Distribution vs Histogram

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.

A histogram cuts the number line into bins of equal width and draws a bar over each bin whose height is the count of values inside it, or the relative frequency. The bars touch because the axis is a continuous scale rather than a list of labels. Every histogram needs a boundary rule, and the usual one puts a value in the bin whose lower bound it meets or exceeds and whose upper bound it falls below.

Thirty exam scores binned by tens give 9 scores in 70 to 80 and 12 in 80 to 90, and three of those students scored exactly 80. Switch to the other convention, where a value belongs to the bin whose upper bound it equals, and those three move down: the counts become 12 and 9, the tallest bar changes, and so does the shape you would report. Nothing in the drawn picture tells you which rule was used.

The misreading is "twelve students scored 85." The bar over 80 to 90 says twelve students scored somewhere in that interval and nothing more. The individual values have been thrown away, which is also why you cannot read a median off a histogram, only the bin the median lands in.

Equal width is doing real work in the definition. When bins differ in width, height stops encoding frequency fairly: a bin of width 20 holding 12 values and a bin of width 10 holding 9 draw bars of height 12 and 9, so the wide one looks bigger, while the densities are $12 / 20 = 0.6$ and $9 / 10 = 0.9$ values per unit and the narrow bin is the more crowded one. With unequal bins the vertical axis has to carry density and you read area, not height.

A histogram also destroys the order the values arrived in, which is what a time plot keeps, and its shape depends on the bin width you choose. Histograms are topic 1.5.