Distribution

A distribution belongs to one variable and answers two questions at once: which values occur, and how often each one does. For a categorical variable that is the count or the share in every category. For a quantitative variable it is the pattern of values you read off a dotplot, histogram, or stemplot. For a random variable it is every possible value paired with its probability, and those probabilities have to sum to 1.

Twelve students report their number of siblings: 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5. The distribution is the whole tally: 0 occurs twice, 1 occurs four times, 2 occurs three times, 3 occurs twice, 4 occurs zero times, and 5 occurs once. That zero at 4 is part of it. The mean, $21/12 = 1.75$, and the median, 1.5, are summaries pulled out of the tally, not the distribution itself.

"Describe the distribution" answered with one number is the most expensive habit on the free-response section. "The mean is 1.75" describes nothing. A description names shape, center, spread, and any unusual features such as outliers, gaps, or clusters, in the context of the variable: skewed right, centered near 1.5 siblings, running from 0 to 5, with the single student who has 5 siblings standing apart from the rest.

One word covers three objects that beginners collapse into one: the distribution of a population, the distribution of a single sample drawn from it, and the sampling distribution of a statistic computed from many such samples. All three are distributions, they have different spreads, and a large share of the confusion in inference comes from reading a sentence about one as a sentence about another.

A distribution is also a one-variable idea. A scatterplot shows two variables at once, so on its own it displays no distribution; you would describe the distribution of each variable separately, or describe the relationship between them. Graphing and describing one quantitative variable are Unit 1 topics 1.5 and 1.6.