A More Precise Elbow Method For Optimum K-means Clustering

Indra Herdiana; Mokhamad Alfin Kamal; Triyani Triyani; Mutia Nur Estri; Renny Renny

doi:10.22342/jims.v32i3.1862

Indra Herdiana ⁽¹⁾ , Mokhamad Alfin Kamal ⁽²⁾ , Triyani Triyani ⁽³⁾ , Mutia Nur Estri ⁽⁴⁾ , Renny Renny ⁽⁵⁾

(1) Mathematics Study Program, Jenderal Soedirman University, Indonesia,

(2) Mathematics Study Program, Jenderal Soedirman University, Indonesia,

(3) Mathematics Study Program, Jenderal Soedirman University, Indonesia,

(4) Mathematics Study Program, Jenderal Soedirman University, Indonesia,

(5) Mathematics Study Program, Jenderal Soedirman University, Indonesia

https://doi.org/10.22342/jims.v32i3.1862

Issue
Vol. 32 No. 3 (2026): SEPTEMBER

Submitted
November 26, 2024

Accepted
March 7, 2026

Published
September 1, 2026

Keywords:

k-means clustering, elbow method, angle between lines

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Abstract

K-means clustering is an unsupervised clustering method that requires an initial decision of number of clusters. One method to determine the number of clusters is the elbow method, a heuristic method that relies on a graphical plot of the within-cluster sum of squares against the number of clusters. The method uses the number based on the elbow point, the point closest to $90^\circ$ that indicates the most optimum number of clusters. This research improves the elbow method such that the selection of cluster numbers is unbiased towards visual interpretation. We use the analytical geometric formula to calculate an angle between lines and real analysis principle of derivative to simplify the elbow point determination. We also consider every possibility of the elbow method graph behaviour such that the algorithm is universally applicable. The result is that the elbow point can be measured precisely with a simple algorithm that does not involve complex functions or calculations. This improved method gives an alternative of more reliable cluster determination method that contributes to more optimum k-means clustering, obtained by the fewest clusters with the lowest error.

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