Histograms

19 min read

Quick Reference

Try This First

Here are the ages, in whole years, of 20 people waiting in a clinic lobby.

4   6   9   11  12  14  18  19  21  22
23  25  27  31  34  36  41  48  52  63

Before reading any definition, do three things on paper.

1. Group the ages into classes 10 years wide: 0 to under 10, 10 to under 20, 20 to under 30, and so on up to 60 to under 70. Count how many ages fall in each class.
2. Draw a horizontal axis for age and a vertical axis for the count. Over each class, draw a bar whose height is its count, and let neighboring bars touch.
3. Predict, before you finish drawing: where will the tallest bar sit, and will the picture lean to one side?

A wrong turn while grouping is common and worth understanding. The most frequent slip is an age landing in two classes at once, or in none, because the class boundaries were not stated clearly. There is no need to rush this. Deciding once, in plain words, which class an exact boundary value belongs to is what makes a histogram unambiguous, and the next section turns that decision into a rule.

The Big Idea: A Histogram Is a Distribution Drawn to Scale

A quantitative variable can take many distinct values, often too many to list one bar per value. A histogram solves that by slicing the number line into adjacent classes of equal width and reporting how the data spread across those classes.

Think of a histogram as a small process. The input is a list of numbers. The process sorts each number into the one class that contains it, counts the numbers in each class, and draws a bar whose height is that count. The output is a single picture of the whole distribution: where the data cluster, how far they reach, and whether they lean to one side.

Two features make this picture honest and readable:

• The classes are adjacent intervals on one numerical scale, so the bars touch with no gaps. A gap on the axis would falsely claim that no data exist in a region the classes actually cover.
• Every class has the same width, so a taller bar always means more data, never a wider window. Equal width is what lets the eye compare heights directly.

Two representations of the same distribution

Translate both directions. Given the table from the opener, you should be able to point to the two tallest bars and the right-leaning tail. Given a finished histogram, you should be able to read each bar height back into a count and rebuild the table.

Prerequisite Hub

Builds on

A histogram pulls together four earlier skills. Each link below opens the node it names.

• categorical-vs-quantitative-variables: a histogram is only for quantitative data; categorical data get a bar graph instead.
• distribution-of-a-variable: a histogram is a distribution made visible, so the idea of a distribution comes first.
• reading-a-graph-and-its-axes: reading a bar height against a labeled vertical axis is the core act a histogram asks of the reader.
• proportions-and-percentages: a percent histogram plots each class’s share of the whole, which is part-to-whole arithmetic.

Unlocks

• describing-distribution-shape-center-spread: once the histogram exists, the next step is naming its shape, locating its center, and measuring its spread.
• density-curves: smoothing a percent histogram into a continuous curve leads directly to the density curve of Section 1.4.

Cross-course prerequisites

• (None.) MATH 380 assumes only the introductory arithmetic and graph-reading review.

The Official Definition

Histogram. A histogram displays the distribution of a quantitative variable by dividing the range of values into classes (intervals) of equal width and drawing a bar over each class whose height is the count or the percent of observations falling in that class. Because the classes are adjacent intervals on a numerical scale, the bars touch with no gaps between them.

Source: Moore, McCabe, Craig, IPS 9th edition, Section 1.2 (Displaying Distributions with Graphs); corroborated by OpenStax Introductory Statistics 2e, histogram defined as an x-y display of a data set where x is the data and y is the frequency or relative frequency (https://openstax.org/books/introductory-statistics-2e/pages/2-key-terms).

There are no named theorems for this skill. What follows directly from the definition are two arithmetic facts worth keeping in view:

• The class counts add to the total number of observations $n$. In the opener, $3 + 5 + 5 + 3 + 2 + 1 + 1 = 20$, the number of people. A column of counts that misses this total signals a miscount or a value that slipped through a class boundary.
• A percent histogram plots the relative frequency of each class, the count in the class divided by $n$ and rescaled to a percent. For a class with count $n_{\text{class}}$ out of $n$ total observations:

\[ \text{percent in class} = 100 \cdot \frac{n_{\text{class}}}{n}. \]

Switching from counts to percents changes only the vertical scale, not the shape. The 20-to-under-30 class above holds $\frac{5}{20} = 0.25$ of the data, which is $25\%$; the bars that were tallest in counts are still tallest in percents.

Choosing Classes and Boundary Rules

A histogram is a choice, not a fixed object. Two reasonable people can draw different histograms of the same data by choosing different class widths, and both can be correct. The choices that have to be made:

• How many classes. Too few classes hide the shape (a single wide bar tells you nothing); too many classes leave most bars at height 1 and the picture looks ragged. A common starting point is somewhere between 5 and 20 classes, then adjust until the shape reads clearly.
• Equal width. Every class spans the same number of units. This is what makes bar heights directly comparable.
• A boundary rule. Decide once where an exact boundary value goes. A standard convention is that each class includes its lower endpoint and excludes its upper endpoint, so the classes read “0 to under 10,” “10 to under 20,” and so on. Under that rule, an age of exactly 10 belongs to the 10-to-under-20 class, never to both and never to neither.

State the boundary rule before counting. The most common counting error is an exact boundary value placed in two classes or in none, which throws off every count and the total.

Histogram Versus Bar Graph

These two pictures look similar and are constantly confused, yet they display different kinds of data and follow different drawing rules.

The single fastest check: if the bars touch and the horizontal axis is a number line, it is a histogram; if the bars are separated and the horizontal axis lists names, it is a bar graph.

Worked Examples

Example 1: From counts to a histogram, and reading it back

Twelve quiz scores out of 20 are recorded:

7   8   9   11  12  12  13  14  15  16  18  19

Build a histogram with classes 5 units wide starting at 5: 5 to under 10, 10 to under 15, 15 to under 20.

Predict first. The scores bunch in the low teens, so expect the middle class to be the tallest.

Build the table. Sort each score into its class using the “include the lower endpoint, exclude the upper” rule.

Draw the bars. Over each class draw a bar at its count height; the bars touch because the classes are adjacent.

count
  5 |        ┌─────┐
  4 |        │     │┌─────┐
  3 |  ┌─────┤     ││     │
  2 |  │     │     ││     │
  1 |  │     │     ││     │
  0 +--┴─────┴─────┴┴─────┴----- score
       5    10    15    20

Check against the prediction. The middle class (10 to under 15) is tallest with a count of 5, matching the prediction. The counts add to $3 + 5 + 4 = 12$, the number of scores, so nothing was dropped. Reading the picture back gives exactly the table, which is the test that the histogram is faithful.

Example 2: The same data as a percent histogram

Convert the quiz-score histogram above to percents. The total is $n = 12$.

Predict first. Switching to percents rescales the vertical axis but keeps the shape, so the middle class stays tallest and the percents must add to 100.

Compute each share.

Check. The proportions add to $0.25 + 0.417 + 0.333 = 1.00$ and the percents add to about 100. The tallest bar is still the middle class. Only the axis label changed, from “count” to “percent.”

Example 3: How the class width changes the picture

Take eight measurements:

2   3   3   4   5   6   8   9

Build two histograms, one with width 2 and one with width 4, and compare.

Width 2 (classes 2 to under 4, 4 to under 6, 6 to under 8, 8 to under 10):

Width 4 (classes 2 to under 6, 6 to under 10):

What changed. Both histograms describe the same eight numbers, and both sets of counts add to 8. The width-2 picture shows a small dip in the middle (the 6-to-under-8 class is lowest); the width-4 picture flattens that detail into two bars and hides the dip. Wider classes smooth the shape and can erase real structure; narrower classes show more detail but can look ragged with few data. Neither is wrong; they answer the question at different resolutions, which is exactly why the class width is a reported choice.

Example 4: Reading a histogram without the raw data

A histogram of the wait times (in minutes) for 50 callers has these bar heights, read off the vertical count axis:

Answer three questions from the picture alone.

(a) Do the counts account for all 50 callers? Sum the heights: $18 + 14 + 9 + 5 + 4 = 50$. Yes, every caller is counted once.

(b) What percent of callers waited under 10 minutes? The first two classes cover 0 to under 10 minutes, with $18 + 14 = 32$ callers. As a percent, $\frac{32}{50} \times 100 = 64\%$.

(c) Which way does the distribution lean? The tall bars sit on the left (short waits) and the bars shrink toward the right (long waits), so the distribution is skewed to the right. A few long waits stretch the tail without a matching number of very short-then-shorter waits on the other side.

Common Misconceptions

Practice Problems

2   2   3   3   4   4   4   5   6   7   8   8   9   9   10  10

Ask Why

• Justify it to a classmate. A friend draws a histogram of exam scores with gaps between the bars, “to make them easier to tell apart.” Explain why the gaps misrepresent quantitative data, and what the touching bars are meant to say about the number line.
• Tie the picture back to the situation. In the clinic-age histogram from the opener, the 20-to-under-30 class is one of the two tallest bars, with a count of 5. Say in plain words what that bar height means about the people in the lobby, and why it does not mean “the most common age is 5.”
• Reverse it (high ceiling). You are handed only a finished percent histogram of 200 observations, with class percents 10, 25, 40, 20, and 5. Recover the count in each class. Then describe a general rule for turning any percent histogram back into counts when the total $n$ is known.

Mastery Checklist

Novice (Level 1-2):

• I can tell a histogram from a bar graph by looking at the horizontal axis and whether the bars touch.
• I can sort a list of numbers into equal-width classes using a stated boundary rule.
• I can build a frequency table and confirm the class counts add to the number of observations.

Competent (Level 3-4):

• I can convert a count histogram to a percent histogram and confirm the percents sum to 100.
• I can read a histogram back into counts and answer “how many” and “what percent” questions from the picture.
• I can describe whether a distribution is roughly symmetric or skewed from the bar heights.

Proficient (Level 5):

• I can explain how the choice of class width changes the visible shape.
• I can decide which class width to report and justify the choice from the data.
• I can reverse a percent histogram back to counts given the total.

Connections

Looking back:

• Categorical vs quantitative variables decides that a histogram is the right display, because the variable is quantitative.
• Distribution of a variable is the idea a histogram makes visible.
• Reading a graph and its axes supplies the axis-reading the histogram depends on.

Looking ahead:

• Describing distribution shape, center, and spread reads those three features off the histogram you now know how to build.
• Density curves smooth a percent histogram into a continuous curve, the next step in Section 1.4.

Real-world connections:

• Test-score reports, income surveys, and wait-time studies are routinely summarized as histograms so a reader sees the spread at a glance.
• The shape a histogram reveals (one peak, two clusters, a long tail) drives which summary number, the mean or the median, describes the center honestly.

Resources

• Textbook (IPS). Moore, McCabe, Craig, Introduction to the Practice of Statistics, 9th edition, Chapter 1, Section 1.2 (Displaying Distributions with Graphs), the histogram for quantitative variables. Page numbers pending faculty verification.
• OpenStax Introductory Statistics 2e, Chapter 2 Key Terms (histogram, frequency, relative frequency): https://openstax.org/books/introductory-statistics-2e/pages/2-key-terms

Last updated: 2026-06-16