- Define the terms survey, population, and sample
- Indicate that valid samples enable inferences to be made about the population
- Indicate that samples should be large enough and representative of the population as a whole
- Use proportional reasoning when making inferences
- Data Sampling: Representing Many by Sampling Some Digital Interactive Tool
- Interactive whiteboard or computer/projector hookup
- Internet connection
- Answer Key for the Digital Interactive Tool
- Taste Testing: Identifying Sample Size and Making Inferences Worksheet printable
- Product Planning: Selecting Valid Samples Worksheet printable
- Cooking Up Success: Interpreting and Presenting Survey Results Worksheet printable
- Answer Key: Data Sampling Worksheets
- Standards Chart: Data Sampling printable
- Optional: Computers for small groups and/or all students
Note: The online and offline components can be flexibly used as appropriate for each classroom. The sequence below is a suggestion only.
- Make class sets of the Taste Testing: Identifying Sample Size and Making Inferences Worksheet printable, the Product Planning: Selecting Valid Samples Worksheet printable, and the Cooking Up Success: Interpreting and Presenting Survey Results Worksheet printable.
- Print a copy of the Answer Key: Data Sampling Worksheets for your use.
- If you are using the Digital Interactive Tool, load the program on your interactive whiteboard or computer.
Introduction to Sampling
Step 1: Announce to the class that the school administration is considering adding an elective course on coding. (Note: Feel free to modify the topic of coding or substitute another if there is a question of greater interest at your school.) School administrators don't want to invest the time and expense of recruiting a teacher, developing curriculum, or buying materials unless a sufficient number of students are interested in taking the class. Without interviewing everyone in the school, is there a way to gauge student interest?
Step 2: Presumably, the suggestion of taking a survey (a method of asking part of a larger group questions to learn about the entire group) of some of the students at school will be made. Ensure that the class understands that the purpose of taking such a survey is to be able to make inferences (i.e., draw conclusions) about the population (the entire group) by obtaining data from a sample (a part) of the population.
Step 3: If necessary, demonstrate how proportional reasoning (using multiplicative relationships to compare things) can be used to utilize the results from the sample to draw a conclusion about the population as a whole. For example, if a sample of 200 students is surveyed out of a school of 2,000 students, and 54 indicate that they would be interested in the course, then (2,000/200) x 54 = 540 students in the entire school who would be interested in the course. Discuss how confident the school administration can be in the result of 540, and how far away from 540 the true number might be. Make sure that students understand that an inference is a prediction, not a fact.
Optional Mini Lesson: Confidence Intervals and Confidence Levels
For advanced classes with prior knowledge of basic statistical concepts, this could be a good time to provide a basic overview of confidence intervals and confidence levels as follows:
Elaborate on the question, "...how far away from 540 the true number might be?" When we surveyed the 200 students in the sample, we found out that 54 were interested in the coding course. If we asked a different sample of 200 students, would we get exactly 54 positive responses? If we took many samples of 200 students, how far from 54 would the number of positive responses from these samples range?
Point out that when news organizations report the results of surveys prior to elections, they always add a plus or minus factor, e.g. the candidate is projected to receive 56% of the vote, ± 4%. That way, they are more confident that the results of their survey, including the ± factor, reflect the views of the population as a whole than they would be without the ± factor. So, would we be more confident reporting that exactly 540 students are interested in the course, or that 540 students ± 4% would be interested? Point out that the confidence interval is a measure of reliability for our estimate.
If we have a degree of confidence that the result of our survey, including the ± factor, reflects the population as a whole, exactly how confident are we? Explain how this can be quantified with a percentage known as the confidence level. If we took many samples, the confidence level would tell us the percentage of samples that would fall within the confidence interval. So if we said that we had a 95% confidence level for our survey, that means that if multiple samples were taken from our population, we would expect that 95% of the time, the results would fall within the confidence interval. Note that a 95% confidence level is the typical, real-world standard for research. Also note that higher levels can be achieved with larger samples.
Step 4: Ask the class how to design the survey of 200 students (or whatever number would be appropriate for your school). Brainstorm ideas. Ask if it would be appropriate to create a sample of varsity athletes. It would be easy to administer because we could survey them prior to practice. However, can we be sure that the results would reflect the school population as a whole? Discuss how overrepresenting one subgroup or another can make the results of a survey less reliable. The sample must be representative (an accurate reflection) of the population as a whole.
Step 5: Ask if it wouldn't be quicker and easier to simply administer the survey to 20 students. Discuss how the size of the sample needs to be large enough if we are to rely on the results.
Step 6: Once it is established that the sample has to be large enough and representative of the population, brainstorm ideas for selecting the students to whom the survey would be administered. Some student suggestions might include asking the first 200 students to arrive in the morning, 8 classes of 25 students each, every 10th student to arrive, etc. Introduce the idea of using student identification numbers and using a random number generator to select the sample. Discuss the pros and cons of each method and ensure that students understand the importance of randomness when selecting a sample, i.e. that each member of the population has an equal chance of being selected.
Step 7: Wrap up, ensuring that students understand the key vocabulary terms introduced and the concept that samples should be randomly selected, representative of the population as a whole, and large enough.
Step 8: Before transitioning to Guided Practice, review these tips for students about exemplary math thinking:
The key skills described below support the Common Core State Standards for Mathematical Practice. You may wish to select one or two skills that are new to students and guide them in incorporating that skill when responding to reflection questions.
- Cite Evidence: Justify conclusions by referring to survey results, both mathematical and contextual (the phrasing of the survey question).
- Point Out Unknowns: Identify relevant unknown information, and suggest follow-up questions to elicit it. Explain how unknown information impacts the precision of the results and the decision-making process.
- Make Connections: Draw conclusions by connecting the results of two or more surveys. Connect survey results or sampling concepts to other texts or experiences.
- Construct Arguments: Use academic, persuasive language to support an evidence-based conclusion. Recognize flaws in others' arguments in a discussion, and politely justify how they came to a different conclusion.
- Identify Assumptions: Identify assumptions when reasoning why results may have turned out as they did (including why opinions may differ between population segments), and suggest follow-up survey questions that could substantiate or disprove the assumptions.
- Search for Patterns: Notice patterns or trends across survey questions (e.g., "Sample sizes of X or lower tend to produce this type of result"), and use these to justify conclusions. Acknowledge or respond to patterns when recommending next steps.
Step 9: Have students use the Data Sampling: Representing Many by Sampling Some Digital Interactive Tool to practice what they have learned, or distribute the practice worksheets.
Using the Digital Interactive Tool*
Data Sampling: Representing Many by Sampling Some is an online, interactive tool designed to help students understand why it is important to select statistically valid representative samples. The tool, with its many variables to adjust and questions to discuss, provides numerous opportunities for active student participation. The questions challenge students to consider not just the results but their meaning. (For answers to the questions, see the Answer Key for the Digital Interactive Tool).
Data Sampling: Representing Many by Sampling Some contains four modules of the same skill level covering four policy issues in a small town: Skate Park, Pet Insurance, Summer Concert Series, and Public Computer Center. After choosing an issue, users select one of four related policy questions. They will then survey three different population samples, the parameters of which the user sets (sample size, population segment, and confidence level are adjustable). Then a statistically valid representative sample as well as the results of the entire population will appear for comparison. Questions are provided to aid users in thinking critically about the meaning of the results.
Select one issue module to model for the class using an interactive whiteboard or via computer/projector hookup. Perform a "Think Aloud" as you adjust survey variables and launch discussion using the question prompts. As you move to the second issue module, encourage students to take the lead and, through discussion, reach a consensus on what variables to adjust and why.
You may wish to launch a discussion of survey results by posing the following:
General Analysis Questions
- What impact does adjusting the sample size have on the results?
- How does limiting a survey to a specific population segment affect the results? Is this ever justified? If so, when?
- What conclusions can you draw about sampling in general, or a survey question in particular, when comparing the results from the three population samples you surveyed?
- What did you notice about the sample size and the results (compared to the population as a whole) when you ran samples with 95% and 99% confidence levels? What is the relationship between confidence level percentage, sample size, and the degree of comfort you have that you can rely on the results?
Answers to General Analysis Questions
- In general, a larger sample size means we can be more confident that the results are reflective of the population as a whole than a smaller sample size. Results from a larger sample tend to be closer to the results of the entire population. Smaller samples tend to be less accurate.
- In looking at the results of individual demographic groups, we usually find that members of some groups have different opinions about certain issues than members of other groups. For example, parents of children who play recreation league sports might more strongly favor an increase in the school district sports budget than taxpayers in general. "Loading up" a sample with an unusually high number of members of one group or the other would give survey results that are not reflective of the population as a whole. However, there are times when it is appropriate to limit a survey to members of specific groups. For example, owners of a music store might want to limit a survey to musicians when determining what new products and services to offer.
- In general, larger sample sizes are more likely to yield survey results that are reflective of the population as a whole than smaller samples. In general, samples that don't overrepresent one demographic group or other give results that are reflective of the population as a whole, versus samples that have an out-of-proportion number of members of one demographic group.
- For advanced classes ready to explore the confidence-level feature of the tool, make sure that the class sees that higher confidence levels require larger sample sizes, meaning a larger investment in time and money to conduct the survey. The benefit, however, is greater confidence in the results. Consider asking the class if the results obtained while using a higher confidence level were worth the added time and expense.
*The Digital Interactive Tool requires Internet access and either an interactive whiteboard or a computer/projector hookup.
Using the Taste Testing: Identifying Sample Size and Making Inferences Worksheet
Complete the Taste Testing: Identifying Sample Size and Making Inferences Worksheet printable as a class. Have students explain their reasoning and methods for coming up with the answers. Ask students:
- What other surveys might they want to conduct within the worksheet narrative?
- How would they set up their surveys to make them representative of the population?
Worksheet 1 Extension: Ask the students what conclusions Snacks Stat! can draw from the graphs. Remind students to cite mathematical evidence (including processing the data into percentages) in their responses. Then, have students come up with additional questions that Snacks Stat! should ask via survey to determine whether they should sell the products or not.
Answers will vary but could include:
Graph #1: Overall, the energy bars were very popular with potential customers so the company should pursue selling it. Only 24% of the people surveyed didn't have a favorite flavor (18 with the "none" response divided by the sample of 75). Snacks Stat! could follow up by asking customers about price points, how often people would buy it, what could be improved, and what other flavors customers think they might also want to buy.
Graph #2: Kale-Coconut-Maple was by far the most popular smoothie; 53% of those surveyed would buy it (40 "yes" responses divided by the sample size of 75). Broccoli-Orange was the least popular smoothie. Snacks Stat! might want to move ahead with developing the three most popular flavors (Kale-Coconut-Maple, Beet-Cashew, and Parsley-Pear) and consider eliminating Broccoli-Orange. The company should also survey customers to find out about price points, locations where the product would be sold, portion sizes, etc.
Step 10: Checking for Understanding: Address any student misconceptions as they occur. Wrap up by having several students each volunteer one fact they learned, share something they'd like to learn more about, or ask a question.
Step 11: Assign students to complete additional practice using the Data Sampling: Representing Many by Sampling Some Digital Interactive Tool or the Product Planning: Selecting Valid Samples Worksheet printable.
Using the Digital Interactive Tool*
Assign small groups a survey issue(s) from the Data Sampling: Representing Many by Sampling Some to work on independently.
*Requires a computer/Internet for each small group.
Using the Product Planning: Selecting Valid Samples Worksheet
Assign the Product Planning: Selecting Valid Samples Worksheet printable for students to complete individually.
Step 12: Checking for Understanding: Have students report their findings/answers to the class and share their conclusions. Address any misconceptions (yourself, or through student volunteers).
Formative Assessment: More About Sampling in the Real World
Step 13: Use the discussion prompt below to further connect sampling to real-world spheres and careers, while evaluating if students understand the concepts enough to apply them to other situations. Clarify any student misconceptions.
Discussion Prompt: Sampling is used in the real world to address financial, political, scientific, and legal concerns. What procedures might organizations use to conduct the business below?
- Conducting market research for a product (Who uses the product? How do users feel about the product?)
- Tracking TV viewership (e.g., Nielsen ratings)
- Medical trials (e.g., testing the safety and efficacy of medicines in FDA drug trials)
- Political campaign polling
- Factory testing of certain mass-produced goods
- Evaluating the impact of a law or policy
Step 14: Follow up with the writing prompt, which can be used as an "exit slip" that each student turns in at the end of class. Review the exit slips for a snapshot of the class's understanding of sampling.
Writing Prompt "Exit Slip": You are the manager of a candy company. You want to make sure that each piece of your delicious kale brittle has that yummy kale taste. To be totally sure that each piece meets standards, you would have to eat every piece you produce, but then you wouldn't have anything left to sell. Using what you learned today, how could you determine that your candy meets your standards without eating each piece? Explain each step of your process.
Step 15: Assign additional practice using the Data Sampling: Representing Many by Sampling Some Digital Interactive Tool or the Cooking Up Success: Interpreting and Presenting Survey Results Worksheet printable.
Using the Digital Interactive Tool*
Direct students to complete the remaining Data Sampling: Representing Many by Sampling Some Data Sampling Displayer issue module independently for homework and answer the question prompts on paper. Note that the data and results vary significantly between modules. If the Skate Park module is worked on during class, the student will find homework assignments for the other modules engaging.
*Requires a computer/Internet access outside of school hours for each student.
Using the Cooking Up Success: Interpreting and Presenting Survey Results Worksheet
Assign the Cooking Up Success: Interpreting and Presenting Survey Results Worksheet printable to students for homework.
Step 16: Checking for Understanding: Review student work for any misconceptions. Address the misconceptions yourself or via student volunteers.
Optional: After reviewing the objectives with students and addressing any misconceptions displayed in prior student work, assign the online or offline components not previously completed for Guided Practice, Independent Practice, Formative Assessment, and Homework.
- Grade 6: Sampling (CCSS 6.SP.5)
- Grade 7: Sampling & Inferences (CCSS 7.SP.1 and 2)
- Grade 8 Advanced: Sampling & Inferences (CCSS HSS.IC.1, 3, and 4)
- Grades 6–8: Reasoning, Constructing an Argument, and Attending to Precision (CCSS MP2, 3, and 6); NCTM Data Analysis and Probability
For more information, download the comprehensive Standards Chart: Data Sampling printable.