Making Inferences from Samples

Active learning works for this topic because comparing data sets requires students to move from abstract calculations to concrete, observable differences. When students physically collect and analyze real data, they can see how measures like mean and spread actually describe two populations, making the concept more meaningful and memorable.

• Grade 7
• Mathematics
• 3 activities
• 20 min–45 min

Learning Objectives

1. Explain how a random sample can represent a larger population for a specific characteristic.
2. Evaluate the reliability of inferences based on different sampling methods, such as convenience or random sampling.
3. Construct an argument to support or refute a claim using data collected from a sample.
4. Calculate the proportion of a characteristic in a sample and use it to predict the proportion in the population.
5. Compare inferences made from multiple random samples of the same population.

Inquiry Circle

⏱ 40 min·Small Groups

Inquiry Circle: Reaction Time Challenge

Students use an online tool to measure their reaction times. They compare the data distribution of the 'left-hand' group vs. the 'right-hand' group, calculating mean and median for both to see if there is a significant difference.

Prepare & details

Explain how a sample can be used to make predictions about an entire population.

Facilitation Tip: In the Reaction Time Challenge, circulate as groups collect data and ask them to predict which team’s results might have less variability before they calculate the mean absolute deviation.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Outlier Effect

Give students a set of 'test scores' where one score is a 0. They calculate the mean and median. Then, they 'remove' the 0 and recalculate. They discuss with a partner which measure of center changed more and why.

Prepare & details

Evaluate the reliability of an inference based on the sampling method used.

Facilitation Tip: During The Outlier Effect, pause pairs after their discussion to ask one group to share how the outlier changed their interpretation of 'typical'.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Weather Watchers

At different stations, students analyze temperature data for two Canadian cities (e.g., Victoria and Winnipeg). They must calculate the mean and the spread (range) to explain which city has more 'predictable' weather and why.

Prepare & details

Construct an argument for or against a claim based on sample data.

Facilitation Tip: In Weather Watchers, set a timer for each station so groups rotate before they become too focused on one type of comparison.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Experienced teachers start with hands-on data collection to build intuition, then introduce the formulas for mean and mean absolute deviation only after students see why they are needed. Avoid rushing to abstract calculations; instead, use real-world examples where the data itself sparks curiosity. Research suggests that students grasp spread better when they physically see the gaps between data points, so encourage them to plot their results whenever possible.

What to Expect

Successful learning looks like students using measures of center and spread to justify comparisons between data sets. They should explain why one set might be more consistent or typical than another, and connect their analysis to real-world decisions, such as choosing the better team or interpreting weather data.

Watch Out for These Misconceptions

Common Misconception

During the Reaction Time Challenge, watch for students defaulting to the mean without considering outliers or variability in their data.

What to Teach Instead

Ask groups to calculate both the mean and median for their data set, then ask them to explain which one better represents a 'typical' reaction time in their sample.

Common Misconception

During The Outlier Effect, watch for students assuming that two sets with the same mean are identical.

What to Teach Instead

Have pairs compare two pre-made dot plots with the same mean but different spreads, then ask them to describe how the sets differ in practical terms.

Assessment Ideas

Quick Check

After the Reaction Time Challenge, present students with a scenario: 'A gym wants to compare the average heights of two basketball teams. Team A’s heights are {170, 175, 180, 185, 190} and Team B’s heights are {160, 170, 180, 190, 200}. Ask students to calculate the mean for each team and explain which team’s mean better represents a 'typical' player height, referencing the spread in their answer.

Exit Ticket

During Weather Watchers, provide students with a small dataset of temperatures from two provinces (e.g., 5 days in summer for Ontario and BC). Ask them to calculate the mean and mean absolute deviation for each, then write one sentence explaining which province had more consistent temperatures and why.

Discussion Prompt

After The Outlier Effect, pose the question: 'If you wanted to compare the average test scores of two classes, would you include a student who scored 0% due to illness? Explain your reasoning by considering how the outlier affects the mean and median for each class.'

Extensions & Scaffolding

• Challenge: Ask students to design their own comparison scenario (e.g., comparing two brands of batteries) and collect their own data to analyze.
• Scaffolding: Provide pre-calculated mean and MAD for one data set and ask students to calculate the other, then compare them in a sentence stem: 'Set A is more consistent because...'
• Deeper exploration: Have students research how statisticians use samples to predict election outcomes or sports performance, then present one example to the class.

Key Vocabulary

Inference	A conclusion reached on the basis of evidence and reasoning, often about a population based on a sample.
Population	The entire group of individuals or objects that you want to know something about.
Sample	A subset of individuals or objects selected from a population to make inferences about the whole group.
Random Sample	A sample where every member of the population has an equal chance of being selected, which helps reduce bias.
Bias	A systematic error introduced into sampling or testing by selecting or encouraging one outcome or answer over others.