Hierarchical cluster analysis can calculate distances using a variety of different distance measures (Euclidean, Euclidean squared, Block etc.), you can pick the distance measure you want to use. This is just how we calculate distances between clusters (or how we tell whatever program we're using to calculate distances).
There are lots of different measures, but generally take the form of:
D$_{xy}$ = ( $\Sigma_i$ |x – y|$^p$ )$^{1/q}$
Where D = distance, x and y are scores for the i-th person on variables x and y, and p and q are different values depending on which distance metric we're using (e.g., both 2 if we're using the Euclidian distance measure).
So, roughly speaking, we can pick which "space" we're going to work in.
After we do that, there is a second thing we have to decide!
Where do we calculate the distances from when we start to amalgamate individual data points together? As, when we amalgamate our data points based on their distances to make clusters, we then have to calculate new distances. We have made a cluster with individual data points, and now have to decide what should count as its new coordinates when calculating distances.
Do we find the centre of the new cluster and measure from that (which would be the centroid method)? Do we just pick the closest member of our cluster to our other data point and measure from there (single linkage)? Do we pick the furthest data point in the cluster and measure from there (complete linkage)?
The method we pick just decides where we measure the distance between clusters from. We then use the distance measure we decided on earlier to calculate those distances. There's no strong reason to think that if we pick the Euclidian distance metric, we have to then use the centroid method. All of these are choices that we can make, and that will impact the clusters that we end up constructing.
