Comparing Two Distributions: MAD-Multiple

Grade 7 Statistics and Probability

In this lesson:

• Compare two data sets placed on a shared number line
• Compute the Mean Absolute Deviation (MAD)
• Express the gap between groups as a multiple of MAD

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Compare two distributions on a shared number line
2. Compute the Mean Absolute Deviation (MAD)
3. Express the difference in means as a MAD-multiple
4. Judge separation from the ratio
5. Connect dot plots to the MAD-multiple

How Different Are These Two Groups?

You already know how to find the mean and read a dot plot.

New question: Does the gap between two groups actually mean something?

"Basketball team mean is 10 cm greater — about twice the typical spread."

This lesson teaches you to produce and justify such statements.

Two Dot Plots, One Shared Axis

Both distributions plotted on the same scale — overlap becomes visible immediately.

Vocabulary for Comparing Two Distributions

Describe Center, Spread, and Overlap

Team A (cm): 162, 165, 165, 168, 170, 172, 175, 178, 180, 185
Team B (cm): 150, 152, 155, 158, 160, 162, 163, 165, 167, 170

• Where does each team cluster?
• Which team is more spread out?
• Do the teams share any heights?

Check-In: Classify the Degree of Overlap

Which statement best describes the Team A and Team B distributions?

A. Complete overlap — the groups are essentially the same.

B. Partial overlap — groups differ, but share a middle region.

C. Noticeable separation — almost no shared territory.

Think first, then we discuss.

MAD Measures Typical Distance from the Mean

Mean Absolute Deviation (MAD): Average distance from the mean.

• Uses every data point (unlike range, which uses only two)
• In the same units as the data — directly comparable to mean differences

MAD Computation Step 1: Find the Mean

Team B data: 150, 152, 155, 158, 160, 162, 163, 165, 167, 170

The mean is our reference point — each deviation is measured from 160.2.

MAD Step 2: Compute Absolute Deviations

| Value | | |
|-------|-------------|---------------|
| 150 | −10.2 | 10.2 |
| 152 | −8.2 | 8.2 |
| 155 | −5.2 | 5.2 |
| 158 | −2.2 | 2.2 |
| 160 | −0.2 | 0.2 |
| 162 | +1.8 | 1.8 |
| 163 | +2.8 | 2.8 |
| 165 | +4.8 | 4.8 |
| 167 | +6.8 | 6.8 |
| 170 | +9.8 | 9.8 |

Sum of raw deviations ≈ 0 → use

MAD Step 3: Average the Absolute Deviations

A typical Team B player is about 5.2 cm from the team mean.

Compute Team A's MAD: Your Turn

Team A: 162, 165, 165, 168, 170, 172, 175, 178, 180, 185

Hint: sum = 60

Team A MAD: Answer Revealed

Deviations: 10, 7, 7, 4, 2, 0, 3, 6, 8, 13 → sum = 60

Check-In: Why Absolute Value Matters

Question: If we averaged the raw deviations (without absolute value), what would we get — and why is that a problem?

Write one sentence explaining why the MAD formula requires absolute value.

A Raw Difference in Means Can Mislead

Same 10 cm gap — very different meanings:

• Spread = 2 cm → dramatic separation
• Spread = 50 cm → barely noticeable

Scale the gap by the typical spread within each group.

Solution: Divide the gap by the MAD.

Step 1: Find the Difference in Means

Team A: 172.0 cm Team B: 160.2 cm

Subtract the smaller mean from the larger — always positive. This is our numerator; we need a denominator to judge whether 11.8 cm is large or small.

Step 2: Choose a MAD as the Ruler

When both MADs are close, any choice gives a similar ratio. We use the average: 5.6 cm.

Step 3: Compute the MAD-Multiple Ratio

The gap between Team A and Team B is about 2.1 times the typical spread within either team — noticeable separation, with partial overlap only near 162–170 cm.

Interpreting the Ratio: Three Zones

Writing a Complete Interpretation Statement

Sentence frame:
"[Group A]'s mean is [X] greater than [Group B]'s — about [ratio] times the MAD. The separation is [adjective]."

Applied:

"Team A's mean height is 11.8 cm greater than Team B's — about 2.1 MADs. The separation is noticeable."

Check-In: Predict from the Ratio

Question: If the MAD-multiple ratio were 0.5 instead of 2.1, what would the dot plot of Team A and Team B look like?

Describe in 1-2 sentences before we discuss.

The CCSS Benchmark: Ratio ≈ 2

The Common Core standard uses this example:

"Basketball team mean is 10 cm greater than soccer team's — about twice the variability (MAD). The separation is noticeable."

At ratio ≈ 2, the dot plot shows mostly distinct groups with only thin boundary overlap.

CCSS Example: Work Backward from Ratio

Given: gap = 10 cm, ratio = 2. What is the MAD?

Each team's MAD ≈ 5 cm — typical heights vary about 5 cm from the team mean. Working backward builds flexibility with the ratio formula.

Contrasting Scenario: Study Hours with Overlap

Difference = 1.1 hr, Average MAD = 1.45 hr

Side-by-Side: Two Ratios, Two Dot Plots

Ratio ≈ 2.1: separation visible — clouds mostly apart.
Ratio ≈ 0.76: heavy overlap — clouds deeply interleaved.

Exit Ticket: Compute and Classify

Plant heights (cm) after 4 weeks:

1. Difference in means?
2. MAD-multiple ratio?
3. Classify the separation.
4. Write one interpretation sentence.

Exit Ticket: Answers and Classification

1. Difference: cm
2. Ratio:
3. Noticeable separation (ratio ≈ 2)
4. "Group P's mean is 6 cm greater — 2 times the MAD. The separation is noticeable: most Group P plants are taller."

Key Takeaways and Misconception Warnings

• Shared axis reveals overlap
• MAD = average deviation from mean
• Ratio = difference ÷ MAD

Watch out:

• Shared values ≠ same groups — look at the bulk
• MAD ≠ range; raw deviations cancel without
• Ratio ~2 allows thin boundary overlap

Coming Up Next: Drawing Inferences

7.SP.B.4 — Informal Comparative Inference

You can now describe how different two distributions are.

Next lesson: use that description to make an inference — a claim about the broader populations the samples came from.

The MAD-multiple you computed today becomes the evidence for tomorrow's comparative inference.