Differences Between K-Means, K-Medoids, and K-Modes

Clustering algorithms form the backbone of unsupervised machine learning, organizing data into meaningful groups without predefined labels. Among the most widely used partitioning methods, k-means, k-medoids, and k-modes appear deceptively similar—all partition data into k clusters and iteratively optimize cluster assignments. However, fundamental differences in how they represent clusters, measure distances, and handle different data types make each algorithm optimal for distinct scenarios. Understanding these differences transforms clustering from a black-box technique into a deliberate tool selection process where you choose the right algorithm based on your data characteristics and business requirements.

The confusion between these algorithms stems from their shared conceptual framework: initialize k cluster centers, assign points to nearest centers, update centers, and repeat until convergence. This common structure obscures critical distinctions in implementation details that determine whether an algorithm succeeds or fails on your specific dataset. K-means excels with numeric data exhibiting spherical clusters but breaks down with categorical data or outliers. K-medoids sacrifices some computational efficiency for robustness to outliers and interpretability. K-modes extends the partitioning approach to categorical data where traditional distance metrics fail. This article explores these differences in depth, providing the understanding needed to select and apply the appropriate algorithm for your clustering tasks.

Core Algorithmic Differences

How K-Means Works

K-means represents each cluster by its centroid—the arithmetic mean of all points assigned to that cluster. The algorithm alternates between two steps: assigning each point to the nearest centroid based on Euclidean distance, then recalculating centroids as the mean of assigned points. This approach assumes numeric data where means are meaningful and distance calculations are straightforward.

The centroid calculation for k-means directly averages coordinate values across all dimensions. For a cluster containing points (2, 3), (4, 5), and (6, 7), the centroid becomes ((2+4+6)/3, (3+5+7)/3) = (4, 5). This mathematical operation produces a point that minimizes the sum of squared distances to all cluster members, providing an optimal center in terms of variance minimization.

Distance measurement in k-means typically uses Euclidean distance, though variations exist. Euclidean distance between points (x₁, y₁) and (x₂, y₂) equals √[(x₂-x₁)² + (y₂-y₁)²], treating all dimensions equally. This metric assumes continuous numeric data and spherical cluster shapes—clusters should be roughly circular in 2D or spherical in higher dimensions for k-means to perform well.

The convergence criterion stops iteration when centroids no longer change significantly between iterations or when a maximum iteration count is reached. Convergence guarantees exist for k-means—it always converges to a local optimum, though not necessarily the global optimum. Multiple runs with different initializations help find better solutions since the algorithm is sensitive to initial centroid placement.

How K-Medoids Works

K-medoids differs fundamentally by representing each cluster with a medoid—an actual data point that minimizes dissimilarity to all other points in the cluster. Rather than computing means, the algorithm selects the most central existing point as the cluster representative. This constraint that cluster centers must be real data points makes medoids more interpretable and robust to outliers than centroids.

The medoid selection process examines all points in a cluster, computing the sum of distances from each candidate point to all other cluster members. The point with the minimum total distance becomes the medoid. For a cluster with points A, B, C, D, the algorithm calculates distance sums: sum_distances(A) = d(A,B) + d(A,C) + d(A,D), and similarly for B, C, D. The point with the smallest sum becomes the medoid.

PAM (Partitioning Around Medoids) is the most common k-medoids algorithm. It differs from k-means not just in medoid selection but in how it evaluates cluster quality. PAM considers the cost of swapping medoids with non-medoid points, accepting swaps that reduce total within-cluster dissimilarity. This process continues until no beneficial swaps exist, ensuring the selected medoids minimize overall cluster cost.

Distance metrics in k-medoids are more flexible than k-means. While Euclidean distance works, k-medoids naturally accommodates Manhattan distance, cosine similarity, or any dissimilarity measure appropriate for your data. This flexibility makes k-medoids suitable for data types where Euclidean distance is inappropriate, such as time series with dynamic time warping or text data with edit distances.

How K-Modes Works

K-modes extends the partitioning approach to categorical data where numeric operations like means are undefined. You cannot average “red,” “blue,” and “green” to get a meaningful cluster center, nor can you calculate Euclidean distance between color categories. K-modes replaces means with modes—the most frequent value in each attribute—and introduces a dissimilarity measure for categorical data.

The mode calculation for each attribute examines the frequency of each category among cluster members. For a cluster with animals [dog, cat, dog, dog, cat], the mode is “dog” as it appears most frequently. Multi-modal situations where multiple values tie for highest frequency are resolved by choosing arbitrarily or using the lexicographically smallest value, depending on implementation.

Simple matching dissimilarity measures distance between categorical points by counting differing attributes. For points (red, large, metal) and (red, small, wood), dissimilarity equals 2 since two attributes differ (size and material). The distance between points equals the number of mismatched categories, normalized by total attributes. This metric treats all mismatches equally, assuming no inherent ordering among categories.

The k-modes algorithm iterates similarly to k-means: assign points to the nearest mode based on simple matching distance, then update modes to the most frequent value in each attribute. Convergence occurs when modes stabilize, though k-modes can be more prone to local optima than k-means due to the discrete nature of mode selection.

Mathematical Foundations

Distance Metrics and Their Impact

The choice of distance metric fundamentally determines clustering behavior. Euclidean distance in k-means computes straight-line distances in multi-dimensional space, making it sensitive to magnitude differences across dimensions. Features with larger numeric ranges dominate distance calculations unless data is normalized. A dataset with age (0-100) and income (0-1,000,000) will cluster primarily by income without standardization.

Manhattan distance sums absolute differences across dimensions: |x₁-x₂| + |y₁-y₂|. This metric, sometimes called city-block distance, proves less sensitive to outliers than Euclidean distance since it doesn’t square differences. K-medoids frequently uses Manhattan distance for this robustness, especially with data containing occasional extreme values that shouldn’t dominate clustering.

Cosine similarity measures the angle between vectors rather than their magnitude, making it popular for high-dimensional data like text where document length shouldn’t affect similarity. Two documents discussing the same topics with different lengths should cluster together based on content, not word count. K-medoids naturally accommodates cosine similarity through its flexible distance framework.

Simple matching for categorical data in k-modes treats all attribute mismatches equally. This assumption may not hold—mismatching “color” might matter differently than mismatching “texture” for product categorization. Weighted variants exist where domain knowledge assigns different importance to attributes, though standard k-modes uses uniform weighting.

Optimization Objectives

K-means minimizes within-cluster sum of squared errors (WCSS), also called inertia. Mathematically: minimize Σᵢ Σₓ∈Cᵢ ||x – μᵢ||², where Cᵢ represents cluster i and μᵢ its centroid. This objective creates compact, spherical clusters by penalizing squared distances. The squared term means outliers contribute disproportionately to the objective function, explaining k-means’ outlier sensitivity.

K-medoids minimizes the sum of dissimilarities between points and their cluster medoids: minimize Σᵢ Σₓ∈Cᵢ d(x, mᵢ), where mᵢ is the medoid of cluster i and d() is the chosen dissimilarity measure. Without squaring distances, outliers affect the objective linearly rather than quadratically, providing natural robustness. The constraint that medoids must be actual data points adds interpretability—you can examine the medoid to understand what a “typical” cluster member looks like.

K-modes minimizes total dissimilarity using simple matching distance: minimize Σᵢ Σₓ∈Cᵢ δ(x, Mᵢ), where Mᵢ is the mode vector for cluster i and δ() counts mismatched attributes. This objective creates homogeneous clusters where members share many categorical values. The discrete nature of categorical data means small changes in cluster assignment can cause abrupt mode shifts, potentially causing instability during convergence.

Computational Complexity

K-means achieves O(nkdi) time complexity per iteration, where n = number of points, k = clusters, d = dimensions, i = iterations. The assignment step computes k distance calculations for each of n points (O(nkd)), and the update step averages assigned points (O(nd)). With typically small k and d, and convergence in few iterations (often 10-50), k-means scales well to large datasets.

K-medoids has O(k(n-k)²) complexity per iteration for the PAM algorithm, reducing to O(n²ki) in practical terms. Each iteration considers swapping each medoid with each non-medoid point, requiring distance recalculations for affected points. This quadratic scaling in n makes vanilla PAM impractical for datasets exceeding thousands of points, though optimized variants like CLARA and CLARANS improve scalability.

K-modes has similar complexity to k-means—O(nkdi)—since computing simple matching dissimilarity and updating modes both scale linearly with data size. However, practical performance depends on the number of categories per attribute. High-cardinality categorical attributes (hundreds of unique values) slow mode calculation and increase memory requirements compared to low-cardinality attributes.

Practical Considerations for Each Algorithm

When to Use K-Means

K-means excels with continuous numeric data exhibiting spherical or convex cluster shapes. Customer segmentation based on purchase amounts, spending frequency, and average order value clusters naturally with k-means. Scientific measurements like temperature, pressure, and concentration also suit k-means when relationships are roughly linear and clusters are compact.

Standardization becomes critical for k-means since features with different scales distort distance calculations. Always scale features to comparable ranges—common approaches include z-score normalization (mean 0, standard deviation 1) or min-max scaling (0-1 range). Without standardization, features with larger numeric ranges dominate clustering regardless of their actual importance.

The algorithm struggles with non-spherical clusters such as elongated or crescent-shaped patterns. Datasets with clear banana-shaped or concentric-ring structures typically frustrate k-means, which will force-fit spherical clusters that split natural groupings. In such cases, consider density-based methods like DBSCAN or spectral clustering instead of k-means.

Outliers severely impact k-means since they pull centroids toward extreme values. A handful of anomalous points can distort multiple clusters. Either remove obvious outliers during preprocessing or choose k-medoids for inherent robustness. The squared distance penalty in k-means’ objective function means single extreme points affect results disproportionately.

When to Use K-Medoids

K-medoids provides the optimal choice when robustness to outliers matters more than computational efficiency. Financial fraud detection, medical diagnosis clustering, or any domain with noisy data and occasional extreme values benefits from k-medoids’ resistance to being pulled by outliers. The linear distance penalty in its objective function reduces outlier influence compared to k-means’ quadratic penalty.

Interpretability requirements favor k-medoids since medoids are actual data points you can examine. In customer segmentation, presenting a specific customer profile (the medoid) as “typical” for each segment is more meaningful than presenting an artificial centroid that may not correspond to any real customer. Stakeholders can understand and relate to concrete examples better than abstract averages.

Custom distance metrics make k-medoids suitable for specialized data types. Time series clustering with dynamic time warping, graph clustering with edit distance, or biological sequence clustering with alignment scores all work naturally with k-medoids. As long as you can define a meaningful dissimilarity measure, k-medoids can cluster your data.

Small to medium datasets (hundreds to low thousands of points) fit k-medoids’ computational constraints. For larger datasets, variants like CLARA (Clustering Large Applications) sample the data and run PAM on samples, providing approximate solutions with better scalability. Alternatively, run k-means for initial clustering, then use k-medoids to refine cluster centers for the interpretability benefits.

When to Use K-Modes

Categorical data without meaningful ordering requires k-modes. Customer demographic attributes (gender, occupation, region, product category preferences) have no inherent numeric ordering that would allow k-means. Education levels might seem ordered, but the distance between “high school” and “bachelor’s” isn’t necessarily equal to the distance between “bachelor’s” and “master’s,” violating k-means’ assumptions.

Market basket analysis and purchase behavior clustering suit k-modes when working with product categories, brand preferences, or shopping channels rather than purchase amounts. Click stream analysis with page types, device categories, and referral sources also fits k-modes’ strengths. Survey data with multiple-choice responses provides another natural application.

Mixed data with both numeric and categorical attributes presents challenges. Pure k-means fails on categorical attributes, while pure k-modes wastes information in numeric attributes. The k-prototypes algorithm combines k-means and k-modes, using Euclidean distance for numeric attributes and simple matching for categorical attributes, weighting each component appropriately.

Low-cardinality categorical attributes work better than high-cardinality ones. With hundreds of unique categories per attribute, mode calculation becomes less meaningful and memory requirements grow. Consider aggregating rare categories into “other” or using dimensionality reduction techniques before applying k-modes. Binary categorical attributes (yes/no, present/absent) particularly suit k-modes’ simple matching dissimilarity.

Strengths and Limitations

K-Means Advantages and Drawbacks

K-means’ primary strength lies in computational efficiency and scalability. The algorithm handles millions of data points in reasonable time, making it the default choice for large-scale clustering problems. Implementations are highly optimized, with parallel versions leveraging multi-core CPUs and GPU implementations for even larger speedups.

Simplicity makes k-means easy to understand, implement, and explain to non-technical stakeholders. The intuitive concept of grouping points around central averages requires minimal mathematical background to grasp. This accessibility has made k-means ubiquitous in data science and machine learning pipelines.

However, k-means assumes spherical clusters of similar size, failing spectacularly when these assumptions violate. Elongated clusters, varying cluster densities, or non-convex shapes produce poor results. The algorithm also requires specifying k in advance, though techniques like the elbow method or silhouette analysis help determine appropriate values.

Initialization sensitivity means different random starts produce different results. K-means++ initialization alleviates this issue by selecting initial centroids that are spread apart, but multiple runs with different initializations remain best practice. The algorithm guarantees convergence to a local optimum, not the global optimum, making solution quality dependent on initialization luck.

K-Medoids Advantages and Drawbacks

Robustness to outliers represents k-medoids’ flagship advantage. The algorithm naturally resists being pulled by extreme values since it minimizes absolute distances rather than squared distances and constrains medoids to be actual data points. This robustness makes k-medoids preferable for real-world messy data where outliers commonly occur.

Interpretability through concrete exemplars helps stakeholders understand clustering results. Presenting actual customers as cluster representatives feels more tangible than abstract centroids. This explainability proves valuable in business contexts where decisions based on clustering need justification and buy-in.

Flexibility in distance metrics enables k-medoids to handle diverse data types and custom similarity measures. Whether using dynamic time warping for time series, Jaccard distance for sets, or domain-specific dissimilarity functions, k-medoids accommodates them naturally. This flexibility extends its applicability far beyond standard numeric data.

Computational complexity limits practical application to smaller datasets. The O(n²) scaling means doubling data size quadruples computation time, quickly becoming prohibitive. CLARA and CLARANS approximations help but sacrifice solution quality. For truly large datasets, k-means remains more practical despite its theoretical limitations.

K-Modes Advantages and Drawbacks

K-modes’ ability to cluster purely categorical data addresses a real need—much real-world data comes in categories rather than numbers. Without k-modes, practitioners must resort to problematic workarounds like arbitrary numeric encoding that introduces false ordering assumptions. K-modes handles categorical data natively and naturally.

Computational efficiency matches k-means, making k-modes scalable to large categorical datasets. The simple matching dissimilarity computes quickly, and mode calculation is straightforward. This efficiency enables clustering transactional databases with millions of categorical records.

Simple interpretability helps explain results. Unlike embeddings or dimensionality reduction that obscure relationships, k-modes clusters based on direct attribute matching that stakeholders understand immediately. A cluster where most members are “female,” “urban,” and “college-educated” communicates clearly without statistical translation.

Limited to categorical data restricts k-modes’ applicability. Datasets with primarily numeric features can’t use k-modes effectively. The algorithm also struggles with high-cardinality attributes—when attributes have hundreds of unique values, finding a meaningful mode becomes difficult and memory requirements explode. Preprocessing to reduce cardinality is often necessary.

Convergence and Initialization

Convergence Guarantees

K-means mathematically guarantees convergence to a local optimum in finite iterations. Each iteration either decreases the objective function (WCSS) or leaves it unchanged, and since there are finitely many ways to assign n points to k clusters, the algorithm must eventually stabilize. However, convergence to a local optimum doesn’t guarantee the global optimum—different initializations lead to different final clusters.

K-medoids through PAM also guarantees convergence since each swap either reduces total dissimilarity or leaves it unchanged. The finite number of possible medoid sets ensures eventual stabilization. Like k-means, convergence is to a local optimum, and solution quality depends heavily on initialization.

K-modes convergence is more complex due to the discrete nature of modes. Cycles can theoretically occur where the algorithm oscillates between cluster configurations without settling. Practical implementations include safeguards like maximum iteration limits and monitoring for cycling. Empirically, k-modes usually converges quickly, though less reliably than k-means.

Initialization Strategies

Random initialization selects k points randomly as initial centers—simple but often produces poor results. K-means++ improves on random selection by choosing initial centers that are spread apart probabilistically, dramatically improving final cluster quality and reducing iterations until convergence. K-means++ has become the default initialization for most implementations.

K-medoids initialization similarly benefits from intelligent selection. K-medoids++ adapts k-means++ concepts, selecting initial medoids that maximize minimum distance to already-selected medoids. BUILD algorithm for PAM greedily selects medoids that minimize total dissimilarity, providing good starting points for the main algorithm.

K-modes initialization faces unique challenges with categorical data. Random mode initialization or selecting k random points as initial modes works but can be unstable. Cao et al.’s initialization method selects initial modes by finding points that maximize total dissimilarity, similar to k-means++ logic but using simple matching distance.

Multiple random restarts remain the most robust strategy across all three algorithms. Running the algorithm 10-50 times with different initializations and selecting the solution with the best objective function value significantly improves results. The computational cost is manageable for k-means and k-modes but can be prohibitive for k-medoids on larger datasets.

Real-World Application Examples

Customer Segmentation Scenarios

B2C retail segmentation with purchase history, browsing behavior, and demographics illustrates algorithm selection nuances. Purchase amounts and frequency are numeric (k-means), preferred product categories are categorical (k-modes), and customer location might use specialized geo-distance metrics (k-medoids). K-prototypes, combining k-means and k-modes, handles the mixed data appropriately.

B2B client segmentation often emphasizes relationship characteristics: industry sector (categorical), annual revenue (numeric), contract length (numeric), and service tier (categorical). The presence of occasional very large clients (outliers) suggests k-medoids for robustness. Alternatively, handle outliers separately and cluster the main client base with k-means for efficiency.

Behavioral segmentation based on app usage patterns involves features like session duration (numeric), favorite features (categorical), device type (categorical), and usage frequency (numeric). Mixed data types suggest k-prototypes, though if interpretability matters most, clustering separately on numeric features (k-means) and categorical features (k-modes), then combining results provides insights.

Healthcare Applications

Patient clustering for treatment personalization uses symptoms (categorical), test results (numeric), medical history (categorical), and severity scores (numeric). K-medoids’ robustness suits clinical data with measurement errors and outliers. The medoid patients provide concrete examples of typical presentations within each cluster, aiding clinical decision support.

Disease outbreak clustering based on infection type (categorical), geographic location (spatial distance metric), transmission vector (categorical), and case counts (numeric) benefits from k-medoids’ flexible distance metrics. Custom distances can incorporate geographic proximity, genetic similarity of pathogens, and temporal clustering patterns simultaneously.

Text and Document Clustering

Topic clustering with bag-of-words representations uses word counts (numeric) or presence/absence (binary categorical). For count vectors, k-means with cosine similarity works well, treating documents as points in high-dimensional space. For binary representations, k-modes with simple matching naturally captures shared vocabulary without weighting word frequency.

Hashtag or keyword clustering for social media analysis involves purely categorical data—tags don’t have numeric meanings. K-modes groups posts by shared tags, revealing communities and discussion themes. The modes represent characteristic tag combinations defining each cluster.

Conclusion

The distinctions between k-means, k-medoids, and k-modes extend far beyond superficial naming differences, reflecting fundamental mathematical and practical considerations that determine success or failure for specific clustering tasks. K-means optimizes computational efficiency for numeric data with clean, spherical clusters. K-medoids trades speed for robustness and interpretability, excelling with outliers and custom distance metrics. K-modes extends partitioning methods to categorical data where traditional approaches fail. Understanding these differences enables deliberate algorithm selection aligned with your data characteristics, quality requirements, and computational constraints rather than defaulting to whichever algorithm is most familiar or convenient.

Successful clustering demands matching algorithm properties to data properties and business needs. Consider your data types, outlier prevalence, cluster shapes, interpretability requirements, and dataset size before committing to an algorithm. Often, experimenting with multiple approaches and comparing results provides insights that guide final selection. The computational cost of running both k-means and k-medoids on a sample, comparing cluster quality and interpretability, frequently justifies the investment when it informs a better final decision. By understanding not just how these algorithms differ but why those differences matter, you transform clustering from trial-and-error into systematic, well-justified analysis.