How to interpret the dendrogram of a hierarchical cluster analysis

Consider the R example below:

plot( hclust(dist(USArrests), "ave") )

1. What exactly does the y-axis "Height" mean?

2. Looking at North Carolina and California (rather on the left). Is California "closer" to North Carolina than Arizona? Can I make this interpretation?

3. Hawaii (right) joins the cluster rather late. I can see this as it is "higher" than other states. In general how can I interpret the fact that labels are "higher" or "lower" in the dendrogram correctly?

• Answers in ?hclust. – Scortchi Jan 15 '14 at 11:15
• The positions of the labels have no meaning. If you don't understand the y-axis then it's strange that you're under the impression to understand well the hierarchical clustering. – Stéphane Laurent Jan 15 '14 at 11:25
• Please also be aware that hierarchical clustering generally does not give you hierarchical (tree) classification. Average method (which you used) does not, in particular. See last point here. – ttnphns Jan 15 '14 at 12:05
• The position of a label has a little meaning though. The higher the position the later the object links with others, and hence more like it is an outlier or a stray one. – ttnphns Jan 15 '14 at 12:18
• @StéphaneLaurent You are right that this sound like a contradiction. On there hand I still think I am able to interpet a dendogram of data that I know well. Furthermore the position of the lables has a little meaning as ttnphns and Peter Flom point out. Finally your comment was not constructive to me. – Richard Jan 15 '14 at 12:39

1) The y-axis is a measure of closeness of either individual data points or clusters.

2) California and Arizona are equally distant from Florida because CA and AZ are in a cluster before either joins FL.

3) Hawaii does join rather late; at about 50. This means that the cluster it joins is closer together before HI joins. But not much closer. Note that the cluster it joins (the one all the way on the right) only forms at about 45. The fact that HI joins a cluster later than any other state simply means that (using whatever metric you selected) HI is not that close to any particular state.

• Thus "height" gives me an idea of the value of the link criterion (as here) - in my case the average distance of clusters to each other. Is this right? Thanks! – Richard Jan 15 '14 at 12:43
• Isn't the y-axis a measure of dissimilarity between clusters and points? I.e. negative the closeness, because it's largest when things are the most dissimilar, not the other way around @PeterFlom – Felipe Almeida Aug 4 '17 at 17:39

I had the same questions when I tried learning hierarchical clustering and I found the following pdf to be very very useful.

http://www.econ.upf.edu/~michael/stanford/maeb7.pdf

Even if Richard is already clear about the procedure, others who browse through the question can probably use the pdf, its very simple and clear esp for those who do not have enough maths background.

• Just want to re-iterate that the linked pdf is very good. – Heisenberg Feb 10 '15 at 8:21
• Reference: Klimberg, Ronald K. and B. D. McCullough. 2013. “Chapter 7: Hierarchical Cluster Analysis.” in Fundamentals of predictive analytics with JMP. Cary, NC: SAS Institute. – jay.sf Feb 11 at 12:36

The horizontal axis represents the clusters. The vertical scale on the dendrogram represent the distance or dissimilarity. Each joining (fusion) of two clusters is represented on the diagram by the splitting of a vertical line into two vertical lines. The vertical position of the split, shown by a short bar gives the distance (dissimilarity) between the two clusters.