Mathematics · Grade 7 · Data Analysis and Statistics · Term 4

Making Inferences from Samples

Using data from a random sample to draw inferences about a population with an unknown characteristic of interest.

• Ontario Curriculum Expectations
• 7.SP.A.2

About This Topic

Comparing data distributions in Grade 7 moves students from looking at single numbers to analyzing whole sets of information. The Ontario curriculum focuses on using measures of center (mean, median) and measures of spread (mean absolute deviation) to compare two different populations. This topic is essential for making evidence-based decisions, such as comparing the performance of two sports teams or the weather patterns in two different Canadian provinces.

Students learn that the 'average' doesn't tell the whole story; the variability or 'spread' of the data is just as important. They investigate how outliers can skew the mean and why the median is often a more reliable measure for things like income or house prices. By engaging in collaborative investigations, students learn to interpret data more holistically. This topic particularly benefits from hands-on, student-centered approaches where students can manipulate data points and see the immediate effect on the distribution.

Key Questions

1. Explain how a sample can be used to make predictions about an entire population.
2. Evaluate the reliability of an inference based on the sampling method used.
3. Construct an argument for or against a claim based on sample data.

Learning Objectives

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

Before You Start

Collecting and Organizing Data

Why: Students need to be able to gather and structure data before they can analyze it for inferences.

Representing Data

Why: Students should be familiar with various ways to display data (e.g., bar graphs, pictographs) to help visualize sample results.

Calculating Averages (Mean)

Why: Understanding how to calculate a mean from a set of data is foundational for making predictions about a population's average characteristic.

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.

Watch Out for These Misconceptions

Common Misconception

The mean is always the best 'average'.

What to Teach Instead

Students often default to the mean. Using a data set with a massive outlier (like 'salaries in a room where one person is a billionaire') helps them see during a group discussion that the median often gives a better 'typical' value.

Common Misconception

If two sets have the same mean, they are the same.

What to Teach Instead

Students may ignore the spread. Comparing two sets like {5, 5, 5} and {0, 5, 10}, both with a mean of 5, helps them realize that the 'variability' makes the two sets very different in practice.

Active Learning Ideas

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.

40 min·Small Groups

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.

20 min·Pairs

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.

45 min·Small Groups

Real-World Connections

• Market researchers use random sampling to survey a small group of consumers to understand preferences for a new product, like a new flavour of ice cream, before launching it nationwide.
• Political pollsters select random samples of voters to predict election outcomes, using the sample results to estimate the overall voting intentions across the country.
• Quality control inspectors in manufacturing plants take random samples of products from an assembly line to assess the defect rate and make inferences about the quality of the entire production batch.

Assessment Ideas

Quick Check

Present students with a scenario: 'A school wants to know the favourite sport of its 500 students. They survey 50 students from the Grade 7 classes only.' Ask students to identify the population and the sample, and explain one potential source of bias in this sampling method.

Exit Ticket

Provide students with a small dataset from a sample (e.g., 20 coloured marbles drawn from a bag). Ask them to calculate the proportion of each colour in their sample and then write one sentence predicting the proportion of each colour in the full bag, explaining why their prediction is reasonable.

Discussion Prompt

Pose the question: 'If you wanted to know the average height of all Grade 7 students in Ontario, would it be better to measure 100 students from your own school or 100 students randomly selected from across the province? Explain your reasoning, considering the concepts of sample and population.'

Frequently Asked Questions

What is the difference between mean and median?

The mean is the average (sum divided by the count), while the median is the middle number when the data is ordered. The mean is sensitive to outliers, while the median is more 'robust' and stays stable even if there are extreme values.

What does 'variability' mean in data?

Variability refers to how spread out the data points are. If all the numbers are close together, there is low variability. If they are far apart, there is high variability. This is often measured using the range or mean absolute deviation.

How can active learning help students compare data?

Active learning, like the 'Reaction Time Challenge,' makes the data personal. When students are analyzing their own results, they are more invested in understanding why their 'spread' is different from another group's. This leads to deeper questions about what the numbers actually represent.

When should I use the median instead of the mean?

You should use the median when your data set has extreme outliers or is 'skewed' (not symmetrical). For example, Canadian real estate prices are usually reported as medians because a few multi-million dollar homes would make the 'mean' price look much higher than what most people actually pay.

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