Mathematics · Grade 6 · Data, Statistics, and Variability · Term 4

Box Plots (Box-and-Whisker Plots)

Creating and analyzing box plots to represent numerical data and summarize distributions.

• Ontario Curriculum Expectations
• 6.SP.B.4

About This Topic

Box plots, or box-and-whisker plots, summarize numerical data distributions using five key values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Grade 6 students create these plots from raw data or five-number summaries and use them to compare data sets, such as test scores from two classes. This approach highlights center (median), spread (range and interquartile range), and potential outliers more clearly than line plots or histograms for small data sets.

In the Ontario curriculum's data strand, box plots build data management skills for interpreting variability and making evidence-based comparisons. Students address key questions like what box plots reveal that histograms cannot, such as precise quartile positions without binning data. These tools foster statistical reasoning, preparing students for secondary math and cross-curricular applications in science experiments or social studies surveys.

Active learning benefits this topic because students collect real classroom data, sort values on number lines in groups, and construct plots collaboratively. Hands-on sorting clarifies quartiles as medians of halves, while comparing side-by-side plots reveals distribution shapes intuitively, making abstract summaries concrete and memorable.

Key Questions

1. Explain what specific information a box plot can show that a histogram cannot.
2. Construct a box plot from a five-number summary.
3. Compare two different data sets using their box plots.

Learning Objectives

• Calculate the five-number summary (minimum, Q1, median, Q3, maximum) from a given data set.
• Construct accurate box plots on a number line using a five-number summary.
• Analyze and interpret the spread and center of a data set as represented by a box plot.
• Compare and contrast the distributions of two different data sets by analyzing their respective box plots.
• Explain the specific advantages of using a box plot over a histogram for displaying certain data characteristics.

Before You Start

Ordering and Representing Data

Why: Students need to be able to order numerical data from least to greatest before they can find medians and quartiles.

Calculating the Median

Why: Finding the median is a core component of calculating the five-number summary and constructing box plots.

Key Vocabulary

Five-Number Summary	A set of five key values that describe a data distribution: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
Median	The middle value in a data set when the data is ordered from least to greatest. It divides the data into two equal halves.
Quartiles	Values that divide a data set into four equal parts. Q1 is the median of the lower half, and Q3 is the median of the upper half.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1), representing the spread of the middle 50% of the data.

Watch Out for These Misconceptions

Common Misconception

The box shows the full range of all data points.

What to Teach Instead

The box represents only the interquartile range from Q1 to Q3, covering the middle 50% of data. Whiskers extend to min and max, excluding outliers. Group sorting activities on number lines help students see this visually as they order values and mark quartiles.

Common Misconception

The median always sits exactly in the center of the box.

What to Teach Instead

The median's position within the box indicates skewness; it may be off-center in asymmetric distributions. Comparing physical card sorts of skewed data sets lets students manipulate halves to find medians, revealing why plots show shape beyond symmetry.

Common Misconception

Points outside the whiskers are errors to ignore.

What to Teach Instead

Outliers are valid data points that fall beyond 1.5 times the interquartile range from quartiles. Hands-on plotting with real data, like unusual heights, teaches students to question sources first, then include them to understand variability fully.

Active Learning Ideas

Pairs Plotting: Student Heights

Pairs measure and record classmates' heights in centimetres, then find the five-number summary. Each pair sketches a box plot on grid paper and compares it to a partner's plot. Discuss differences in medians and spreads during share-out.

30 min·Pairs

Small Groups: Sports Data Comparison

Provide data sets on basketball free throws for two teams. Groups calculate summaries, draw box plots side by side, and note which team has greater consistency via interquartile range. Present findings to class with one insight each.

45 min·Small Groups

Whole Class: Poll and Plot

Conduct a class poll on minutes spent on screens daily. Record data on board, compute summary as a class using a number line. Draw a large box plot together and interpret median versus range.

35 min·Whole Class

Individual: Mystery Data Analysis

Give five-number summaries for three data sets. Students construct box plots individually, then rank sets by spread. Pair up to verify calculations and explain one comparison.

25 min·Individual

Real-World Connections

• Sports analysts use box plots to compare player statistics, such as the distribution of points scored per game by two different basketball players, to assess performance.
• Financial advisors might use box plots to visualize the range and spread of historical stock prices for different investment funds, helping clients understand potential risk and return.

Assessment Ideas

Quick Check

Provide students with a small data set (e.g., 15-20 numbers). Ask them to calculate the five-number summary and then draw the corresponding box plot on a provided number line template. Check for accuracy in calculations and plot construction.

Discussion Prompt

Present two side-by-side box plots representing student scores on two different math quizzes. Ask students: 'Which quiz appears to have had more consistent scores? How can you tell from the box plots? What does the median tell you about the typical score on each quiz?'

Exit Ticket

Give students a box plot. Ask them to write down: 1) The minimum value, 2) The median, 3) The range of the middle 50% of the data (IQR), and 4) One observation about the spread of the data.

Frequently Asked Questions

How do you construct a box plot from a five-number summary?

Order the data to find min, Q1 (median of lower half), median (middle value), Q3 (median of upper half), and max. Plot a number line scale, draw box from Q1 to Q3 with median line inside, add whiskers to min and max. Practice with 15-20 data points ensures accuracy; digital tools like Google Sheets verify student work quickly.

What information does a box plot show that a histogram cannot?

Box plots display exact quartile positions, median, and range without grouping into bins, making them ideal for small data sets or precise comparisons. Histograms show shape via frequencies but obscure exact values. Side-by-side box plots quickly reveal which set has tighter spread or higher center, supporting clear data talks.

How can active learning help students understand box plots?

Active methods like sorting data cards into ordered lists clarify quartiles as medians of halves, while group plotting emphasizes steps collaboratively. Real data collection, such as measuring jumps, connects summaries to context, reducing abstraction. Comparing plots physically or digitally builds comparison skills, with discussions uncovering misconceptions through peer explanations.

How to compare two data sets using box plots?

Draw plots on the same number line scale. Compare medians for center, box lengths for interquartile ranges (spread of middle 50%), and whisker lengths for full ranges. Note outliers and skewness via median position. Classroom examples like comparing boys' versus girls' run times make interpretations relevant and discussion-rich.