Histograms

Learn how to read and create histograms to display continuous data distributions.

Intermediate25 minLesson

Definition

A histogram is a type of graph that shows the distribution of numerical data. It uses bars to display how often data values fall within specific ranges called intervals or bins.

Key features of a histogram:

The x-axis shows the intervals (ranges of values)
The y-axis shows the frequency (how many data points fall in each interval)
Bars touch each other (no gaps!) because the data is continuous

\text{Frequency} = \text{Number of data points in an interval}

Try it now

A histogram shows test scores. The interval

60-70

has a bar height of

8

. What does this mean?

Worked Examples

A histogram shows test scores. The interval

70-80

has a bar reaching up to

12

. What does this tell us?

Identify the interval

The interval is

70-80

, meaning scores from 70 up to (but not including) 80 → Score range: 70-79

Read the frequency

The bar height is

12

on the y-axis → Frequency = 12

Interpret the meaning

12 students scored between 70 and 79 points → 12 students in this range

Answer: 12 students scored between 70 and 79 on the test.

Common Mistakes

Leaving gaps between bars

Why it's wrong: In histograms, bars touch because the data is continuous. Gaps would imply missing data ranges.

Correct: Always draw histogram bars touching each other. Bar graphs (for categorical data) have gaps; histograms do not.

Confusing histograms with bar graphs

Why it's wrong: Bar graphs show categories (like favorite colors). Histograms show continuous numerical ranges.

Correct: Histogram = continuous data with intervals, bars touch. Bar graph = categories, bars have gaps.

Using unequal interval widths

Why it's wrong: Unequal widths make it hard to compare frequencies fairly - a wider bar naturally captures more data.

Correct: Keep all intervals the same width (e.g., all 10 units wide: 0-10, 10-20, 20-30).

Miscounting which interval a value belongs to

Why it's wrong: Boundary values can be confusing. Is 20 in the 10-20 or 20-30 interval?

Correct: Use a consistent rule: intervals include the lower bound but exclude the upper bound. So 20 goes in 20-30, not 10-20.

Interactive Visual

Histogram

Test scores

50-60(5)

60-70(8)

70-80(12)

80-90(3)

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

16 problems

Problem 1 of 16

Easy

What makes a histogram different from a bar graph?

Practice Problems: Histograms

1. What makes a histogram different from a bar graph?
2. A histogram shows test scores. The interval 60-70 has a bar height of 8. What does this mean?
3. A histogram has 3 bars with frequencies 7, 12, and 5. What is the total number of data points?
4. Which type of data is best displayed using a histogram?
5. In a histogram, the intervals are: 0-10, 10-20, 20-30. What is the width of each interval?
6. A histogram shows ages of visitors: 0-10 (15), 10-20 (28), 20-30 (32), 30-40 (18), 40-50 (7). How many visitors are under 30 years old?
7. A value of 25 needs to be placed in a histogram. The intervals are 10-20, 20-30, 30-40. Which interval should it go in?
8. Data ranges from 12 to 47. Create 5 equal intervals for a histogram.
9. A histogram shows weights in kg: 50-60 (8), 60-70 (15), 70-80 (22), 80-90 (10), 90-100 (5). What percentage of data is in the 70-80 interval? Round to the nearest whole number.
10. A histogram has most of its bars on the right side with a tail extending to the left. What is this distribution called?
11. A histogram shows exam scores: 0-20 (3), 20-40 (7), 40-60 (18), 60-80 (15), 80-100 (7). Find the modal class (interval with highest frequency).
12. A histogram shows daily temperatures in °C: 10-15 (4), 15-20 (9), 20-25 (12), 25-30 (8), 30-35 (2). What is the estimated mean temperature? Use midpoints.
13. A histogram shows running times in minutes: 20-25 (6), 25-30 (14), 30-35 (18), 35-40 (9), 40-45 (3). How many runners finished in less than 35 minutes?
14. Two histograms show test scores. Class A: 0-20 (5), 20-40 (15), 40-60 (20), 60-80 (8), 80-100 (2). Class B: 0-20 (2), 20-40 (6), 40-60 (12), 60-80 (18), 80-100 (12). Which class performed better and why?
15. A histogram shows the ages of people at a birthday party: most bars are on the left with one tall bar around 10-15 and a small bar far on the right around 70-80. What type of event is this likely describing?
16. A histogram has 6 equal intervals. The frequencies are: 4, 9, 15, 15, 9, 4. What fraction of the total data falls in the middle two intervals? Express as a simplified fraction.

Why It Matters

Histograms help us understand how data is distributed:

Test Scores: Teachers use histograms to see how a class performed - are most scores clustered in the middle, or spread out?
Heights: Scientists study height distributions to understand growth patterns in populations
Wait Times: Businesses analyze customer wait times to improve service
Weather: Meteorologists display temperature ranges to show climate patterns

Unlike simple lists of numbers, histograms give us an instant visual picture of our data!

Real World Applications

Analyzing Test Scores

Teachers use histograms to understand class performance at a glance.

Example:

A histogram of 30 test scores shows: 0-50 (2 students), 50-60 (5), 60-70 (8), 70-80 (10), 80-90 (4), 90-100 (1).

1Try It Yourself

Looking at the histogram above, which interval has the most students?

What is the frequency of the most common interval?

Step 1: Write the mathematical expression

Find the highest bar in the histogram:

Customer Wait Times

Restaurants track how long customers wait to be seated to improve service.

Example:

A histogram shows wait times: 0-5 min (15 customers), 5-10 min (25), 10-15 min (18), 15-20 min (7), 20+ min (3).

2Try It Yourself

Using the wait time histogram, find the total number of customers surveyed.

How many customers were included in this data?

Step 1: Write the mathematical expression

Add all the frequencies: $15 + 25 + 18 + 7 + 3$

Student Heights in PE Class

Physical education teachers track student heights to group students fairly for activities.

Example:

Heights in cm: 140-150 (4), 150-160 (12), 160-170 (18), 170-180 (8), 180-190 (3).

3Try It Yourself

What percentage of students are between 160 and 170 cm tall?

Calculate the percentage of students in the 160-170 cm range.

Step 1: Write the mathematical expression

Use the formula: $(\text{frequency} \div \text{total}) \times 100$

Key Takeaways

1A histogram displays the distribution of continuous numerical data using bars
2The x-axis shows intervals (bins), the y-axis shows frequency (count)
3Bars in a histogram touch each other because the data is continuous
4The total number of data points equals the sum of all frequencies
5The shape of a histogram reveals patterns: symmetric, left-skewed, or right-skewed

Frequently Asked Questions

A histogram shows continuous numerical data with touching bars, while a bar graph shows categorical data with gaps between bars. Histograms use intervals; bar graphs use distinct categories.

A common guideline is 5-15 intervals. Too few intervals hide details; too many create noise. The square root of the number of data points is a good starting point.

Use a consistent rule. Typically, intervals include the lower bound but exclude the upper bound. So 50 would go in the 50-60 interval, not the 40-50 interval.

Glossary

Histogram: A graph that shows the distribution of numerical data using bars where each bar represents an interval
Frequency: The number of data points that fall within a specific interval
Interval (Bin): A range of values used to group data in a histogram (e.g., 10-20, 20-30)
Distribution: The pattern of how data values are spread across different intervals
Skewed: When data is not symmetric and leans more toward one side