Understanding Statistics and Sampling

In this lesson:

• Distinguish population from sample
• Identify biased and representative sampling
• Evaluate statistical claims using sampling quality

What You Will Learn Today

By the end of this lesson, you will be able to:

1. Distinguish population from sample
2. Explain why statisticians use samples
3. Define a representative sample
4. Explain why random sampling reduces bias
5. Identify biased vs. representative sampling methods
6. Evaluate statistical claims by sample quality

How Would We Find Out?

What is the most popular lunch item in the school cafeteria?

Population: The Entire Group Under Study

• A population is the entire group you want to learn about
• The key question: "Who are we really interested in?"
• Populations can be huge — all voters, all lightbulbs, all patients

Example: To learn about lunch preferences, the population is all students at our school.

Sample: The Subset We Study

• A sample is a subset of the population that you actually study
• The goal of a sample is to give information about the population, not just itself
• Samples are chosen because studying the whole population is impractical

Why sample? Too many, too slow, too expensive, some unavailable.

Seeing Population and Sample Together

• The sample is always chosen from the population
• Conclusions from the sample are about the population

Spotting Population and Sample in Context

Quick Check: Identify Population and Sample

A school principal surveys 30 randomly chosen students to learn about homework habits across the school.

What is the population? What is the sample?

Think for a moment before the next slide…

From Definitions to Sample Quality

We know what a sample is — but not all samples are equally useful.

Some samples give us accurate information about the population.
Others mislead us badly.

The difference: how the sample was selected.

Biased Sample: Misleading by Design

• A biased sample does not accurately reflect the population
• It systematically over-represents or under-represents part of the population
• Bias comes from the selection process — not from lying or bad intentions

Key idea: Even well-meaning surveys can produce biased results.

Representative Sample: Reflecting the Population

• A representative sample accurately reflects the characteristics of the population
• It gives every subgroup a fair chance of being included
• When a sample is representative, we can trust its results to generalize

Goal: Select a sample that looks like a "miniature version" of the population.

Biased Sample: Asking the Soccer Team

Scenario: A student asks soccer teammates: "What sport do 7th graders prefer?" She finds 90% prefer soccer.

• Population: All 7th graders
• Sample: Soccer team members
• Problem: Only soccer players were asked — of course they prefer soccer!
• Conclusion: This estimate likely misrepresents all 7th graders.

Biased Sample: Surveying the Toy Store Line

Scenario: A company surveys kids at a toy store to learn what toys children prefer.

• Population: All children
• Sample: Kids at a toy store
• Problem: These kids already love toys — not typical of all children
• Missing: Kids who never visit toy stores

Biased vs. Representative Side by Side

Does the selection method give everyone a fair chance?

Classify Each Sampling Method as Biased or Representative

Quick Check: Online Survey Bias

A city posts a survey on its website asking residents whether they support a new park.

Is this sampling method biased or representative? Why?

Consider: Who responds to voluntary online surveys?

Random Sampling: The Solution to Bias

• Convenient, volunteer, and self-selected samples are almost always biased
• Random sampling gives every member an equal chance of selection
• No subgroup is systematically favored or excluded

Three Ways to Sample Randomly

All three give every member an equal chance of selection.

Comparing Random and Non-Random Methods

Quick Check: Is Volunteer Selection Random?

A teacher wants to know how many hours 7th graders sleep. She asks for volunteers to share their sleep times.

Is this random? Which students might volunteer — and why does that matter?

Does Random Always Equal Perfect?

No — random samples vary naturally.

• A fair coin flipped 10 times may land heads 7 times — not biased, just variation
• Over many random samples, results converge on the true population value

A large biased sample is worse than a small random one.

Applying Your Toolkit to Real Claims

We now have all three pieces:

• What a population and sample are
• What makes a sample biased or representative
• Why random sampling reduces bias

Four Questions for Evaluating Any Claim

Evaluating Claim One: Magazine Survey

Claim: "75% of adults prefer brand X soap." (500 magazine subscribers polled online)

Evaluating Claim Two: Health Study

Claim: "Adults sleeping 7–8 hours have fewer health problems." (2,000 randomly selected from national registry)

Evaluating Claim Three: Five Friends

"80% of 7th graders prefer cats to dogs." (5 friends)

Your Turn: Apply the Four Questions

Claim A: Restaurant voluntary survey — 92% rate food excellent.

Claim B: Random sample of 1,000 households across all 50 states.

Which is more trustworthy? Apply the 4 Questions.

Key Ideas: Population, Sample, and Bias

✓ Population = entire group; Sample = subset
✓ Only useful when representative
✓ Biased = over- or under-represents systematically
✓ Random = equal chance for every member

Method quality beats sample size.

Common Mistakes to Watch Out For

Larger sample ≠ better — method matters more
Random sample ≠ perfect mirror — variation is normal
Open to everyone ≠ unbiased — voluntary skews to extremes
Statistical "random" = equal probability — not casual haphazard
Say "our sample suggests…" not "students prefer…"

What's Next: Making Inferences from Samples

Coming up: 7.SP.A.2

• Use sample data to estimate population characteristics
• Compare two samples to draw conclusions
• Understand how sample size affects reliability

Today's lesson was the foundation for all statistical inference.