Summarizing a feature's distribution: mean, median, skew

The problem, from first principles

Before training anything, a practitioner's first real look at a new feature is almost never a full plot, it's a handful of summary numbers that answer "what does this column roughly look like," fast. Mean and median (02-expectation-variance, 02-imputing-missing-values) each answer "where's the center," but they can disagree, and when they disagree substantially, that disagreement itself is informative: it's the signature of a skewed distribution, one that's lopsided rather than symmetric.

This question builds the third piece of that quick-glance toolkit, skewness, a single number that quantifies exactly that lopsidedness, and packages all three (mean, median, skew, alongside std) into one summary a practitioner would actually reach for first.

From theory to code

Theory defines skewness as the average CUBED z-score of every value (cubing, unlike variance's squaring, preserves sign, capturing which direction the distribution leans), and mean/median/std/skew together as the standard quick-summary bundle.

Implement skewness(x) first, then summarize_distribution(x), which bundles it with mean, median, and std.

Constraints

• skewness returns a plain float.
• summarize_distribution returns a dict with exactly the keys "mean", "median", "std", "skew".
• summarize_distribution must call skewness, not reimplement its formula inline.

Hints

Open one at a time. Each gives away a little more than the last.