The Distribution of a Variable

14 min read

Quick Reference

Section 1.1 defines individuals and variables, then names the central object that the rest of Chapter 1 studies: the distribution of a variable. Every display in Section 1.2 (bar graphs, pie charts, histograms, stemplots) and every number in Section 1.3 (center, spread) is a way of describing a distribution, so this definition is the hinge the chapter turns on.

Try This First

Twelve students were asked how many siblings they have. Here are the answers, in the order the students were asked:

2  0  1  3  2  1  0  2  4  1  2  1

Before reading any further, work with this list on paper:

1. Write down the different values that appear in the list.
2. Next to each value, write how many students gave that answer.
3. Which answer was given most often? Which value did no one give above 4?

Hold your small table. You have just built, by hand, the distribution of a variable. That table has a name, and it is the starting point for every graph and summary that follows. (A worked version appears in the first example, so you can check your counts.)

Counting Before Naming

A raw list of data answers a narrow question: what did each individual report. Read straight across, the sibling list is twelve separate facts and not much more. The list becomes informative the moment the question changes from “what did each student say” to “what values show up, and how often.”

That second question is the one worth asking, because it does not depend on the order the students were asked, and it does not grow longer when more students are added. Two values appear: a short collection of possible answers (0, 1, 2, 3, 4) and a count attached to each. A long list of individual responses collapses into a compact summary of values-and-counts.

Think of a variable as a process that takes each individual and gives back a value. The sibling variable takes the first student and returns 2, takes the second and returns 0, and so on. Asking for the distribution runs that process across every individual at once and records not the order of the outputs but the tally of them: how many times each possible output came back. That tally, the values together with how often each one occurs, is the distribution.

This is why the distribution comes before any picture. A bar graph, a histogram, and a stemplot are three different drawings of the same underlying values-and-counts. Build the tally first and each drawing is just a way to show it.

Prerequisite Hub

The distribution is the first object the course studies in depth, so only two short pieces of vocabulary come before it.

Builds on

• Individuals and variables. A distribution is always the distribution of one variable, measured on a set of individuals. The individuals are who is measured; the variable is what is recorded. Without that pair, there is nothing to tally.
• Categorical versus quantitative variables. The kind of variable decides what its distribution looks like and how it is displayed. A categorical variable distributes over named groups; a quantitative variable distributes over numbers.

Unlocks

Once the distribution is the object, every display and summary in Chapter 1 is a way to describe it:

• Displaying categorical data with bar and pie charts
• Histograms
• Stemplots
• Describing a distribution: shape, center, and spread

Prerequisite map

The Official Definition

Distribution of a variable. The distribution of a variable tells us what values the variable takes and how often it takes these values.

Source: Moore, McCabe, Craig, Introduction to the Practice of Statistics, 9th edition, Section 1.1 (Data); corroborated by IPS summary sources, where a distribution tells which values occur and how often.

There is no theorem here. The definition has two parts, and naming them separately keeps the idea sharp.

The two parts of every distribution

A distribution is incomplete with only one part. A list of possible values with no counts does not say what is common; a pile of counts with no labeled values cannot be read. The distribution is the pairing.

Counts and percents say the same thing two ways

How often a value occurs can be reported as a count (the number of individuals with that value) or as a percent, also called a relative frequency (that count divided by the total, times 100). Counts answer “how many”; percents answer “what share.” Both describe the same distribution, and percents make two distributions of different sizes comparable.

The relative frequency of a value is $$\text{relative frequency} = \frac{\text{count for that value}}{\text{total number of individuals}}.$$

Because every individual lands at exactly one value, the counts add to the total and the relative frequencies add to 1 (the percents add to 100).

Worked Examples

Example 1: Building a quantitative distribution by hand

Take the sibling list from the opener:

2  0  1  3  2  1  0  2  4  1  2  1

Predict first. Scan the list. The answers 1 and 2 seem to show up often; 3 and 4 look rare. Expect a distribution that piles up at 1 and 2 and thins out toward 4.

Now build it. Sweep through the list once and tally each value:

Check the total. The counts add to $2 + 4 + 4 + 1 + 1 = 12$, which is the number of students, and the relative frequencies add to $1.000$. Both checks pass, so no student was missed or double counted.

Report and compare to the prediction. The most common answer is a tie between 1 and 2 (four students each); 3 and 4 are rare (one student each). The prediction (a pile at 1 and 2, thinning toward 4) matches the built distribution. The table is the distribution of the variable “number of siblings”: the values are 0 through 4, and the counts say how often each occurs.

Example 2: A categorical distribution

A first-year seminar of 20 students records each student’s class standing. The recorded values are:

first-year  first-year  sophomore  first-year  junior
first-year  first-year  sophomore  first-year  senior
first-year  junior      first-year  first-year  first-year
sophomore   first-year  first-year  junior      first-year

Predict first. The variable is class standing, which is categorical (named groups, no numerical order required). “First-year” appears far more than any other label, as expected in a first-year seminar, so its share should be large.

Now build it. Tally each category:

Check the total. Counts: $13 + 3 + 3 + 1 = 20$. Percents: $65 + 15 + 15 + 5 = 100$. Both pass.

Report. The distribution of class standing has four values (first-year, sophomore, junior, senior) with first-year holding 65 percent of the seminar. The prediction (a large first-year share) matches. The same two parts appear as in Example 1: the values are the class names, and “how often” is the count or percent in each.

Example 3: Why percents make two groups comparable

A second seminar of 50 students reports its class standing as 30 first-year, 9 sophomore, 8 junior, and 3 senior. Is the second seminar “more first-year” than the first seminar in Example 2?

Predict first. The second seminar has 30 first-year students against the first seminar’s 13, so counting raw heads, the second seminar has more first-year students. But the seminars are different sizes (50 against 20), so the raw counts alone may mislead. Compare the shares.

Now compute the shares.

Report and resolve the prediction. By raw count, Seminar 2 has more than twice as many first-year students (30 against 13). By share, Seminar 1 is slightly more first-year (65 percent against 60 percent). The two readings disagree because the seminars differ in size. The distribution stated as percents answers “what share of this group,” which is what the comparison needs; the distribution stated as counts answers “how many,” which depends on the group size. Both are the same distribution, reported two ways.

Common Misconceptions

Practice Problems

3  6  3  1  4  6  3  2  6  4

Resources

• Textbook section. Moore, McCabe, Craig, Introduction to the Practice of Statistics (IPS), 9th edition, Chapter 1, Section 1.1 (Data), subsection “Data: the distribution of a variable.” Page numbers pending faculty verification.
• OpenStax Introductory Statistics 2e, Chapter 2 Key Terms. The open-text definitions of frequency and relative frequency that match the “how often” part of a distribution: openstax.org/books/introductory-statistics-2e/pages/2-key-terms.

Mastery Checklist

Novice (Level 1-2):

• I can state, in plain words, what the distribution of a variable tells me
• I can separate the two parts of a distribution: the values, and how often each occurs
• I can build a small distribution from a raw list by tallying each value

Competent (Level 3-4):

• I can convert a distribution from counts to relative frequencies and check that they add to 1
• I can explain why percents, not raw counts, compare two groups of different sizes fairly
• I can tell whether a piece of information is a value or a count

Proficient (Level 5):

• I can reconstruct missing counts from a total and a stated percent
• I can state a full distribution as both counts and relative frequencies and verify consistency
• I can explain why a distribution does not depend on the order of the data list

Connections

Looking back:

• Individuals and variables supplies the who (individuals) and the what (the variable) that a distribution summarizes.
• Categorical versus quantitative variables decides whether a distribution spreads over named groups or over numbers, which sets the right display.

Looking ahead:

• Bar and pie charts draw the distribution of a categorical variable.
• Histograms and stemplots draw the distribution of a quantitative variable.
• Shape, center, and spread describes a distribution in words and numbers, once it has been displayed.

Real-world connections:

• A poll result (“42 percent favor the proposal”) is one value-and-share from the distribution of a survey variable.
• A weather summary of “days of rain per month” is the distribution of a count variable across the year.
• A grade report of how many students earned each letter grade is the distribution of the grade variable.

Last updated: 2026-06-16