# Is clustering within cluster a sensible analysis?

Using K-means I have clustered my data into 3 clusters. After further analysis on these clusters I found the following:

• Cluster 1 - Low
• Cluster 2 - Medium
• Cluster 3 - High

Again, I clustered the data within Cluster 3 (High) into 3 clusters.

Now, I was asked how feasible/rational is this analysis? Why not just derive 6 clusters in the first place?

The objective of k-means, for a given $k$, is to find a partition $S = S_1 \bigcup \cdots \bigcup S_k$ which is
$$\arg \min_{S} \sum_{i = 1}^k \sum_{x \in S_i} \left| x - \mu_i \right|^2 . \; (1)$$
In your second, non-hierarchical, case, $k = 6$. In your first, hierarchical case, you first use $k = 3$, then continue until $k = 6$. This means that you are solving
$$\arg \min_{S; H(S)} \sum_{i = 1}^k \sum_{x \in S_i} \left| x - \mu_i \right|^2 , \; (2)$$
where $H(S)$ is a constraint on the hierarchy of the sets. Since a constraint cannot lower the minimum, (2) cannot be lower than (1). If your k-mean applications actually approximate the minimums, then it is unlikely that (2) will work better than (1) in terms of the minimization objective.