Mathematics · 7th Grade · Probability and Statistics · Weeks 28-36

Random Sampling and Bias

Understanding that statistics can be used to gain information about a population by examining a sample.

• Common Core State Standards
• CCSS.Math.Content.7.SP.A.1
• CCSS.Math.Content.7.SP.A.2

About This Topic

Once students understand the population-sample distinction, they move to the question of how a sample should be selected. Random sampling is the standard method for avoiding bias because it gives every member of the population an equal chance of being included, removing the influence of researcher preference or self-selection. This topic connects directly to the broader CCSS standard of using statistics to gain information about a population.

Students explore different sampling scenarios and evaluate whether a given method produces a random sample. They examine why voluntary response surveys (where participants choose to respond) consistently skew toward people with strong opinions, and why convenience samples (selecting whoever is easiest to reach) often leave out important parts of the population. The concept of bias is examined not as intentional wrongdoing but as a structural feature of how data is collected.

Small sample sizes introduce another layer of complexity: even a well-designed random sample may produce unreliable inferences if too few people are sampled. Active learning activities that simulate multiple small samples and compare their results give students concrete experience with sampling variability before they formalize the idea.

Key Questions

1. What makes a sample representative of a population?
2. Why is random sampling the best way to avoid bias in data collection?
3. How much confidence can we have in an inference made from a small sample size?

Learning Objectives

• Analyze given sampling methods to identify potential sources of bias.
• Compare the results of random samples to voluntary response samples for a given population question.
• Explain why random sampling is essential for making valid inferences about a population.
• Evaluate the impact of sample size on the reliability of statistical inferences.
• Design a simple random sampling plan for a specified scenario.

Before You Start

Data Collection and Representation

Why: Students need to be familiar with collecting and organizing data before they can analyze different sampling methods.

Basic Probability Concepts

Why: Understanding the concept of equal chance is foundational to grasping the principles of random sampling.

Key Vocabulary

Population	The entire group of individuals or objects that we want to study or draw conclusions about.
Sample	A subset of individuals or objects selected from a population to represent the entire group.
Random Sampling	A method of selecting a sample where every member of the population has an equal and independent chance of being chosen.
Bias	A systematic error introduced into sampling or testing by selecting or encouraging any one outcome or answer over others.
Sampling Variability	The natural variation in results that occurs when different samples are taken from the same population.

Watch Out for These Misconceptions

Common Misconception

Any sample drawn without purposeful selection is random.

What to Teach Instead

Random sampling requires a systematic process that gives every member an equal chance of selection. Grabbing the first 20 people in a hallway is convenience sampling, not random sampling. Simulation activities make this distinction visceral.

Common Misconception

If enough people respond to a voluntary survey, the results are reliable.

What to Teach Instead

High response volume doesn't correct for selection bias. Voluntary responders systematically differ from non-responders, meaning even thousands of responses can produce a biased picture. Historical examples like the Literary Digest 1936 poll help ground this.

Active Learning Ideas

Simulation Game: Bag of Colored Tiles

Each group receives a bag with an unknown proportion of two colors of tiles. They draw 5-tile samples repeatedly, record results, and predict the full bag composition. Groups compare predictions and then reveal the true proportion to discuss sample size effects.

30 min·Small Groups

Think-Pair-Share: Is This Sample Random?

Present five sampling methods for a school-wide survey. Students individually classify each as random or biased, then discuss with a partner. Whole-class discussion focuses on the two or three cases where partners disagreed.

15 min·Pairs

Formal Debate: Should We Trust This Survey?

Share a 'news story' reporting survey results based on a clearly described sample. Groups are assigned to argue either that the sample supports the conclusion or that it doesn't. After 5 minutes of prep, groups debate while the class votes on the most convincing argument.

25 min·Small Groups

Real-World Connections

• Political pollsters use random sampling to survey likely voters across a district or state to predict election outcomes. They must ensure their sample accurately reflects the demographic makeup of the voting population to avoid biased results.
• Market researchers for companies like Nike or Apple use random sampling to gather feedback on new product ideas or advertising campaigns. This helps them understand consumer preferences without surveying every potential customer, saving time and resources.
• Scientists conducting environmental studies, such as tracking endangered species populations in a national park, employ random sampling techniques to estimate population sizes and health. This allows them to make informed conservation decisions based on representative data.

Assessment Ideas

Quick Check

Present students with three scenarios: a convenience sample (e.g., surveying friends), a voluntary response sample (e.g., an online poll), and a random sample (e.g., using a random number generator to select names from a class list). Ask students to identify which is which and explain one reason why the random sample is most likely to be unbiased.

Discussion Prompt

Pose the question: 'Imagine you want to know the favorite lunch item of students in your entire school. If you only survey students in the cafeteria during 7th-grade lunch, what problems might arise?' Guide students to discuss potential biases related to convenience and self-selection.

Exit Ticket

Give students a scenario where a sample of 10 students is taken from a class of 30 to represent the class. Ask them to write two sentences explaining why the results from this small sample might not perfectly match the results if all 30 students were surveyed.

Frequently Asked Questions

Why is random sampling important in statistics?

Random sampling ensures every member of a population has an equal chance of being selected, which removes systematic bias from the data. This makes the sample more likely to reflect the population accurately, so any inferences drawn from it are more trustworthy.

What is the difference between random sampling and convenience sampling?

Random sampling uses a process that gives all population members an equal selection chance. Convenience sampling picks whoever is easiest to reach. Convenience samples are often biased because the accessible group may differ systematically from the broader population.

How does sample size affect the accuracy of inferences?

Larger random samples tend to produce inferences that are closer to the true population value because they capture more of the population's variability. Small samples, even when randomly selected, may produce estimates that are far from the actual population characteristic by chance.

How can active learning activities help students understand random sampling?

Hands-on simulations, like drawing tiles from a bag, let students experience sampling variability directly. When students run multiple trials and compare results, they build intuition for why random selection matters and how sample size affects reliability , concepts that are hard to grasp from definitions alone.