally, we would obtain very different means for most, if not all, dimensions used in the analysis. The magnitude of the F values from the analysis of variance performed on each dimension is another indication of how well the respective dimension discriminates between clusters.

See also, Differences in k-Means Algorithms in Generalized EM & k-Means Cluster Analysis vs. Cluster Analysis and Exploratory Data Analysis and Data Mining Techniques.