Describing a distribution completely

Describing the distribution of one quantitative variable means reporting four things:

1. Shape. Symmetric or skewed, how many peaks, anything the silhouette announces.
2. Center. A typical value: where the distribution balances or splits in half, read as a ballpark from a graph.
3. Variability. How spread out the values are: roughly the interval that holds most of the data, or the full range.
4. Unusual features. Outliers, gaps, clusters: whatever refuses to fit the summary above.

Each part is stated in context, with units: not "center about 14" but "the typical backpack weighs about 14 pounds". Context is not politeness. A sentence with no variable name and no units is checkable against any data set whatsoever, which means it says nothing about this one.

Shape takes at most a phrase, drawn from a short list. Skewed right: the longer, thinner tail points toward larger values. Skewed left: the tail points toward smaller values. The tail names the direction, never the pile: incomes crowd the low end and straggle high, and that is skewed right.

Approximately symmetric: the left half roughly mirrors the right. The word "approximately" is doing work: real data are never perfect mirrors, and near enough counts.

Peak-counting has its own words. One main peak is unimodal, two prominent peaks are bimodal, and bars of roughly constant height with no prominent peak are approximately uniform. These are alternatives to skew talk, not flavors of it: a bimodal distribution is not "skewed toward its bigger peak", and uniform is not "symmetric with extra steps". Choosing the wrong family of words hides the finding the display was making.

The fourth part of the description is where the interesting statistics tends to live.

1. An outlier is a value unusually small or large relative to the rest of the data. Unit 1 will later attach fences to that judgment; for now, a point standing far from the pack gets named, in context: "one house sold near 2 million dollars, far above the rest".
2. A gap is a stretch of the number line, inside the data's range, holding no observations at all.
3. A cluster is a concentration of values, usually separated from another cluster by a gap. Two clusters often mean two kinds of thing were measured together: two eruption styles, two species, two sections of a course.

None of these get deleted or smoothed over. An outlier might be a typo, or it might be the discovery; either way the description reports it and the investigation follows. A description that ignores a screaming gap in the middle of the data has failed at its one job: saying what the data look like.

Topic 1.7 computes center and variability precisely. From a graph alone, the description wants defensible ballparks:

1. Center: the value with about half the area on each side. "The typical commute is around 20 minutes" is a complete claim of center; a bare "around 20" is not.
2. Variability: an interval statement. "Most days used between 55 and 105 gallons" or "usage spans about 100 gallons from lowest to highest day" both work; pick the one the graph supports.

Two habits finish the job. First, sweep the checklist: shape, center, variability, unusual features, one clause each; the common failure is not a wrong clause but a missing one, and a missing clause loses the point even when everything written is true. Second, describe the data rather than the drawing. "The graph goes up and then down" is about ink; "session lengths are unimodal, typically around 8 minutes" is about the variable. Every sentence should survive the question: what does this tell me about the thing measured?