Can the extent of variability within a dataset be reflected through clustering?

Question

As an example: I need to compare the extent of variability amongst houses belonging to 4 different architectural eras - I want to see how different the houses are within each group and then compare such variability between groups (example of hypothetical conclusion: the houses from the baroque era had the most variability when compared to the other 3 groups).

Different descriptive variables on all houses are available (area, number of rooms, number of floors etc).

I wish to use a technique which can take into account all variables.

Would it make sense to perform clustering within each individual group, on the basis that the higher the optimal number of clusters, the larger the extent of variability between houses within each group? Would hierarchical clustering fit?

If not, what would you suggest please?

Answer 1

I think it would make sense to compare the distribution (and means) of standard deviations for each variable between the groups. So the mean of the standard deviation for each era would be something like:

$ \overline{\sigma}_{Era} = \frac{\sum_{i = (features)} \sigma_{i}}{n} $ where $n$ is the number of features for each era, and $ \sigma_{i} $ is the standard deviation for one feature of the era.

I think interpreting the results of this would be more straight forward and would be simple to compute. It would also be easy to visualize, as you could just make a box plot for the feature standard deviations for each architectural era and you'd be able to easily see which era has the greatest variance.