Univariate Data vs Bivariate Data

Univariate data records one variable per observational unit, so what you hold is a single column of values and what you can describe is a single distribution. That fixes the available questions: shape, center, spread and unusual values. Nothing else is on the table, because there is no second measurement on the same individuals to set each value against.

Nine patients wait 4, 6, 7, 7, 9, 11, 12, 15 and 34 minutes. The total is 105, so $\bar{x} = 105/9 = 11.67$ minutes, and the median is the fifth value in order, 9 minutes. The gap between those two centers, and the 34 sitting clear of everything else, is the kind of thing a univariate description reports. What it cannot report is why that patient waited 34 minutes, since no other variable was recorded.

The wrong sentence is "our survey collected age, income and city, so nothing about it is univariate." Univariate describes the question, not the file. Almost every real data table has many columns, and you work through them one at a time: the distribution of age is univariate, and so is the distribution of income. The moment you ask whether income changes with age you have left univariate analysis, and no new data were collected at that point.

That second question is bivariate data, where a scatterplot or a two-way table replaces the dotplot and direction, form and strength become askable. Comparing one quantitative variable across groups sits in between: side-by-side boxplots of wait time at three clinics still describe one variable's distribution within each group, which the course treats as comparing distributions, topic 1.9.

Dotplots, histograms, stemplots and boxplots are univariate displays, topics 1.5 and 1.8, and describing what one of them shows is topic 1.6. The single variable can be categorical instead, in which case the display is a bar graph and the summary is counts and proportions rather than center and spread.

Bivariate data is a set of ordered pairs, one pair per observational unit, so the two values inside a pair were recorded on the same individual. That pairing is the entire content of the data set. Two columns of equal length are not bivariate data unless row $i$ of one column describes the same unit as row $i$ of the other. When both variables are quantitative the pairs go on a scatterplot; when both are categorical they go in a two-way table.

Five people record hours slept, $x$, and errors on a proofreading task, $y$: $(4, 12)$, $(5, 10)$, $(6, 9)$, $(7, 6)$, $(8, 3)$. The correlation is $r = -0.9839$ and the least-squares line is $\hat{y} = 21.2 - 2.2x$ (y-hat). More sleep, fewer errors.

"I have a sleep column and an error column, so I have bivariate data." Not until the rows line up. Sort the error column ascending in a spreadsheet while the sleep column stays where it is, and you now hold $(4, 3)$, $(5, 6)$, $(6, 9)$, $(7, 10)$, $(8, 12)$. Same ten numbers. Each column still has the mean and standard deviation it had before, so nothing in either one alone looks wrong. And $r = +0.9839$: the finding reversed by one careless sort.

Two lists from two different groups are not bivariate either. Twenty men's heights and twenty-five women's heights are one variable measured on two samples, which is univariate data compared across groups. Pairs also have to be complete: a unit missing one of its two values contributes to neither the scatterplot nor $r$. And pairing is not causation, only the arithmetic that lets you ask about association at all.

The Fall 2026 course framework defines a bivariate quantitative data set as ordered pairs from two quantitative variables collected from the same individuals, in topic 5.1, Graphical Representations Between Two Quantitative Variables. Two categorical variables get their own treatment in topic 2.1.