My apologies for the mixing of the terms quartile and quantile below. I am interested in the general case of quantiles, but I'm using a quartile as a specific example. Also feel free to clarify any points I seem to be confused or unclear about. Most of my exposure with the subject of quantiles has been using software tools generating partitions, so I'm looking for clarification of these issues from a stricter mathematical formulation.
Suppose we have the following population and that we want to take the quartiles of it:
1
1
1
1
What would the usual formal definition of a quartile result in? Would each quartile point be 1? Often times, we want to examine the partitions of data divided by the quantile points. What would we expect to form in this case with formal definitions? 4 partitions containing a single 1?