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Understand
Random Sampling

Learn how random samples help us draw valid conclusions about entire populations: from surveys to science experiments.

7th Grade Math

What Is a Population?

A population is the entire group you want to learn about. It could be every student in your school, every tree in a forest, or every candy bar produced in a factory.

POPULATION Each dot = one individual in the population

What Is a Sample?

A sample is a smaller group chosen from the population. We study the sample and then use our findings to make generalizations about the whole population.

POPULATION SAMPLE Study sample → make conclusions about population

Why Sampling Matters

★ Saves Time
Surveying 200 students is much faster than surveying 2,000.
★ Saves Money
Testing every product in a factory would be too costly.
★ Often Necessary
Some tests are destructive: you can't crash-test every car.

What Makes a Sample Representative?

A representative sample accurately reflects the characteristics of the population. If \(40\%\) of a school is in 7th grade, then roughly \(40\%\) of a representative sample should be 7th graders too.

Population
\(40\%\)
\(35\%\)
\(25\%\)
 similar proportions Representative Sample
\(\sim 40\%\)
\(\sim 35\%\)
\(\sim 25\%\)
 7th Grade 8th Grade 6th Grade

The Power of Random Sampling

Random Selection POPULATION SAMPLE Every member has an equal chance = Representative Sample

How Random Sampling Works

1
Define your population. Who or what do you want to learn about?
2
List every member. Create a complete list (or numbering) of the population.
3
Use a random method to select members: draw names from a hat, use a random number generator, or roll dice.
4
Collect data from your sample and use it to make inferences about the whole population.

Random vs. Biased Samples

✓ Random Sample
Every member has an equal chance of being selected. Results can be generalized to the whole population.
✕ Biased Sample
Some members are more likely to be chosen than others. Results may be misleading and cannot be generalized.

Types of Biased Sampling

✕ Convenience Sampling
Choosing whoever is easiest to reach. Example: surveying only your friends.
✕ Voluntary Response
Only people who choose to respond are counted. Example: an online poll.
✕ Systematic Exclusion
Leaving out a group entirely. Example: surveying only 8th graders about a school policy.

Making Valid Inferences

An inference is a conclusion drawn from data. A valid inference is one that can be trusted because it comes from a representative sample.

Random Sample Representative Sample Valid Inference Biased Sample Not Representative Invalid Inference vs.

How to Apply Random Sampling

1
Identify the population. "All \(600\) students at Lincoln Middle School."
2
Choose a sample size. Larger samples give more reliable results. A sample of \(60\) (\(10\%\)) is better than \(6\) (\(1\%\)).
3
Select randomly. Assign each student a number \(1\) to \(600\), then use a random number generator to pick \(60\) numbers.
4
Analyze and generalize. If \(45\) out of \(60\) sampled students prefer pizza, you can infer that about \(75\%\) of all \(600\) students prefer pizza.

Example 1: School Lunch Survey

A school has \(500\) students. The principal wants to know how many students want a salad bar. She randomly selects \(50\) students and finds that \(30\) of them want a salad bar.

Example 2: Is This Sample Biased?

A teacher wants to know if students enjoy reading. She asks the first \(15\) students who arrive at the library after school. \(12\) out of \(15\) say they enjoy reading.

Library Visitors Already like reading ≠ All Students Mixed preferences

Example 3: Scaling Up from a Sample

A city has \(12{,}000\) households. A random sample of \(200\) households finds that \(56\) of them have solar panels. How many households in the city likely have solar panels?

1
Find the sample proportion: \(\frac{56}{200} = 0.28 = 28\%\)
2
Apply to the population: \(0.28 \times 12{,}000 = 3{,}360\) households
3
State the inference: About \(3{,}360\) households in the city likely have solar panels.

Common Mistakes

✕ Convenience = Random
Asking whoever is nearby is not random. Every member needs an equal chance.
✕ Small = Bad
A small random sample can still be valid. Size matters, but randomness matters more.
✕ Ignoring the Population
Always check: does the sample actually come from the population you're studying?

Sampling Methods Compared

Random Sample Biased Sample
Selection Method Every member has equal chance Some members more likely
Representative? Yes, tends to be No
Valid Inferences? Yes No
Example Random number generator Surveying only friends
Generalize to Population? ✓ Can generalize ✕ Cannot generalize

Question 1 of 5

A school wants to know students' favorite sport. Which method produces a random sample?

Question 2 of 5

In a town of \(5{,}000\) people, a random sample of \(100\) finds that \(35\) prefer public transit. About how many people in the town likely prefer public transit?

Question 3 of 5

A researcher surveys every \(5^{\text{th}}\) person leaving a grocery store about food preferences. Is this sample likely to support valid inferences about the entire city's food preferences?

Question 4 of 5

Why does random sampling tend to produce representative samples?

Question 5 of 5

A random sample of \(80\) students out of \(800\) shows that \(20\) walk to school. Another random sample of \(80\) from the same school shows \(24\) walk. What should you conclude?