Sampling and making inferences about a population
Understanding how a random sample can represent a whole population, and what makes a conclusion drawn from a sample valid
About two to three lessons of 45 to 60 minutes
You cannot ask everyone, so who you ask matters enormously
A city cannot survey every single resident before making a decision, a factory cannot test every light bulb it makes, and a pollster cannot call every voter. Instead, they study a SAMPLE, a smaller group, and use it to make an inference, an educated estimate, about the whole POPULATION. The entire idea only works if the sample is chosen randomly, so that it is not secretly stacked toward one kind of answer.
This unit is about the difference between a sample that supports a trustworthy conclusion and one that does not, and about understanding that even a good random sample gives an ESTIMATE, not the exact truth about the whole population.
- A poll of 500 randomly selected votersused to estimate an election result for millions of voters
- Testing 50 light bulbs from a factory batch of 10,000a sample used to estimate the whole batch's quality, without destroying every bulb
- Surveying your basketball team about the school's favourite sporta biased, non-random sample that would NOT support a valid schoolwide conclusion
- Counting red candies in a scoop from a huge jara random handful can estimate the whole jar's mix without counting every candy
What students will be able to do
Students will understand that a random sample can be used to draw valid conclusions about a larger population, explain why the method of selecting a sample (not just its size) determines whether it is representative, and recognize that a sample gives an estimate of the true population value, which naturally varies somewhat from sample to sample.
- I can explain what makes a sample 'random', and why that matters for drawing a valid conclusion.
- I can identify whether a described sample is likely to be representative of its population, or is likely biased.
- I can find a sample proportion from data and compare it to a known or estimated population proportion.
- I can explain why different random samples from the same population can give somewhat different results, without either one being 'wrong'.
Standards this unit teaches
- 7.SP.A.1Common Core (US)Sampling and inference
Understand that a random sample can represent a population and support valid conclusions about it.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Population
- the entire group a study wants to learn about, such as every resident of a city
- Sample
- a smaller group selected from the population, studied to learn about the whole population
- Random sample
- a sample where every member of the population has an equal chance of being selected
- Representative
- a sample that reflects the key characteristics of the whole population it was drawn from
- Bias (in sampling)
- a systematic tendency for a sample to over- or under-represent parts of the population, usually from how it was selected
- Sampling variability
- the natural tendency for different random samples from the same population to give somewhat different results
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What makes a sample random, and why it matters
ConcreteA sample supports a valid conclusion about a population only if it was chosen so that every member of the population had an equal chance of being picked. Convenience, choosing whoever is easiest to reach, is the opposite of random, and can badly mislead a conclusion no matter how many people it includes.
A school with 1,200 students wants to estimate what fraction like pizza for lunch, but can only survey 100 students. Randomly selecting 100 names from the FULL enrollment list gives every student an equal chance of being chosen, so the resulting estimate is likely to reflect the whole school reasonably well. Surveying the first 100 students to walk into the cafeteria on pizza day, by contrast, is a convenience sample that is almost guaranteed to overestimate how much the school likes pizza.
- Why would surveying the first 100 students into the cafeteria on pizza day give a misleading estimate?
- What has to be true about EVERY member of the population for a sample to be called 'random'?
2. Samples give estimates, and estimates vary
PictorialEven a properly random sample rarely lands exactly on the true population value; it gives an ESTIMATE. Different random samples of the same population naturally give somewhat different estimates, called sampling variability, and that is expected, not a sign anything is wrong.
A bag has 1,000 marbles: 600 red and 400 blue, so the true population proportion of red is 600/1000 = 0.60 (60%). Four different random samples of 20 marbles are drawn, and the number of red marbles is counted each time: 11, 13, 12, and 14.
A bag has 1,000 marbles, 600 of which are red (a true proportion of 60%). Four random samples of 20 marbles contain 11, 13, 12 and 14 red marbles. Find each sample's proportion of red marbles, and compare them to the true population proportion.
- Sample 1: 11/20 = 0.55 (55%).
- Sample 2: 13/20 = 0.65 (65%).
- Sample 3: 12/20 = 0.60 (60%).
- Sample 4: 14/20 = 0.70 (70%).
Answer: The four sample proportions are 55%, 65%, 60% and 70%. They cluster around the true population proportion of 60%, but only one (60%) matches it exactly; the others vary above and below, illustrating normal sampling variability.
- Does the fact that Sample 1 gave 55%, not exactly 60%, mean something went wrong with that sample?
- Would you expect samples of 200 marbles each to vary MORE or LESS from the true 60% than these samples of 20?
3. Drawing valid conclusions from a sample
AbstractA valid conclusion from a sample is stated as an ESTIMATE, acknowledges that a different random sample might have given a somewhat different number, and is more trustworthy when the sample was both random AND reasonably large.
Using a random sample of 20 students, 12 said they walk to school, an estimate of 12/20 = 60%. Using a different random sample of 200 students, 108 said they walk to school, an estimate of 108/200 = 54%. Both are valid random samples, but the LARGER sample (200) generally gives a more reliable estimate of the true school-wide percentage, because larger random samples tend to have less sampling variability.
A random sample of 20 students finds 12 walk to school. A different random sample of 200 students finds 108 walk to school. Which estimate is more likely to be close to the true percentage for the whole school, and why?
- Sample of 20: 12/20 = 0.60, or 60%.
- Sample of 200: 108/200 = 0.54, or 54%.
- Compare reliability: larger random samples reduce the effect of chance variation, so their estimates tend to land closer to the true population value.
Answer: The estimate from the sample of 200 students (54%) is generally more reliable, because larger random samples typically give estimates closer to the true population value than smaller ones.
- Why does sample SIZE affect how much you can trust an estimate, even when both samples are random?
- If a sample were large but NOT random (for example, only students who stay after school for a club), would its size make up for the bias?
Common misconceptions and how to address them
MisconceptionA convenience sample (like surveying only nearby friends, or one classroom) is just as good as a random sample, as long as it includes enough people.
Why it happens: Students focus on sample SIZE as the main thing that makes a sample trustworthy, and overlook how it was selected.
How to address it: Sample size alone does not fix bias. HOW a sample is chosen matters: if some members of the population had little or no chance of being picked, the sample can be misleading no matter how large it is.
MisconceptionA single random sample gives the EXACT true value for the whole population, treated as a fact rather than an estimate.
Why it happens: Students are used to computed answers being exactly right, and extend that expectation to sample statistics.
How to address it: A sample gives an ESTIMATE that varies somewhat from sample to sample (sampling variability). It is unlikely to land exactly on the true population value, even when it was chosen randomly and carefully.
MisconceptionA bigger sample automatically fixes bias, so a large but non-random sample is assumed to be reliable.
Why it happens: Students conflate sample SIZE with sample RANDOMNESS, treating them as the same kind of quality.
How to address it: A large but still biased (non-random) sample can be just as misleading as a small biased one. Randomness in HOW the sample is selected, not size alone, is what supports a valid inference; size then improves how PRECISE a valid, random estimate is.
MisconceptionIf two different random samples from the same population give different results, one of them must be 'wrong'.
Why it happens: Students expect a single 'correct' answer, the way a computed worksheet problem has one.
How to address it: Different random samples naturally give somewhat different results; this is sampling variability, not an error. Neither sample is wrong, both are valid estimates with some expected natural variation between them.
Guided practice (with answers)
1. A city wants to know if residents support a new park. Which is more likely to give a valid estimate: surveying 500 randomly selected residents citywide, or surveying the first 500 people who visit the mayor's website? Explain.
Answer: The random citywide sample of 500. Website visitors are a convenience/self-selected sample that may not represent all residents (for example, people without internet access, or those less politically engaged, would be underrepresented).
2. A bag has 500 marbles, 300 of which are green (60%). A random sample of 25 marbles contains 14 green marbles. Find the sample proportion and compare it to the true population proportion.
Answer: 14/25 = 0.56, or 56%, close to but not exactly the true 60%, which is expected sampling variability.
3. Would you trust an estimate more from a random sample of 30 people or a random sample of 300 people, all else being equal? Why?
Answer: The sample of 300. Larger random samples tend to have less sampling variability, so their estimates are generally more reliable.
4. A student surveys only their basketball teammates about their favourite sport, to estimate the favourite sport of the whole school. Is this a valid random sample? Why or why not?
Answer: No. It is a convenience sample of one narrow group (teammates, likely all interested in basketball already), not a random sample of the whole school population, so it would not support a valid schoolwide conclusion.
5. Two random samples of 50 students each are drawn from the same school to estimate the percentage who bring lunch from home. Sample A gives 44%, Sample B gives 39%. Does this mean one sample is wrong? Explain.
Answer: No. This is normal sampling variability between two valid random samples; both are reasonable estimates of the true school-wide percentage, which likely lies somewhere near both values.
Independent practice worksheets
Reason through sampling and representativeness questions, and practise finding sample proportions with computed, never-wrong answer keys.
Differentiation
- Use very concrete, familiar population/sample pairs first (a jar of candy, a class of students) before larger or more abstract ones (a whole city, an election).
- Give two sampling methods side by side (one random, one a named convenience sample) and have students identify which is which before asking them to explain why.
- For sample-proportion calculations, provide the division already set up (count / total) so the focus stays on interpretation, not the arithmetic.
- Use the dot-plot figure from section 2 as a physical, hands-on activity: have small groups each draw their own sample from a class set of marbles or counters and add their proportion to a shared class dot plot.
- Research and compare named sampling methods (random, systematic, convenience, voluntary-response) and rank them from most to least likely to be representative, with reasons.
- Design a sampling plan for a real question about the school (e.g. favourite lunch option) that would be genuinely random, and explain how it avoids the bias of a convenience sample.
- Investigate how much a sample proportion is likely to vary by simulating many samples (physically or with a spreadsheet/random-number tool) and observing the spread of results.
- Critique a real or invented news headline that draws a conclusion from a small or non-random sample, identifying exactly what is missing for the conclusion to be valid.
Assessment: exit ticket
A three-question exit ticket sampling what makes a sample random, interpreting a sample estimate, and explaining why larger samples are more reliable.
1. What makes a sample 'random'?
Answer: Every member of the population has an equal chance of being selected.
2. A random sample of 40 out of 800 students found 25 prefer online classes. Is 62.5% exactly the percentage for all 800 students? Explain.
Answer: Not necessarily exactly; 25/40 = 0.625 (62.5%) is the sample's estimate, but the true percentage for all 800 students could be somewhat different due to sampling variability.
3. Why does a larger random sample generally give a more reliable estimate of a population than a smaller one?
Answer: Larger random samples reduce the effect of chance variation, so the sample statistic tends to land closer to the true population value.
Teacher notes and timings
- Rough timing across two to three lessons: Lesson 1 what makes a sample random (section 1), Lesson 2 sampling variability (section 2), Lesson 3 valid conclusions plus the exit ticket (section 3 and assessment).
- This is a shorter, single-standard unit (7.SP.A.1 stands alone in the Grade 7 gap-coverage audit), so it is deliberately more concept- and reasoning-focused than the site's typical computation-heavy math units; several checks and guided-practice items ask students to EXPLAIN, not just calculate.
- This unit assumes comfort with basic data summaries (the Year 7 mean/median/mode/range unit). Revisit that first if reading a small data set is still shaky.
- Honest coverage note: this standard's closest EXACT-topic worksheet (naming and comparing sampling methods) currently sits at Grade 8, not Grade 7, on this site. The independent practice above leads with the genuine Grade 7 statistical-investigation worksheet first, and links the Grade 8 sampling worksheet as a clearly labelled stretch resource rather than pretending it is at-grade.
- Language to keep repeating: random means EQUAL CHANCE for everyone; a sample gives an ESTIMATE, not the exact population value; bigger random samples are more reliable, but size alone never fixes a biased selection method.
- Present mode and print both work: use Present to build the dot plot of sample proportions live with the class in section 2 (ideally from a real hands-on sampling activity), then print the worksheets for independent practice.