A nested tree built by merging cases step by step. Why the merge rule changes the answer more than anything else you choose, and how to report a clustering honestly.
Hierarchical clustering groups cases by repeatedly merging whichever two are most similar, then merging the merged groups, until everything is joined in one tree. You then cut the tree at a chosen number of clusters to get your grouping.
Unlike k-means, it does not require you to decide the number of groups before running. The tree contains every solution from one cluster to n, and cutting is a separate decision made afterwards.
That flexibility is the appeal. The catch is that the tree's shape depends entirely on how you define the distance between two groups, which is the linkage rule, and different rules produce genuinely different trees from identical data.
This is not a technical footnote. Here is one dataset, 250 cases, cut at three clusters, under four merge rules:
Single 248 / 1 / 1 · Average 109 / 5 / 136 · Ward 64 / 51 / 135 · Complete 115 / 23 / 112
Single linkage defines group distance by the two closest members, which lets clusters grow by absorbing whatever is nearest. The result above is the classic failure mode, called chaining: one cluster holding 248 of 250 cases and two lone outliers. It is not wrong arithmetic; it is what the rule does when data has no clean gaps.
Complete linkage uses the two furthest members, producing compact, similarly sized clusters and a strong sensitivity to outliers.
Average linkage uses the mean distance between all pairs across two groups, sitting between the two extremes.
Ward merges whichever pair increases within-cluster variance least, and tends to produce balanced clusters of comparable size. It is the most common default in social science and usually the sensible starting point.
The practical rule: run more than one linkage and see whether the structure survives. A grouping that appears under Ward, average, and complete is worth interpreting. One that appears under a single rule is a property of that rule.
Clustering depends on distance, and distance depends on scale. A variable measured 0 to 100 will dominate one measured 1 to 5 simply because its numbers are bigger, regardless of which matters to your question.
Quanta reports whether the analysis was standardized directly in the summary tiles, so the choice is visible on the result rather than remembered.
Standardize unless every variable is already on the same scale and you specifically want the raw spread to count. Failing to standardize mixed-scale variables is the most common way a clustering answers a question about measurement units rather than about people.
Lay the data out one row per case, one column per clustering variable, all numeric.
Open Data Reduction and choose Hierarchical Clusters, then set your variables, the linkage rule, whether to standardize, and where to cut the tree in the Setup panel on the right.
The summary tiles report the cut, the linkage, the case count, and whether variables were scaled, and the Cluster sizes card gives the size of each cluster at your chosen k.
Change the linkage and rerun before you interpret anything. It takes seconds and it is the single most informative robustness check available for this method.
Use the action bar to add the result to a report, copy the table, or copy the APA text.
Clustering is descriptive. There is no p value here and no test of whether the clusters are real, so report it as an exploratory grouping.
A hierarchical cluster analysis of 18 standardized variables (N = 250) using Ward linkage produced three clusters of 64, 51 and 135 cases. Average and complete linkage produced comparable structure; single linkage did not, yielding a single cluster of 248 cases.
Report the linkage, the distance measure, whether variables were standardized, and how you chose the number of clusters. Without those four, the result cannot be reproduced.
Report the alternative linkages you tried and what they showed. Reporting only the rule that gave the tidiest answer is selective reporting.
Describe the clusters substantively before naming them. A cluster label is an interpretation, and readers will treat a named group as a discovered type unless you are careful to say it is a grouping produced by an algorithm.
Every result carries a ReliCheck Intelligence card with one action: explain this result in plain language. It runs entirely on your Mac with Apple Intelligence. No upload, no API key, nothing to configure. The model only explains the result Quanta has already computed and validated. It never computes a statistic, and it never applies a decision on your behalf.
From the action bar you can add the result to a report, copy the table, or copy the APA text. Exported reports carry the APA table, its note, and a caution specific to that analysis beneath it.
Quanta's engine is pure Swift on Apple's Accelerate framework, with no R or Python at runtime. Every engine is checked against an independent reference rather than internal consistency alone, and the automated suite exceeds 500 checks. Full record on the validation page.
Data Reduction is part of every Quanta subscription, $9.99 a month or $79.99 a year after the 30-day trial. One price for everyone. See pricing.
The analysis, the report builder, and the on-device explanations all work with the network off. Your dataset is never uploaded.
K-means clustering when you already know how many groups you want · Principal components reducing variables rather than grouping cases · Chi-square testing whether cluster membership relates to something else · the complete list on the analyses page.