Summarizing Data Sets in Context

Grade 6 Statistics | 6.SP.B.5

In this lesson:

• Compute IQR and MAD as measures of variability
• Choose the right pair of measures based on data shape
• Write a complete statistical summary

What You Will Learn Today

By the end of this lesson, you will:

1. Report n, attribute, method, and units
2. Compute and interpret the IQR
3. Compute and interpret the MAD
4. Describe overall pattern and striking deviations
5. Choose mean + MAD vs. median + IQR by shape
6. Write a complete summary paragraph

What Does This Tell You?

"The average is 58.2."

• Average of what?
• Based on how many measurements?
• In what units?

A summary that can't be answered is not a summary.

Two Summaries: One Useful, One Not

Useless: "The average is 58.2."

Complete: "We measured the heights of 25 sixth graders in our school, in inches, using a tape measure to the nearest inch. The mean height was 58.2 inches."

What changed? What made the second one useful?

n, Attribute, and Units Defined

• n — the number of observations (n = 25 for heights)
• Attribute — what was measured, how, and with what instrument
• Units — inches, pounds, seconds, dollars — always required

Without these, no number in the summary means anything.

Worked Example: Completing a Summary

Sloppy: "The average price was $12."

Complete: "We recorded the prices of 18 lunch orders at the school cafeteria, in US dollars, using the point-of-sale register. The mean price was $12.00."

• n = 18 ✓
• Attribute: lunch order price, measured by register ✓
• Units: US dollars ✓

What Is This Summary Missing?

A student writes: "The average number of books read was 4.3."

What are the three things this summary is missing?

Think about it before the next slide.

Check Answer: Three Missing Pieces

• n — How many students?
• Attribute — Books read when? How long? Which books?
• Units — self-reported or verified?

Improved: "We surveyed 32 students about books finished in the past month, by self-report. The mean was 4.3 books."

Metadata Named — Now Measure Spread

• You can write the opening sentence of a summary
• Next: describe how spread out the data is
• Mean and median describe the center — you know those
• We need two measures of variability

Coming up: IQR and MAD

IQR: Spread of the Middle Half

• IQR = Q3 − Q1: the span from Q1 to Q3 — the middle 50%
• Divides data into four equal quarters
• For our heights: , , so IQR = 4.5 inches

Why IQR Is Better Than Range

• Range goes from 13 to 23 — nearly doubles when a 75" student is added
• IQR stays close to 4.5 — barely changes
• IQR is resistant to extreme values; range is not

Worked Example: Compute IQR Step by Step

Data (n = 7): 2, 4, 5, 7, 9, 10, 12

Step 1: Median = position 4 → 7

Step 2: Lower half: 2, 4, 5 → Q1 = 4

Step 3: Upper half: 9, 10, 12 → Q3 = 10

Your Turn: Compute the IQR

Data (n = 6): 3, 5, 5, 6, 7, 9

1. Find the median
2. Find Q1 and Q3
3. Compute IQR

Try it, then advance for the answer.

IQR Worked Out: Splitting Even Data

• Median = (5 + 6) ÷ 2 = 5.5
• Lower half: 3, 5, 5 → Q1 = 5
• Upper half: 6, 7, 9 → Q3 = 7
• IQR = 7 − 5 = 2

Even n: split directly in half; no value excluded.

Why Signed Deviations Always Sum to Zero

Data: 4, 7, 5, 3, 6 — Mean = 5

Why We Need Absolute Values

• Signed deviations always sum to zero — their average is always 0
• Zero says nothing about how spread out data is

Fix: Take the absolute value of each deviation.

• Turns distances positive — cancellation stops
• The average is now meaningful distance from the mean

Computing MAD Step by Step

• Data: 4, 7, 5, 3, 6 (mean = 5); absolute deviations: 1, 2, 0, 2, 1
• Sum of absolute deviations = 6
• units

MAD for the Heights Data

Heights (n = 25), mean = 58.2 in — same four steps:

1. Subtract 58.2 from each value
2. Take absolute values
3. Sum → divide by 25

MAD ≈ 2.4 inches — a typical height is 2.4 in from the mean

Your Turn: Computing MAD from Scratch

Data (n = 3): 6, 8, 10 — Mean = 8

1. Compute signed deviations
2. Verify their sum = 0
3. Take absolute values
4. Compute MAD

Try it, then advance.

Predict: What Happens Without Absolute Value?

A student computes MAD for 6, 8, 10 but forgets the absolute value bars.

What answer do they get?

• A) MAD = 0
• B) MAD = 2
• C) Between 0 and 2

Commit to A, B, or C before advancing.

Skipping Absolute Value Always Gives Zero

Without absolute value: −2, 0, +2

Answer: A — MAD = 0

• Zero means "no spread" — clearly wrong
• Sign cancellation is always exact — by definition of the mean
• Absolute values are never optional

Describing Pattern and Striking Deviations

You now have IQR and MAD — next, describe the data qualitatively:

• Overall pattern — shape (symmetric / skewed), center, typical spread
• Striking deviations — values that stand clearly apart from the bulk

Identify outliers by inspection at grade 6 — "does it look separated?" is enough.

Heights Data: Pattern and Deviation

Overall pattern:

• Roughly symmetric, peak around 56–58 inches
• Most values within 4–5 inches of center

Striking deviation:

• 65 inches — ~3 inches above next-tallest (62, 63)
• Real data; on the upper edge of typical for the grade

Homework Hours Data: Pattern and Deviation

Data: 0, 0.5, 0.5, 1, 1, 1, 1, 1.5, 1.5, 2, 2, 6 (n=12)

Overall pattern:

• Right-skewed; peak around 1 hour; long right tail

Striking deviation:

• Value 6 — next highest is 2 (gap: 4 hours)
• Real data, but very unusual; may reflect a heavy assignment day

Context Determines What to Do With Outliers

Describe the deviation — never silently remove it.

Both Data Sets Described — Now Choose

Which pair fits each data set — and why?

Shape Determines Which Measures to Report

These are heuristics — judgment, not formula. Always explain your choice.

Apply the Rule: Two Data Sets

Heights (symmetric):

• Mean 58.2 ≈ median 57 — use mean + MAD

Homework hours (right-skewed):

• Mean 1.46 pulled up by 6-hour value; median = 1.0
• Use median + IQR — resistant to the extreme

Context Layer: When to Use Mean Anyway

• Typical individual? → Median
• Total amount? → Mean (even with skewed data)

Homework hours: mean × 12 ≈ 17.5 hrs total; median × 12 = 12 hrs — not the actual total

Complete Summary: Heights Data (Symmetric)

"Heights of 25 sixth graders, in inches, tape measure. Roughly symmetric, peak 56–58. Mean = 58.2 in; MAD ≈ 2.4 in. Value of 65 sits slightly apart — upper edge of typical."

n ✓ attribute ✓ method ✓ shape ✓ center ✓ variation ✓ deviation ✓

Complete Summary: Homework Hours (Skewed)

"12 students, homework hours, nearest half-hour. Right-skewed — most 0.5–2 hrs, one reported 6. Median = 1 hr; IQR = 1 hr. The 6-hour value may reflect a heavy assignment day."

n ✓ attribute ✓ method ✓ shape ✓ center ✓ variation ✓ deviation ✓

Your Turn: Write a Full Summary

Bakery order costs (dollars), n = 10:
3, 5, 5, 6, 7, 8, 9, 10, 12, 35

1. Compute median, Q1, Q3, IQR
2. Describe shape and deviations
3. Choose measures — give a reason
4. Write the complete summary

No scaffolding — this is the real test.

Bakery Exit Task: Worked Answer

1. Median = $7.50; Q1 = $5; Q3 = $9.50; IQR = $4.50

2. Right-skewed; $35 far above next highest ($12)

3. Median + IQR — $35 distorts the mean

4. "10 bakery orders, dollars. Right-skewed — most $3–$12, one $35. Median $7.50; IQR $4.50. The $35 order may reflect a group purchase."

What You Have Learned Today

• Lead with n, attribute, units — required for any summary
• IQR = Q3 − Q1: middle 50%; resistant to extremes
• MAD: average |value − mean|; absolute value required
• Symmetric: mean + MAD; skewed: median + IQR
• IQR ≠ range; omit abs value: MAD = 0

Comparing Two Distributions in Grade Seven

• Grade 7: compare two distributions side by side
• Which class scored more consistently?
• Which neighborhood has the more typical commute?

The measures you choose here (7.SP.B.3) determine how you'll compare.