Histograms
Learn how to read and create histograms to display continuous data distributions.
Definition
- 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
Try it now
Worked Examples
A histogram shows test scores. The interval has a bar reaching up to . What does this tell us?
Identify the interval
The interval is , meaning scores from 70 up to (but not including) 80 → Score range: 70-79
Read the frequency
The bar height is 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
Bar Chart
Bar Chart
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
16 problemsWhat makes a histogram different from a bar graph?
Why It Matters
- 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
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).
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).
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:
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).
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:
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
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