How to explain it
At this standard, students use measures of center and variability from random sample data to draw informal comparative inferences about two populations, using precise inference language.
The anchor students hold onto: Find center and spread for each sample. Express the center gap in MAD units. Then use both statistics together to draw an inference about which population is typically higher.
Informal comparative inference with center and variability prepares students for 8.SP.A.1, where scatter plots and lines of best fit support predictions about bivariate data.
Worked examples
Example 1
MAD Ratio + Inference
mean A=20, B=26, MAD=3 each
Step 1Center gap = 26 − 20 = 6.
Step 2MAD ratio = 6 / 3 = 2 MADs apart → little overlap.
Step 3Population B is typically higher (based on samples).
Answer6/3 = 2 MADs apart; little overlap; Population B is typically higher
Example 2
Write Full Inference
A mean=40, B mean=52; 3 MADs
Step 13 MADs apart: clearly different; little to no overlap.
Step 2Population B has the higher center (mean=52 vs. 40).
Step 3"Based on samples, Population B is typically higher than A."
AnswerPopulation B is typically higher (3 MADs; clearly different)
Common mistakes
What students write
The population with the higher sample mean is always higher than the other population in every case.
The fix
An informal comparative inference says one population is TYPICALLY higher — not always. Overlapping distributions mean some members of the lower-center group may still score above members of the higher-center group. Use "typically" or "tends to be," never "always."
Try this
Two 7th-grade classes each had 30 randomly selected students take a math assessment. Class A: mean=82 pts, MAD=10 pts. Class B: mean=75 pts, MAD=10 pts. Rivera writes: "Class A has a mean of 82 and Class B has a mean of 75. Class A is 7 points higher. Therefore, Class A students are always better at math than Class B students." Identify and correct Rivera's two errors.
What students write
Comparing means alone is enough to draw a valid comparative inference.
The fix
Both center AND variability are required. A center gap of 6 means very different things when MAD=2 (3 MADs — clearly different) vs. MAD=12 (0.5 MADs — much overlap). Always express the center gap in MAD units before drawing an inference.
Try this
A researcher compared two groups on a science test using random samples. Group 1: mean=65 pts, MAD=3 pts. Group 2: mean=58 pts, MAD=8 pts. Tran writes: "Group 2 has a larger MAD (8 > 3), so Group 2's scores are typically higher than Group 1's scores." Identify Tran's error and write the correct inference.
Teacher tip
Head off the two predictable errors before they happen. First: An informal comparative inference says one population is TYPICALLY higher — not always. Overlapping distributions mean some members of the lower-center group may still score above members of the higher-center group. Use "typically" or "tends to be," never "always." Second: Both center AND variability are required. A center gap of 6 means very different things when MAD=2 (3 MADs — clearly different) vs. MAD=12 (0.5 MADs — much overlap). Always express the center gap in MAD units before drawing an inference.