I have a dataset that's nominally 16-dimensional. I have about 100 samples in one case and about 20,000 in another. Based on various exploratory analyses I've conducted using PCA and heat maps, I'm convinced that the true dimensionality (i.e. the number of dimensions needed to capture most of the "signal") is around 4. I want to create a slide to that effect for a presentation. The "conventional wisdom" about this data, which I'm looking to disprove, is that the true dimensionality is one or two.
What's a good, simple visualization for showing the true dimensionality of a dataset? Preferably it should be understandable to people who have some background in statistics but are not "real" statisticians.