7.SP.A.1Common CoreMathStatistics and ProbabilityGrade 7
7.SP.A.1: Populations, Samples and Random Sampling
In plain English: 7.SP.A.1 is the Common Core grade 7 math standard that asks students to understand how a sample, a smaller group, can give information about a whole population. A conclusion about the population is valid only if the sample is representative, meaning it is like the population. Random sampling, where every member has the same chance of being picked, tends to give representative samples.
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Use random sampling to draw inferences about a population. Also written as 7.SP.1 · Official standard
Students learn how to find out about a large group by studying a smaller part of it. The population is the whole group a question is about, such as all 600 students at a school. A sample is the part of the population that is actually measured or asked. Asking everyone is called a census, and it is often too slow, too costly or impossible. Statistics uses the sample to make a generalization (also called an inference): a conclusion about the whole population, such as "about 30% of the students walk to school."
A generalization is valid only if the sample is representative, which means it is like the population in the ways that matter for the question. A biased sample leans one way: it has too many or too few of some kind of member. In a random sample, every member of the population has the same chance of being chosen, for example by drawing names from a hat. Random samples tend to be representative, so they support valid inferences, though no single sample matches the population exactly. Using many samples to judge how much estimates vary is the next standard, 7.SP.A.2.
Learning Objectives
By the end of this lesson, students will be able to:
Name the population and the sample in a statistics question
Use results from a random sample to estimate a fact about a population, and describe the estimate with "about"
Decide whether a sample is representative, and explain how a convenience or volunteer sample can be biased
Describe how to choose a random sample, and explain why random sampling tends to give representative samples
Prior Knowledge Required
Students should already be comfortable with:
Telling a statistical question, one that expects answers to vary, from other questions 6.SP.A.1
Finding the mean of a data set 6.SP.B.5
Finding a percent of a quantity 6.RP.A.3
Using ratios and proportions to scale an amount up 7.RP.A.2
Show this invented ad on the board and ask pairs to discuss it for two minutes:
Warm-Up Prompt
"A cereal company says: '9 out of 10 kids love our new cereal!' The company asked 50 kids at its own booth at a fair, where it was giving away free bowls of the cereal. Do you believe that 9 out of 10 of all kids love it? What questions would you ask the company?"
Collect questions. Students often ask who was asked, how many, and how they were chosen. Kids who walk up to a free cereal booth probably already like cereal, and some may say yes to be polite. Introduce the words: the company wants to know about all kids (the population), but it asked only 50 kids at its booth (the sample). Ask: "Is the sample like all kids?" This lesson is about when a sample tells the truth about a population.
Direct Instruction20 minutes
Part 1: Population and sample. Write the two words side by side and give everyday examples. A cook tastes one spoonful (the sample) to learn about the whole pot of soup (the population). The spoonful works only if the soup has been stirred. That is the key idea of the lesson: a sample tells about the population only if it is like the population.
Part 2: Ways to choose a sample. Show three ways, and ask which ones could be biased:
Convenience sample: ask whoever is easiest to reach, such as your friends or one class. It is often biased.
Volunteer sample: people choose to answer, as in an online poll. People with strong opinions answer more, so it is often biased.
Random sample: every member of the population has the same chance of being picked, for example by numbering a list and using a random number generator. It tends to be representative.
Show Diagram 1. The invented population is 80 grade 7 students, and 32 of them play on a school sports team (40%). A random sample of 10 had 3 players (30%), which is close to 40% but not exact. A sample of 10 students standing at the gym door after practice had 9 players (90%), far from the truth. Then show Diagram 2: the computer took 25 random samples and 25 gym-door samples. The random samples cluster around 40%, and the gym-door samples cluster around 87%. Random sampling does not make every sample perfect, but it tends to give samples that are like the population.
Part 3: Using a sample. When a sample is random, use its percent to estimate the population: find the percent in the sample, then take that percent of the population. Always say "about," because a different random sample would give a slightly different answer. A larger random sample usually gives a closer estimate, but a large biased sample is still biased.
Population and sample
A park ranger wants to know the typical length of the trout in a lake. She nets 40 trout in different parts of the lake, measures them and lets them go.
Equation: The population is all the trout in the lake. The sample is the 40 trout she measured. If her net lets small fish slip through, the sample has too few small trout, and her typical length would be too big.
Estimating from a random sample
A computer picks 50 of the 400 students at a school at random. 9 of them say they would join a chess club. About how many students in the whole school would join?
Equation: 9 ÷ 50 = 0.18, or 18%. 18% of 400 = 0.18 × 400 = 72. About 72 students would join. It is an estimate, so a real count could be a little more or less.
A convenience sample
To estimate how many hours grade 7 students spend on homework each night, Kai asks the 25 students in his after-school homework club.
Equation: This is a convenience sample: Kai asked the people easiest to reach. Students in a homework club may do more homework than other students, so the sample is likely biased, and a generalization to all grade 7 students is not valid.
A volunteer sample
A town website asks, "Should the town build a skate park?" 320 people answer, and 85% say no.
Equation: People chose to answer, so this is a volunteer sample (also called a voluntary response sample). People with strong feelings, and people who visit the town website, are more likely to answer. The 85% may not describe the whole town.
Choosing a random sample
A school has 600 students. The student council wants to ask 60 of them about the school dance. Which plan gives a random sample?
Equation: Give every student a number from 1 to 600 on the school list. Use a random number generator to pick 60 different numbers, skipping any repeats, and ask those students. Every student has the same chance of being picked. Asking the first 60 students at the door is not random.
Guided Practice15 minutes
Pairs read each plan, name the population and the sample, and decide whether the sample is likely representative. Then they share one reason with the class.
Guided practice plans with answers
Plan
Answer
To find the favorite school lunch, Ana asks the 30 students at her lunch table.
Population: all students at the school. Sample: her 30 tablemates. Likely biased: friends at one table may like the same foods.
The cafeteria manager enters all 450 student ID numbers into a computer, which picks 45 at random to answer a lunch survey.
Population: all 450 students. Sample: the 45 picked. Random, so likely representative.
A radio station asks listeners to call in and rate a new song.
Population: all listeners. Sample: the callers. A volunteer sample, likely biased toward people who love or hate the song.
To check the quality of a day's 3,000 phone chargers, a factory numbers them and tests 30 chosen by a random number generator.
Population: all 3,000 chargers. Sample: the 30 tested. Random, so likely representative.
For each biased plan, ask: "How could you change it to make it random?" Listen for plans that start with a list of the whole population, such as the school roster, and use chance to pick from it.
Independent Practice15 minutes
Students answer on their own. (1) A random sample of 40 of the 720 students at a school shows that 15 bring lunch from home. Estimate the number of students in the school who bring lunch. (15 ÷ 40 = 37.5%, and 37.5% of 720 = 270, so about 270 students.) (2) A random sample of 200 light bulbs from a batch of 5,000 has 6 bulbs that do not work. Estimate the number of bulbs in the batch that do not work. (6 ÷ 200 = 3%, and 3% of 5,000 = 150, so about 150 bulbs.) (3) To estimate how many people in a town own a dog, Rosa asks people at the dog park. Explain why her sample is biased. (Everyone at a dog park is likely to own a dog, so her estimate would be far too high.) (4) Describe a better plan for Rosa. (Pick households at random from a list of all addresses in the town.)
Closure5 minutes
Exit ticket: (1) In your own words, what does it mean for a sample to be representative? (2) A random sample of 25 of the 300 students at a school shows that 5 play a musical instrument. Estimate how many students in the school play one. (5 ÷ 25 = 20%, so about 60 students.) (3) Name one way to choose a random sample of students in our class.
Differentiation Strategies
For Struggling Students
Use the soup picture: a spoonful from a stirred pot tells about the whole pot, and a spoonful from the unstirred top does not
Give a two-column chart, "Population" and "Sample," and have students fill it in for every problem
Keep the numbers friendly at first, such as samples of 10, 20, 25 or 50, so the percents are easy
For Advanced Students
Ask students to find the bias in a real headline or ad, and to write a better sampling plan
Ask why a random sample of 1,000 people can describe a whole country, while 100,000 volunteers from one website may not
Have students take several random samples of 10 from the Activity 3 bag and describe how much the percent of red cubes changes (a preview of 7.SP.A.2)
Assessment Guidance
What to Look For
Check that students name both the population and the sample, and that they explain bias with a reason tied to the question ("students in a homework club may do more homework"), not just "the sample is too small." Estimates should use the sample percent, be applied to the population size, and be stated with "about." Listen for the idea that a random sample can still miss by a little, while a biased sample tends to miss in the same direction every time.
02
Classroom Activities
3 Activities
1
Pick by Eye or Pick at Random
20 minPairs
The population is the 60 words of a short paragraph, and the question is: what is the mean length of a word, in letters (the total number of letters divided by the number of words)? Pairs estimate it twice, once with a sample they choose by eye and once with a random sample, then compare with the true mean.
The Paragraph (the population)
Our school garden began as a patch of weeds behind the gym. Three years ago, a group of students asked the principal if they could plant tomatoes, beans and sunflowers. Parents gave tools and soil, and a local nursery donated seeds. Now every class visits the garden in the spring, and most of the fresh vegetables go to the cafeteria.
Procedure
Number the words 1 to 60. Do not count the letters in every word yet
Sample A: choose 10 words that you think are typical. Count the letters in each and find the mean
Sample B: use a random number generator (or 60 numbered slips in a bag) to pick 10 different numbers. Find the mean length of those 10 words
Post both means on a class dot plot (a number line with one dot for each value), one row for Sample A and one row for Sample B
The teacher reveals the true mean of all 60 words: 268 letters ÷ 60 words, about 4.5 letters
Discussion Questions
Which row of the dot plot is centered closer to 4.5 letters? Many classes find that the by-eye means are too high. Why might that happen?
Only 9 of the 60 words have 7 or more letters, but 25 words have 3 letters or fewer. Did your by-eye sample have more long words than that?
Did every random sample land exactly on 4.5? What does that tell you about a single random sample?
Modification for Distance Learning
Share the paragraph on a slide and a link to a random number generator. Pairs type both means into a shared spreadsheet, and the teacher shows the two dot plots.
2
Sampling Plan Card Sort
15 minPairs
Pairs sort 8 cards into three piles, convenience, volunteer and random, and mark each plan as likely representative or likely biased. For every biased plan, they write a random plan that answers the same question.
Plan Cards
Card 1: To learn how late students stay up, a teacher asks the students in her first-period class.
Card 2: A library draws 40 names at random from its list of card holders to ask about opening hours.
Card 3: A pet food website posts a poll: "Do you feed your pet twice a day?"
Card 4: To find the most popular game, a student asks everyone at the chess club.
Card 5: A school numbers its 900 students and uses a random number generator to pick 90 to ask about a new schedule.
Card 6: A mayor asks people who come to a town meeting about a new tax.
Card 7: A farmer picks 30 apples from all over the orchard by drawing tree numbers and branch numbers at random.
Card 8: A TV show asks viewers to text in their vote for the best singer.
Answer Key
Convenience, likely biased: Cards 1, 4 and 6
Volunteer, likely biased: Cards 3 and 8
Random, likely representative: Cards 2, 5 and 7
Discussion Questions
Card 4 asks the chess club about games. Which way would its answers lean?
Card 5 asks 90 students and Card 2 asks 40 people. Are both samples random? Does the larger one have to be better?
For Card 6, who is likely to come to a town meeting about a new tax?
Challenge Variation
Pairs write two new cards of their own, one biased and one random, and trade them with another pair to sort.
3
Shake the Bag
15 minGroups of 3-4
Each group gets a paper bag with 50 cubes. The bag is the population. Groups do not know that it holds 15 red cubes and 35 blue cubes (30% red). They sample to estimate the number of red cubes.
Procedure
Round 1 (not mixed): before class, the teacher puts the 35 blue cubes in first and the 15 red cubes on top, and does not shake the bag. One student takes 10 cubes from the top without looking. Record the number of red cubes, then put them back on top
Round 2 (random): shake the bag well, take 10 cubes without looking, record the number of red cubes and put them back. Repeat 3 times, shaking each time
For each sample, estimate the red cubes in the bag: (red in sample ÷ 10) × 50
Post every estimate on a class dot plot, one row for each round, then open the bags and count
Discussion Questions
A top-of-the-bag sample could hold all red cubes, which gives an estimate of 50 red cubes. How far off is that from the truth, 15?
A shaken sample with 3 red cubes gives an estimate of 15. Did every shaken sample give 15? Why not?
Which round is like surveying only the students nearest the door? Which round is like drawing names from a hat?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: One Population, Two Samples
Each circle is one student in an invented population of 80 grade 7 students. Filled circles are the 32 students who play on a school sports team (40%). The 10 ringed students were picked by a computer at random: 3 of them play, or 30%, which is close to the population's 40%. A sample of 10 students at the gym door after practice had 9 players, or 90%, because team players are much more likely to be there.
Diagram 2: Random Samples Tend to Be Representative
A computer drew 25 random samples of 10 students from the population in Diagram 1, and 25 gym-door samples, in which a team player was 12 times as likely as another student to be at the door. Each dot is one sample. The random samples vary, from 10% to 70%, but they center near the true 40%. The gym-door samples all miss in the same direction, from 70% to 100%. The results come from a real computer simulation of the invented population.
04
Homework Assignment
~30 min
7.SP.A.1 Homework: Populations, Samples and Random Sampling
Directions: Answer in full sentences. For every sample, name the population and the sample. Show your work for every estimate, and use the word "about."
Part 1: Populations and Samples (Problems 1-2)
Name the population and the sample in each study. (a) A librarian checks 25 books chosen at random from the library's 3,000 books for torn pages. (b) A coach times 8 of the 40 runners on her track team in the mile. (c) A city measures air quality at 12 places around the city.
Decide whether each sample is likely representative or likely biased, and explain why. (a) To learn how students get to school, a principal surveys the students on one bus. (b) To learn what grade 7 students think about the dress code, a teacher draws 35 names from a bag holding the names of all grade 7 students. (c) To learn how much time teens spend reading, a bookstore surveys its teen customers.
Part 2: Estimates from Random Samples (Problems 3-4)
In a random sample of 80 of the 1,200 students at a middle school, 34 take a bus to school. Estimate the number of students at the school who take a bus.
A farmer checks 50 apples chosen at random from a harvest of 6,000 apples. 4 of them have bruises. Estimate the number of bruised apples in the harvest. Would you trust this estimate more or less if the farmer had checked only the 50 apples on top of one crate? Explain.
Part 3: Judging and Planning Samples (Problems 5-6)
Two students want to know what share of the 500 students at their school want a longer lunch. Jen asks 50 students picked at random from the school list, and 46% say yes. Omar asks 100 students in the lunch line, and 78% say yes. Whose result is a better estimate for the school? Explain, even though Omar asked more students.
Design your own survey. Write a statistical question about the students at your school, name the population, and describe step by step how you would choose a random sample of 40 students. Then describe one way someone could choose a biased sample for the same question.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Population and Sample
Both named correctly for every study
One or two named incorrectly
Most missing or incorrect
Representative or Biased
Correct choice with a reason tied to the question
Correct choice, reason unclear
Incorrect or no reason
Estimates
Sample percent applied to the population, stated with "about"
One computing error
Estimates missing or incorrect
Sampling Plan
Plan uses a list of the whole population and chance
Plan is random but steps are unclear
Plan is not random
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A principal randomly chooses 50 of the 800 students at her school and asks whether they want school to start later. What is the population?
Answer: B
The population is the whole group the question is about: all 800 students at her school. Choice A is the sample, the students who were asked. Choice C is only part of the population. Choice D is a much larger group than the one she is studying.
Question 2 of 20 · Multiple Choice
A baker tests 12 cookies from a batch of 2,000 to check how sweet they are. What is the sample?
Answer: A
The sample is the part that is actually measured: the 12 tested cookies. Choice B is the population. Choice C describes a result, not the group that was tested. Choice D is a larger population than the batch.
Question 3 of 20 · Multiple Choice
Which plan gives a random sample of 30 students from a school of 600?
Answer: C
Drawing names from a hat that holds every name gives each student the same chance. Choice A is a convenience sample: early arrivals may have something in common, such as riding the same bus. Choice B is a volunteer sample. Choice D asks one class, a convenience sample.
Question 4 of 20 · Multiple Choice
To learn the favorite sport of students at her school, Priya asks 40 students at a soccer game. Why is her sample likely biased?
Answer: B
People at a soccer game are more likely than other students to name soccer, so the sample leans one way. Choice A is not a source of bias. Choice C does not make a sample biased. Choice D, asking friends, is another convenience sample and would not fix the problem.
Question 5 of 20 · Multiple Choice
A newspaper prints a coupon asking readers to mail in their vote on whether the city should build a new stadium. Of the 300 readers who mail it back, 70% say no. Why is this not a good estimate for all city residents?
Answer: A
This is a volunteer sample: people chose to mail in a vote, and those with strong opinions are more likely to do so. Also, only people who read that newspaper could answer. Choice B is wrong because the problem is how the voters were chosen, not how many there were. Choice C is false. Choice D would make the bias worse.
Question 6 of 20 · Multiple Choice
A random sample of 60 of the 900 students at a school shows that 27 walk or bike to school. About how many students in the school walk or bike?
Answer: C
27 ÷ 60 = 0.45, or 45%, and 0.45 × 900 = 405, so about 405 students. Choice A is the count in the sample, not the school. Choice B uses the 33 students who do not walk or bike: 33 ÷ 60 × 900 = 495. Choice D writes the percent, 45%, as a number of students.
Question 7 of 20 · Multiple Choice
A factory makes 10,000 phone cases. In a random sample of 250 cases, 6 are cracked. About how many cracked cases are in the whole batch?
Answer: D
6 ÷ 250 = 0.024, or 2.4%, and 2.4% of 10,000 = 240, so about 240 cases. Choice A is the count in the sample. Choice B divides 10,000 by 250, which counts the groups of 250, not the cracked cases. Choice C reads 0.024 as 24%: 0.24 × 10,000 = 2,400.
Question 8 of 20 · Multiple Choice
In a random sample of 80 of the 1,100 students at a school, 24 say they read before bed. Which conclusion is valid?
Answer: A
24 ÷ 80 = 30%, and a random sample supports an estimate for the school: about 30%, or about 330 students. Choice B says "exactly," but a sample gives only an estimate. Choice C generalizes to a different population, the whole state, which was not sampled. Choice D is wrong because a random sample does support a conclusion about the school.
Question 9 of 20 · Multiple Choice
To estimate how many hours a week the adults in a town exercise, which sample is best?
Answer: B
The random sample of 50 gives every resident the same chance. Choice A is large but biased: people leaving a gym exercise more than most adults. Choice C is a volunteer sample from gym fans. Choice D uses chance, but only among runners, who also exercise more than most adults. A bigger sample does not fix bias.
Question 10 of 20 · Multiple Choice
A survey plan says: "Ask the first 30 students who get off the buses." Which change gives a random sample of all the students at the school?
Answer: C
Picking from the list of all students with chance gives everyone the same chance, including students who walk or get a ride. Choice A makes the sample larger but still leaves out students who do not ride a bus. Choice B uses chance, but only among band members, so no other student can be picked. Choice D changes which bus riders are asked but still leaves out everyone else.
Question 11 of 20 · Multiple Choice
Leo picks 3 students at random from his grade of 150, and all 3 have a dog. He says 100% of the grade has a dog. What is the main problem?
Answer: A
Random samples tend to be representative, but a sample of 3 can easily land far from the truth. A larger random sample would give a closer estimate. Choice B is wrong: random choice is the right way to pick. Choice C is not a problem with the sample. Choice D would make the sample biased.
Question 12 of 20 · Multiple Choice
Which sample is most likely to be representative for finding the mean height of the 7th graders at a school?
Answer: D
Picking at random from the list of all 7th graders gives every student the same chance. Choice A uses chance, but only among basketball players, who tend to be taller, so it is biased. Choice B is a convenience sample, and students in front rows may be seated by height or by class. Choice C is a volunteer sample, and taller or shorter students may be more or less willing to be measured.
Question 13 of 20 · Multiple Choice
Two groups estimate the share of students who want a later bedtime on school nights. Group 1 asks a random sample of 60 students: 55% say yes. Group 2 asks 60 students at a sleepover party: 90% say yes. Which estimate should the class trust?
Answer: B
Group 1's sample was chosen at random, so it tends to be representative. Choice A is not a reason: a higher percent does not make an estimate better. Choice C is wrong because a random sample of 60 can give a useful estimate. Choice D describes why Group 2 is biased: students at a sleepover are already staying up late.
Question 14 of 20 · Multiple Choice
A company makes 50,000 light bulbs a day. To check how long they last, it keeps 100 bulbs on until they burn out. Why does it test a sample instead of every bulb?
Answer: D
This test uses up each bulb it checks, so a census would destroy the whole day's bulbs. Choice A is false: a sample gives only an estimate. Choice B is false, since bulbs vary, which is why the company tests them. Choice C is wrong because the company wants to learn about all 50,000 bulbs, the population.
Question 15 of 20 · Short Answer
A random sample of 120 of the 1,350 students at a school shows that 48 plan to go to the spring fair. Estimate how many students at the school plan to go, and explain why the estimate is reasonable.
48 ÷ 120 = 0.4, or 40%, and 40% of 1,350 = 540. About 540 students plan to go. The estimate is reasonable because the sample was random, so it tends to be like the whole school, but the true number could be a little more or less.
Question 16 of 20 · Short Answer
To learn how often students use the school library, a teacher surveys the students who are studying in the library after school. Is the sample representative? Explain, and describe a better plan.
No, it is likely biased. Students who are in the library after school probably use it more than other students, so the estimate would be too high. A better plan: number every student on the school list and use a random number generator to pick the sample.
Question 17 of 20 · Short Answer
Two random samples of 50 students are taken from a school of 700. In the first, 18 students say they want a school garden. In the second, 22 say so. Why are the results different? Use both samples together to estimate the number of students in the school who want a garden.
Each random sample includes different students by chance, so the results vary a little: 36% and 44%. Together, 40 of 100 students, or 40%, want a garden, and 40% of 700 = 280. About 280 students want a garden.
Question 18 of 20 · Short Answer
A city chooses 200 homes at random from its list of all 18,000 home addresses. 130 of the 200 homes have at least one pet. Estimate the number of homes in the city that have a pet, and say whether the generalization is valid.
130 ÷ 200 = 65%, and 65% of 18,000 = 11,700. About 11,700 homes have a pet. The generalization is valid because the homes were chosen at random from a list of every home in the city, so the sample is likely representative.
Question 19 of 20 · Short Answer
Describe step by step how to choose a random sample of 30 students from a school of 450 students.
Get the list of all 450 students and give each one a number from 1 to 450. Use a random number generator (or 450 numbered slips in a bag, well mixed) to pick numbers until you have 30 different numbers, skipping repeats. Ask the students with those numbers. Every student has the same chance of being picked.
Question 20 of 20 · Short Answer
A news website reports: "72% of parents want a four-day school week!" The result comes from an online poll that 5,000 of the site's readers chose to answer. Give two reasons why this generalization about all parents may not be valid.
It is a volunteer sample: readers chose to answer, and people with strong opinions are more likely to vote. It samples the wrong population: the site's readers, who may not all be parents and may differ from other parents. A large number of answers does not fix either problem.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.SP.A.1 mean?
7.SP.A.1 means students understand how a sample can tell us about a population, and when it cannot. A sample is a smaller part of a group, and the population is the whole group. A conclusion about the population is valid only when the sample is representative, and random sampling is the usual way to get one.
What grade is 7.SP.A.1, and what comes next?
It is a grade 7 standard in the Statistics and Probability domain. The next standard, 7.SP.A.2, uses data from random samples to make estimates and takes many samples to see how much estimates vary. In high school statistics, students treat sampling as the basis of all inference (HSS.IC.A.1) and compare surveys, experiments and observational studies (HSS.IC.B.3).
What is the difference between a population and a sample?
The population is the whole group you want to learn about, and the sample is the part you actually study. If a school wants to know how its 600 students feel about the dance, the 600 students are the population. The 60 students it asks are the sample.
What makes a sample representative?
A sample is representative when it is like the population in the ways that matter for the question. For a question about sports, a representative sample has about the same share of athletes as the whole school. The way the sample is chosen matters most: random choice tends to give representative samples, and choosing whoever is easy to reach often does not.
Is a bigger sample always better?
No, a bigger sample helps only if it is chosen well. A large biased sample is still biased: in 1936, the Literary Digest magazine collected more than 2 million mailed-in ballots and still predicted the wrong winner of the presidential election. A well-chosen random sample of a few hundred or a few thousand people is used by many polls today.
Why don't we just ask everyone?
Asking everyone, a census, is often too slow or too costly. A national survey of every person every month would take huge amounts of time and money. A good random sample gives a close answer much faster, which is why most surveys use samples.
How can students choose a random sample in class?
Number every member of the population, then let chance choose. Students can draw numbered slips from a well-mixed bag, or use a random number generator on a calculator or a website and skip any repeats. What matters is that every member has the same chance of being picked.
What mistakes do students make with 7.SP.A.1?
A common mistake is thinking that a larger sample is always representative, even when it was chosen in a biased way. Other mistakes are mixing up the population and the sample, calling any sample "random" because nobody planned it, such as the first people in line, and expecting one random sample to match the population exactly.
Does 7.SP.A.1 include margin of error?
No. Grade 7 describes estimates from samples with words like "about" and "close to." Measuring how far off an estimate might be with a margin of error comes in high school statistics (HSS.IC.B.4). In grade 7, students see the spread of estimates informally by comparing several random samples in 7.SP.A.2.
How can parents help with samples and surveys at home?
Parents can look at claims in ads and news with their child. Ask: "Who was asked? How were they chosen? Are they like everyone the claim is about?" For example, a restaurant that asks only its own customers will hear from people who already like its food.
07
Related Standards
6 standards
These standards connect to 7.SP.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.SP.A.1Prerequisite
Recognize a statistical question as one that expects variability in the data
