Chapter 5 · Lesson 3 of 8 · Grade 7 · Statistics & Probability · MAT-07-SP-003
Compare Two Populations with Measures of Center and Spread
Represent and solve compare two populations with measures of center and spread problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 5: Sampling, Probability, and Statistical Inference
Statistics & Probability
Essential question: How can samples and probability models support careful claims about uncertainty and populations?
Where this lesson fits
Lesson 3 of 8. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of sampling, probability, and statistical inference, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve compare two populations with measures of center and spread problems using precise mathematical notation, models, and reasoning.
- Explain why a method for compare two populations with measures of center and spread works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 5: Sampling, Probability, and Statistical Inference
Statistics & Probability. Lesson 3 of 8 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Compare Two Populations with Measures of Center and Spread describes data or chance while accounting for variability. Calculations must be interpreted in the context of data collection or a probability model.
Represent the relationship
The Math Box uses statistics plot to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify calculations from the raw values, then decide whether the sample, display, center, spread, or probability model actually supports the claim.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Statistics Lab
Change the data and watch center and spread respond.
- Model compare two populations with measures of center and spread in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
Population A has median 10; B has median 14. Which has the larger median?
Population BCompare center, then also consider spread for a complete comparison.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 11; B has median 15. Which has the larger median?
Population BThe representation should show the same mathematical relationship as the calculation. Compare center, then also consider spread for a complete comparison.
Explain why a valid method works, then solve: Population A has median 12; B has median 16. Which has the larger median?
Population BA complete explanation names the relationship or property being preserved. Compare center, then also consider spread for a complete comparison.
Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 13; B has median 17. Which has the larger median?
Population BThe estimate is a reasonableness check, not a replacement for the exact result. Compare center, then also consider spread for a complete comparison.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 14; B has median 18. Which has the larger median?
Population BVerification should independently support the result. Compare center, then also consider spread for a complete comparison.
Interpret the answer in context after solving. What does the result mean here? Population A has median 15; B has median 19. Which has the larger median?
Population BState the result with its meaning, units, direction, or comparison—not only a number. Compare center, then also consider spread for a complete comparison.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: Population A has median 16; B has median 20. Which has the larger median?
Interpret center with spread, shape, and sampling method. Correct solution: Population BError analysis requires identifying the broken idea, not merely replacing the final answer. Compare center, then also consider spread for a complete comparison.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Population A has median 17; B has median 21. Which has the larger median? Case B: Population A has median 26; B has median 30. Which has the larger median?
Case A: Population B Case B: Population BCompare the structure, representation, units, and result rather than only the numbers. Compare center, then also consider spread for a complete comparison.
Population A has median 18; B has median 22. Which has the larger median?
Population BCompare center, then also consider spread for a complete comparison.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 19; B has median 23. Which has the larger median?
Population BThe representation should show the same mathematical relationship as the calculation. Compare center, then also consider spread for a complete comparison.
Explain why a valid method works, then solve: Population A has median 20; B has median 24. Which has the larger median?
Population BA complete explanation names the relationship or property being preserved. Compare center, then also consider spread for a complete comparison.
Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 21; B has median 25. Which has the larger median?
Population BThe estimate is a reasonableness check, not a replacement for the exact result. Compare center, then also consider spread for a complete comparison.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 22; B has median 26. Which has the larger median?
Population BVerification should independently support the result. Compare center, then also consider spread for a complete comparison.
Interpret the answer in context after solving. What does the result mean here? Population A has median 23; B has median 27. Which has the larger median?
Population BState the result with its meaning, units, direction, or comparison—not only a number. Compare center, then also consider spread for a complete comparison.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: Population A has median 24; B has median 28. Which has the larger median?
Interpret center with spread, shape, and sampling method. Correct solution: Population BError analysis requires identifying the broken idea, not merely replacing the final answer. Compare center, then also consider spread for a complete comparison.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Population A has median 25; B has median 29. Which has the larger median? Case B: Population A has median 34; B has median 38. Which has the larger median?
Case A: Population B Case B: Population BCompare the structure, representation, units, and result rather than only the numbers. Compare center, then also consider spread for a complete comparison.
Population A has median 26; B has median 30. Which has the larger median?
Population BCompare center, then also consider spread for a complete comparison.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 27; B has median 31. Which has the larger median?
Population BThe representation should show the same mathematical relationship as the calculation. Compare center, then also consider spread for a complete comparison.
Explain why a valid method works, then solve: Population A has median 28; B has median 32. Which has the larger median?
Population BA complete explanation names the relationship or property being preserved. Compare center, then also consider spread for a complete comparison.
Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 29; B has median 33. Which has the larger median?
Population BThe estimate is a reasonableness check, not a replacement for the exact result. Compare center, then also consider spread for a complete comparison.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 30; B has median 34. Which has the larger median?
Hint: Choose a representation before computing.
Answer: Population B
02Interpret the answer in context after solving. What does the result mean here? Population A has median 31; B has median 35. Which has the larger median?
Hint: Explain why the method works and name a verification strategy.
Answer: Population B
03Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: Population A has median 32; B has median 36. Which has the larger median?
Hint: Choose a representation before computing.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: Population B
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Population A has median 33; B has median 37. Which has the larger median? Case B: Population A has median 42; B has median 46. Which has the larger median?
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: Population B Case B: Population B
05Population A has median 34; B has median 38. Which has the larger median?
Hint: Choose a representation before computing.
Answer: Population B
06Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 35; B has median 39. Which has the larger median?
Hint: Explain why the method works and name a verification strategy.
Answer: Population B
Practice
Independent practice
01Explain why a valid method works, then solve: Population A has median 36; B has median 40. Which has the larger median?
Answer: Population B
02Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 37; B has median 41. Which has the larger median?
Answer: Population B
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 38; B has median 42. Which has the larger median?
Answer: Population B
04Interpret the answer in context after solving. What does the result mean here? Population A has median 39; B has median 43. Which has the larger median?
Answer: Population B
05Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: Population A has median 40; B has median 44. Which has the larger median?
Answer: Interpret center with spread, shape, and sampling method. Correct solution: Population B
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Population A has median 41; B has median 45. Which has the larger median? Case B: Population A has median 50; B has median 54. Which has the larger median?
Answer: Case A: Population B Case B: Population B
07Population A has median 42; B has median 46. Which has the larger median?
Answer: Population B
08Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 43; B has median 47. Which has the larger median?
Answer: Population B
Common mistakes
Learn to catch the error, not just the answer.
Interpret center with spread, shape, and sampling method.
Finite experimental frequencies vary even when a model is correct.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Summarize data, compare groups, evaluate survey claims, and describe uncertainty.
- Build probability models and simulations for repeated random processes.
Challenge problems
Explain why a valid method works, then solve: Population A has median 44; B has median 48. Which has the larger median?
Population B
Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 45; B has median 49. Which has the larger median?
Population B
Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 46; B has median 50. Which has the larger median?
Population B
Interpret the answer in context after solving. What does the result mean here? Population A has median 47; B has median 51. Which has the larger median?
Population B
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 5 Reasoning Lab
Solve compare two populations with measures of center and spread tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: Population A has median 48; B has median 52. Which has the larger median?
Answer: Interpret center with spread, shape, and sampling method. Correct solution: Population B
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Population A has median 49; B has median 53. Which has the larger median? Case B: Population A has median 58; B has median 62. Which has the larger median?
Answer: Case A: Population B Case B: Population B
03Population A has median 50; B has median 54. Which has the larger median?
Answer: Population B
04Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: Population A has median 51; B has median 55. Which has the larger median?
Answer: Population B
05Explain why a valid method works, then solve: Population A has median 52; B has median 56. Which has the larger median?
Answer: Population B
06Estimate or predict first, then calculate and decide whether the result is reasonable: Population A has median 53; B has median 57. Which has the larger median?
Answer: Population B
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Population A has median 54; B has median 58. Which has the larger median?
Answer: Population B
08Interpret the answer in context after solving. What does the result mean here? Population A has median 55; B has median 59. Which has the larger median?
Answer: Population B
Terminology
Words to know
- distribution
- The pattern of values including center, spread, shape, and unusual features.
- variability
- The degree to which observations differ.
- sample
- A subset of a population used to collect information.
- probability
- A number from 0 to 1 describing likelihood.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.