  With histograms as its foundation, we develop Categorical Exploratory Data
Analysis (CEDA) under the extreme-$K$ sample problem, and illustrate its
universal applicability through four 1D categorical datasets. Given a sizable
$K$, CEDA's ultimate goal amounts to discover by data's information content via
carrying out two data-driven computational tasks: 1) establish a tree geometry
upon $K$ populations as a platform for discovering a wide spectrum of patterns
among populations; 2) evaluate each geometric pattern's reliability. In CEDA
developments, each population gives rise to a row vector of categories
proportions. Upon the data matrix's row-axis, we discuss the pros and cons of
Euclidean distance against its weighted version for building a binary
clustering tree geometry. The criterion of choice rests on degrees of
uniformness in column-blocks framed by this binary clustering tree. Each
tree-leaf (population) is then encoded with a binary code sequence, so is
tree-based pattern. For evaluating reliability, we adopt row-wise multinomial
randomness to generate an ensemble of matrix mimicries, so an ensemble of
mimicked binary trees. Reliability of any observed pattern is its recurrence
rate within the tree ensemble. A high reliability value means a deterministic
pattern. Our four applications of CEDA illuminate four significant aspects of
extreme-$K$ sample problems.
