How to explain it
At this standard, students compare two numerical data distributions by describing the degree of visual overlap and by expressing the center difference as a multiple of the MAD.
The anchor students hold onto: To compare distributions: find the center gap, then divide by the MAD. A ratio of 1 or less means much overlap. A ratio of 2 or more means the groups are clearly separated.
Expressing center differences as MAD multiples leads to 7.SP.B.4, where students use center and variability measures to draw formal comparative inferences about two populations.
Worked examples
Example 1
Describe Visual Overlap
A: 4,5,7,9,10 B: 5,6,8,10,11
Step 1A range: 4 to 10. B range: 5 to 11.
Step 2Shared range: 5 to 10 — both sets have values here.
Step 3Overlap level: much overlap (centers only 1 apart).
AnswerMuch overlap — A (4-10) and B (5-11) share nearly the same range
Example 2
Find the MAD
A: 4, 5, 7, 9, 10 (mean = 7)
Step 1Deviations: |4-7|=3, |5-7|=2, |7-7|=0, |9-7|=2, |10-7|=3.
Step 2Sum of deviations = 3+2+0+2+3 = 10.
Step 3MAD = 10/5 = 2.
AnswerMAD = 10/5 = 2
Example 3
Express Gap in MADs
A mean=7, B mean=13, MAD=2.
Step 1Center gap = 13 - 7 = 6.
Step 2MADs apart = 6 / 2 = 3.
Step 3The centers are 3 MADs apart — little overlap.
Answer6 / 2 = 3 MADs apart; little visual overlap
Common mistakes
What students write
A large center gap alone proves the two groups are clearly different.
The fix
The center gap must be measured in MAD units. A gap of 10 points means little if MAD is also 10, but signals clear separation if MAD is only 2. Always divide the center gap by the MAD.
Try this
Two teams competed in a math relay. Team Blue scores: 10, 11, 13, 15, 16. Team Gold scores: 4, 5, 7, 9, 10. Rivera calculates the center gap and wants to express it as a multiple of a measure of variability. Rivera writes: "Team Blue mean = 13. Team Gold mean = 7. Center gap = 6. The range of Team Gold is 10-4 = 6, so the gap is 6 divided by 6 = 1 range unit. The teams scored about the same." Identify and correct Rivera's error.
What students write
Two distributions that share some range are automatically similar groups.
The fix
Distributions can overlap visually yet still have very different centers. Always report BOTH the visual overlap description AND the MAD ratio for a complete comparison.
Try this
Two classes recorded hours of weekly screen time. Class P: mean=24 hours, MAD=6 hours. Class Q: mean=30 hours, MAD=6 hours. Tran writes: "The center gap is 30-24 = 6 hours. Six is a fairly big number, so Class Q has significantly more screen time than Class P - the distributions are clearly different." Identify and correct Tran's error.
Teacher tip
Head off the two predictable errors before they happen. First: The center gap must be measured in MAD units. A gap of 10 points means little if MAD is also 10, but signals clear separation if MAD is only 2. Always divide the center gap by the MAD. Second: Distributions can overlap visually yet still have very different centers. Always report BOTH the visual overlap description AND the MAD ratio for a complete comparison.