Chapter 6 · Lesson 7 of 12 · Grade 6 · Statistics & Probability · MAT-06-SP-007
Use Mean Absolute Deviation
Represent and solve use mean absolute deviation problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 6: Statistics, Variability, and Mathematical Modeling
Using data and reasoning to support claims
Essential question: How can data, models, and mathematical arguments be used responsibly to describe variation and evaluate conclusions?
Where this lesson fits
Lesson 7 of 12. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Collect a small data set, choose appropriate displays and summary measures, critique a misleading claim, and present a defensible conclusion.
Learning objectives
What you should be able to do
- Represent and solve use mean absolute deviation problems using precise mathematical notation, models, and reasoning.
- Explain why a method for use mean absolute deviation works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Chapter 6: Statistics, Variability, and Mathematical Modeling
Using data and reasoning to support claims. This is Lesson 7 of 12 in Chapter 6. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes all previous chapters. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Use Mean Absolute Deviation 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 one example of use mean absolute deviation in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
For 2, 4, 6, 8, 10, find MAD.
mean = 6; MAD = 2.4MAD averages absolute distances from the mean.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 3, 5, 7, 9, 11, find MAD.
mean = 7; MAD = 2.4The representation should show the same mathematical relationship as the calculation. MAD averages absolute distances from the mean.
Explain why a valid method works, then solve: For 4, 6, 8, 10, 12, find MAD.
mean = 8; MAD = 2.4A complete explanation names the relationship or property being preserved. MAD averages absolute distances from the mean.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 5, 7, 9, 11, 13, find MAD.
mean = 9; MAD = 2.4The estimate is a reasonableness check, not a replacement for the exact result. MAD averages absolute distances from the mean.
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 6, 8, 10, 12, 14, find MAD.
mean = 10; MAD = 2.4Verification should independently support the result. MAD averages absolute distances from the mean.
Interpret the answer in context after solving. What does the result mean here? For 7, 9, 11, 13, 15, find MAD.
mean = 11; MAD = 2.4State the result with its meaning, units, direction, or comparison—not only a number. MAD averages absolute distances from the mean.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 8, 10, 12, 14, 16, find MAD.
Interpret center with spread, shape, and sampling method. Correct solution: mean = 12; MAD = 2.4Error analysis requires identifying the broken idea, not merely replacing the final answer. MAD averages absolute distances from the mean.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 9, 11, 13, 15, 17, find MAD. Case B: For 18, 20, 22, 24, 26, find MAD.
Case A: mean = 13; MAD = 2.4 Case B: mean = 22; MAD = 2.4Compare the structure, representation, units, and result rather than only the numbers. MAD averages absolute distances from the mean.
For 10, 12, 14, 16, 18, find MAD.
mean = 14; MAD = 2.4MAD averages absolute distances from the mean.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 11, 13, 15, 17, 19, find MAD.
mean = 15; MAD = 2.4The representation should show the same mathematical relationship as the calculation. MAD averages absolute distances from the mean.
Explain why a valid method works, then solve: For 12, 14, 16, 18, 20, find MAD.
mean = 16; MAD = 2.4A complete explanation names the relationship or property being preserved. MAD averages absolute distances from the mean.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 13, 15, 17, 19, 21, find MAD.
mean = 17; MAD = 2.4The estimate is a reasonableness check, not a replacement for the exact result. MAD averages absolute distances from the mean.
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 14, 16, 18, 20, 22, find MAD.
mean = 18; MAD = 2.4Verification should independently support the result. MAD averages absolute distances from the mean.
Interpret the answer in context after solving. What does the result mean here? For 15, 17, 19, 21, 23, find MAD.
mean = 19; MAD = 2.4State the result with its meaning, units, direction, or comparison—not only a number. MAD averages absolute distances from the mean.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 16, 18, 20, 22, 24, find MAD.
Interpret center with spread, shape, and sampling method. Correct solution: mean = 20; MAD = 2.4Error analysis requires identifying the broken idea, not merely replacing the final answer. MAD averages absolute distances from the mean.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 17, 19, 21, 23, 25, find MAD. Case B: For 26, 28, 30, 32, 34, find MAD.
Case A: mean = 21; MAD = 2.4 Case B: mean = 30; MAD = 2.4Compare the structure, representation, units, and result rather than only the numbers. MAD averages absolute distances from the mean.
For 18, 20, 22, 24, 26, find MAD.
mean = 22; MAD = 2.4MAD averages absolute distances from the mean.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 19, 21, 23, 25, 27, find MAD.
mean = 23; MAD = 2.4The representation should show the same mathematical relationship as the calculation. MAD averages absolute distances from the mean.
Explain why a valid method works, then solve: For 20, 22, 24, 26, 28, find MAD.
mean = 24; MAD = 2.4A complete explanation names the relationship or property being preserved. MAD averages absolute distances from the mean.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 21, 23, 25, 27, 29, find MAD.
mean = 25; MAD = 2.4The estimate is a reasonableness check, not a replacement for the exact result. MAD averages absolute distances from the mean.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: For 22, 24, 26, 28, 30, find MAD.
Hint: Choose a representation before computing.
Answer: mean = 26; MAD = 2.4
02Interpret the answer in context after solving. What does the result mean here? For 23, 25, 27, 29, 31, find MAD.
Hint: Explain what relationship or property makes your method valid.
Answer: mean = 27; MAD = 2.4
03Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 24, 26, 28, 30, 32, find MAD.
Hint: Choose a representation before computing.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: mean = 28; MAD = 2.4
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 25, 27, 29, 31, 33, find MAD. Case B: For 34, 36, 38, 40, 42, find MAD.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: mean = 29; MAD = 2.4 Case B: mean = 38; MAD = 2.4
05For 26, 28, 30, 32, 34, find MAD.
Hint: Choose a representation before computing.
Answer: mean = 30; MAD = 2.4
06Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 27, 29, 31, 33, 35, find MAD.
Hint: Explain what relationship or property makes your method valid.
Answer: mean = 31; MAD = 2.4
Practice
Independent practice
01Explain why a valid method works, then solve: For 28, 30, 32, 34, 36, find MAD.
Answer: mean = 32; MAD = 2.4
02Estimate or predict first, then calculate and decide whether the result is reasonable: For 29, 31, 33, 35, 37, find MAD.
Answer: mean = 33; MAD = 2.4
03Solve and verify the result with a second method, inverse operation, or equivalent representation: For 30, 32, 34, 36, 38, find MAD.
Answer: mean = 34; MAD = 2.4
04Interpret the answer in context after solving. What does the result mean here? For 31, 33, 35, 37, 39, find MAD.
Answer: mean = 35; MAD = 2.4
05Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 32, 34, 36, 38, 40, find MAD.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: mean = 36; MAD = 2.4
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 33, 35, 37, 39, 41, find MAD. Case B: For 42, 44, 46, 48, 50, find MAD.
Answer: Case A: mean = 37; MAD = 2.4 Case B: mean = 46; MAD = 2.4
07For 34, 36, 38, 40, 42, find MAD.
Answer: mean = 38; MAD = 2.4
08Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 35, 37, 39, 41, 43, find MAD.
Answer: mean = 39; MAD = 2.4
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: For 36, 38, 40, 42, 44, find MAD.
mean = 40; MAD = 2.4
Estimate or predict first, then calculate and decide whether the result is reasonable: For 37, 39, 41, 43, 45, find MAD.
mean = 41; MAD = 2.4
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 38, 40, 42, 44, 46, find MAD.
mean = 42; MAD = 2.4
Interpret the answer in context after solving. What does the result mean here? For 39, 41, 43, 45, 47, find MAD.
mean = 43; MAD = 2.4
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 6 Reasoning Lab
Solve use mean absolute deviation problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 40, 42, 44, 46, 48, find MAD.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: mean = 44; MAD = 2.4
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 41, 43, 45, 47, 49, find MAD. Case B: For 50, 52, 54, 56, 58, find MAD.
Answer: Case A: mean = 45; MAD = 2.4 Case B: mean = 54; MAD = 2.4
03For 42, 44, 46, 48, 50, find MAD.
Answer: mean = 46; MAD = 2.4
04Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 43, 45, 47, 49, 51, find MAD.
Answer: mean = 47; MAD = 2.4
05Explain why a valid method works, then solve: For 44, 46, 48, 50, 52, find MAD.
Answer: mean = 48; MAD = 2.4
06Estimate or predict first, then calculate and decide whether the result is reasonable: For 45, 47, 49, 51, 53, find MAD.
Answer: mean = 49; MAD = 2.4
07Solve and verify the result with a second method, inverse operation, or equivalent representation: For 46, 48, 50, 52, 54, find MAD.
Answer: mean = 50; MAD = 2.4
08Interpret the answer in context after solving. What does the result mean here? For 47, 49, 51, 53, 55, find MAD.
Answer: mean = 51; MAD = 2.4
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 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.