One way to quantify the usefulness of each feature (= variable = dimension), from the book Burns, Robert P., and Richard Burns. Business research methods and statistics using SPSS. Sage, 2008. (mirror), usefulness being defined by the features' discriminative power to tell clusters apart.
We usually examine the means for each cluster on each dimension using ANOVA to assess how distinct our clusters are. Ideally, we would obtain significantly different means for most, if not all dimensions, used in the analysis. The magnitude of the F values performed on each dimension is an indication of how well the respective dimension discriminates between clusters.
Another way would be to remove a specific feature and see how this impact internal quality indices. Unlike the first solution, you would have to redo the clustering for each feature (or set of features) you want to analyze.
FYI:
- Can a useless feature negatively impact the clustering?
- Can the choice of the measurement units of the features impact the clustering?
- Why vector normalization can improve the accuracy of clustering and classification?Why vector normalization can improve the accuracy of clustering and classification?
- What are the most commonly used ways to perform feature selection for k-means clustering?