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Teacher Guide: Box Plots (Box and Whisker)

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

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Statistical Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify the five components of a box plot (min, Q1, median, Q3, max)
  • Calculate the five-number summary from a data set
  • Construct a box plot from given data
  • Calculate and interpret the interquartile range (IQR)
  • Identify outliers using the 1.5 ×\times IQR rule
  • Compare two or more data sets using box plots
Prerequisites
  • • Understanding of median as a measure of center
  • • Ability to order numbers from least to greatest
  • • Basic understanding of percentages
  • • Familiarity with data sets and their representation
Discussion Starters
  • 1. Why might a scientist prefer box plots over bar graphs when comparing experimental results?
  • 2. If two box plots have the same median but one has a much larger box, what does that tell you?
  • 3. How could outliers affect business decisions differently than if we only looked at averages?
  • 4. When might you want data with a large IQR versus a small IQR?
Common Misconceptions

The box contains most of the data

Remediation: Clarify that the box contains exactly 50% of the data (middle half). Each quartile section represents 25% of the data, no matter how wide or narrow it appears.

A wider section means more data points in that region

Remediation: Use concrete examples: show that each quarter has the same number of points. Width indicates spread, not frequency. Draw a number line with actual data points to demonstrate.

The median is always in the middle of the box visually

Remediation: Show examples of skewed data where the median line is closer to one end of the box. Explain this indicates the data is not evenly spread.

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives (index cards with numbers) to sort data
  • • Color-code each section: minimum (red), Q1 (orange), median (yellow), Q3 (green), maximum (blue)
  • • Start with small data sets (7-9 values) before moving to larger sets
  • • Provide a step-by-step checklist for finding the five-number summary

For On-Level Students:

  • • Create box plots from data sets with 15-20 values
  • • Compare two box plots and write conclusions
  • • Identify and calculate outliers using the 1.5 ×\times IQR rule
  • • Interpret real-world data from sports or science

For Advanced Students:

  • • Analyze the effect of outliers on box plots vs. other representations
  • • Create modified box plots showing both mild and extreme outliers
  • • Compare box plots with histograms and discuss when each is more appropriate
  • • Research and present on how box plots are used in a specific field
Standards Alignment
  • 6.SP.B.4 (CCSS.MATH.CONTENT.6.SP.B.4)

    Display numerical data in plots on a number line, including dot plots, histograms, and box plots

  • 6.SP.B.5c (CCSS.MATH.CONTENT.6.SP.B.5c)

    Summarize numerical data sets in relation to their context by giving quantitative measures of center and variability

  • S-ID.A.1 (CCSS.MATH.CONTENT.HSS.ID.A.1)

    Represent data with plots on the real number line (dot plots, histograms, and box plots)

Lesson Resources
  • visualInteractive Box Plot Builder

    Students enter data and see the box plot update in real-time

  • activityClass Height Comparison

    Collect class heights and create box plots to compare with other classes

  • worksheetSports Statistics Analysis

    Analyze and compare player statistics using box plots

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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 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.

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

Why use box plots instead of just the mean and range?

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.

What if two box plots have the same median but different box sizes?

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.

How do I know if my data has outliers?

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)

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