Box Plots (Box and Whisker)

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

Intermediate25 min16 problems
Practice Problems

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. 1.Minimum - the smallest value (excluding outliers)
  2. 2.Q1 (First Quartile) - the median of the lower half (25th percentile)
  3. 3.Median (Q2) - the middle value (50th percentile)
  4. 4.Q3 (Third Quartile) - the median of the upper half (75th percentile)
  5. 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.
How to find the quartiles (the method used in this lesson): order the data and find the median. Q1 is the median of the values below the median, Q3 is the median of the values above it. With an odd number of values, the median itself belongs to neither half.

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
1

Order the data from least to greatest

Already ordered: 12, 18, 22, 25, 28, 30, 35, 42, 45 → 9 values in order
2

Find the minimum and maximum

Minimum = 12, Maximum = 45 → Min = 12, Max = 45
3

Find the median (middle value)

With 9 values, the median is the 5th value: 28 → Median = 28
4

Find Q1 (median of lower half)

Lower half: 12, 18, 22, 25. Q1 = (18 + 22) / 2 = 20 → Q1 = 20
5

Find Q3 (median of upper half)

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

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 of the four sections of a box plot (lower whisker, the two parts of the box, upper whisker) holds about 25% of the data, no matter how wide or narrow it looks.

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

12Q1: 2028Q3: 38.545
Min
12
Q1
20
Median
28
Q3
38.5
Max
45
IQR (Q3 - Q1):18.5
Range:33
Sorted data:
12; 18; 22; 25; 28; 30; 35; 42; 45
n = 9

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

Box Plot Builder

52Q1: 6572Q3: 8095
Min
52
Q1
65
Median
72
Q3
80
Max
95
IQR (Q3 - Q1):15
Range:43
Outliers:38

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

Interactive Sandbox

Expression Calculator

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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, race times)
  • 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 usually scores more (higher median), but Player A is more consistent (smaller IQR).

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

First, calculate Class A's IQR:

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 × IQR below Q1 or above 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.
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.
The median tells you the typical value, but the box size (IQR) tells you about consistency. A smaller box means the data is more clustered around the median (more consistent), while a larger box means more variability.
Calculate the fences: Lower fence = Q1 - 1.5 ×\times IQR, Upper fence = Q3 + 1.5 ×\times IQR. Any data points outside these fences are outliers and should be plotted as individual points.

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)
Today's challengeOne problem, three tries

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