Straw-K: Geometric Optimization of Cluster-Number Selection Using a ShortStraw-Inspired Elbow Criterion

Background: Choosing the number of clusters is still one of the least settled steps in a K-means analysis. The within-cluster sum of squares (WCSS) elbow plot is easy to understand, yet visual inspection is difficult to reproduce, and different validity indices often favor different values because they measure different aspects of the partition. Methods: We propose and evaluate Straw-K, a geometric cluster-number selection method that scores local bends on the normalized WCSS curve. Straw-K is a coined method name, not an acronym. Its construction adapts the local-window and chord-to-path geometry used in ShortStraw corner detection to discrete cluster-number selection; the contribution is the WCSS-specific criterion and its minimization over candidate values of K, rather than a claim that the chord-to-path relation itself is new. Twelve empirical datasets were analyzed without a general sample-size cap. Seven selectors were compared over eight K-means random-state seeds: Straw-K, the official Kneedle implementation, Silhouette, Calinski–Harabasz, Davies–Bouldin, the Gap statistic with B=50, and the L-method. Agreement with the recorded class count, absolute error, seed stability, runtime, the Friedman test, Holm-adjusted paired Wilcoxon tests, an exploratory ambiguity margin, and a separate Digits/PCA analysis were examined. Results: Across the 96 dataset–seed evaluations available for each method, Straw-K matched the recorded class count in 44.8% of cases and produced a mean absolute error of 1.54, the highest exact-match rate and lowest mean error in the descriptive table. The Friedman test detected an overall difference among the seven selectors (χ2=13.51, p=0.036), but none of the 21 paired Wilcoxon comparisons remained significant after Holm correction (minimum adjusted p=0.123). Gap was the slowest method in the independent uncached timing analysis. The ambiguity index showed only a weak, non-significant association with seed instability (Spearman ρ=0.33, p=0.303) and is therefore treated as descriptive rather than calibrated uncertainty. Conclusions: Taken together, the experiments suggest a fairly specific use for Straw-K: it offers an inspectable ranking of bends once the short WCSS support curve has been computed. The method did not recover a unique cluster count in every dataset, and Digits remained particularly difficult. The class labels used in the benchmark are best understood as external comparison points; they do not by themselves define the geometry that K-means will uncover.