Box Plots (Box and Whisker)

Learn to create and interpret box plots to visualize data distribution using the five-number summary.

Intermediate25 minLesson

Definition

A box plot (also called a box and whisker plot) is a visual display of data that shows how the data is spread out. It uses five key values called the five-number summary:

1. Minimum - the smallest value (excluding outliers) 2. Q1 (First Quartile) - the median of the lower half (25th percentile) 3. Median (Q2) - the middle value (50th percentile) 4. Q3 (Third Quartile) - the median of the upper half (75th percentile) 5. Maximum - the largest value (excluding outliers)

The box shows where the middle 50% of data lies (from Q1 to Q3). The whiskers extend to the minimum and maximum values. Points outside the whiskers are outliers.

Try it now

For the data set: 5, 8, 12, 15, 18, 22, 25, what is the median?

Worked Examples

Find the five-number summary for this data set: 12, 18, 22, 25, 28, 30, 35, 42, 45

Order the data from least to greatest

Already ordered: 12, 18, 22, 25, 28, 30, 35, 42, 45 → 9 values in order

Find the minimum and maximum

Minimum = 12, Maximum = 45 → Min = 12, Max = 45

Find the median (middle value)

With 9 values, the median is the 5th value: 28 → Median = 28

Find Q1 (median of lower half)

Lower half: 12, 18, 22, 25. Q1 = (18 + 22) / 2 = 20 → Q1 = 20

Find Q3 (median of upper half)

Upper half: 30, 35, 42, 45. Q3 = (35 + 42) / 2 = 38.5 → Q3 = 38.5

Answer: Five-number summary: Min = 12, Q1 = 20, Median = 28, Q3 = 38.5, Max = 45

Common Mistakes

Including the median when finding Q1 and Q3

Why it's wrong: When you have an odd number of data points, the median is a single value. When finding Q1 and Q3, you should NOT include this median value in either half.

Correct: Split the data at the median, leaving it out of both halves, then find the median of each half separately.

Thinking a longer whisker means more data points

Why it's wrong: Whisker length shows the range of data, not the amount. A long whisker might represent just a few spread-out values.

Correct: Each section of a box plot (each quartile) contains 25% of the data, regardless of how wide or narrow it appears.

Confusing median with mean

Why it's wrong: The line inside the box shows the median (middle value), not the mean (average). These can be very different, especially with outliers.

Correct: Box plots use the median because it's not affected by extreme values, making it a better measure of the typical value.

Interactive Visual

Box Plot Builder

Min

23.5

Median

Max

IQR (Q3 - Q1):16.5

Range:33

Outliers:78

Sorted data:

12, 18, 22, 25, 28, 30, 32, 35, 38, 42, 45, 78

n = 12

Click on parts of the box plot to learn what they represent.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

16 problems

Problem 1 of 16

Easy

What does the line inside a box plot represent?

Why It Matters

Box plots are powerful tools used everywhere:

Science: Comparing experimental results across different groups
Sports: Analyzing player performance statistics (points scored, batting averages)
Business: Comparing salaries, sales figures, or customer satisfaction across departments
Healthcare: Comparing patient recovery times or test results
Education: Visualizing test score distributions across classes

Unlike bar graphs or pie charts, box plots show you the spread and center of data at a glance, making it easy to spot unusual values and compare groups.

Real World Applications

Comparing Sports Performance

Coaches use box plots to compare player statistics and identify consistent performers.

Example:

Two basketball players' points per game are shown in box plots. Player A: median 18, IQR 4. Player B: median 20, IQR 12. Player B scores higher on average but Player A is more consistent.

1Try It Yourself

A soccer team's goals per match last season: 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6

Find the five-number summary to analyze the team's scoring pattern.

Step 1: Write the mathematical expression

First, find the median (middle value of 11 data points):

Analyzing Test Scores

Teachers use box plots to understand class performance and identify students who need extra help.

Example:

If a class has median 75%, Q1 at 68%, and Q3 at 85%, about 25% of students scored below 68% and may need additional support.

2Try It Yourself

Two classes took the same test. Class A: Min 45, Q1 60, Median 72, Q3 82, Max 98. Class B: Min 55, Q1 70, Median 75, Q3 80, Max 90.

Which class performed better overall? Which had more consistent scores?

Step 1: Write the mathematical expression

Calculate IQR for each class:

Key Takeaways

1A box plot displays data using the five-number summary: minimum, Q1, median, Q3, maximum
2The box shows the middle 50% of data (interquartile range = Q3 - Q1)
3The median line shows where the center of the data lies
4Whiskers extend to the smallest and largest values that are not outliers
5Outliers are values more than 1.5 times the IQR beyond Q1 or Q3
6Box plots are excellent for comparing distributions between groups

Frequently Asked Questions

Box plots show much more information: the median (which isn't affected by outliers), how spread out the middle 50% is, whether the data is symmetric or skewed, and which values are unusual outliers. The mean and range can be misleading when extreme values exist.

Glossary

Box plot: A graph that displays the five-number summary of a data set
Five-number summary: The minimum, Q1, median, Q3, and maximum of a data set
Quartile: Values that divide data into four equal parts (Q1 = 25%, Q2/Median = 50%, Q3 = 75%)
IQR (Interquartile Range): The range of the middle 50% of data, calculated as Q3 - Q1
Outlier: A data value that is unusually far from the rest of the data
Whisker: The lines extending from the box to the minimum and maximum values (excluding outliers)