Summarizing Data with Center and Variation

Lesson 1 of 1: Statistics and Probability

In this lesson:

• A measure of center summarizes all values with one number
• A measure of variation describes how spread out the values are
• Both measures together give a complete statistical summary

What You Will Learn Today

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

1. Explain the role of a measure of center: it summarizes all values with a single representative number
2. Explain the role of a measure of variation: it describes how spread out values are
3. Recognize that both measures are needed for a complete summary

What One Number Would You Choose?

Your class just finished a test. Scores (out of 20): 12, 14, 15, 15, 16, 17, 18

Someone asks: "How did the class do?" You can only say one number.

• What number would you choose?
• What makes that number a good representative?
• What does it leave out?

What a Measure of Center Does

A measure of center gives a single number that represents the typical value in a data set.

• It collapses the whole distribution into one representative number
• It answers: "What is the typical value?"

Mean and Median as Center Measures

Both summarize the typical value — but handle outliers differently.

Same Mean — Very Different Data

Both data sets have a mean of 5. Are they the same?

Can the Mean Tell the Difference?

Set A: {5, 5, 5, 5} — mean = 5

Set B: {2, 4, 6, 8} — mean = 5

• The mean is 5 for both — it cannot tell these sets apart
• To describe the difference, ask: how spread out are the values?

What a Measure of Variation Does

A measure of variation describes how spread out values are around the center.

• It answers: "How far from the center do values fall?"
• Small variation → values clustered close together (consistent)
• Large variation → values spread widely (less consistent)

Grade 6 Measures of Variation Compared

Calculations come in 6.SP.B.5.

Same Sets Revisited — Now with Variation

Both sets have mean = 5. Watch what the variation measures reveal:

• Set A {5,5,5,5}: range = 0, MAD = 0 → no variation
• Set B {2,4,6,8}: range = 6, MAD = 2 → clear variation

What Does Variation Tell You?

Set A {5, 5, 5, 5}: range = 0, MAD = 0

Set B {2, 4, 6, 8}: range = 6, MAD = 2

• Set A: mean perfectly predicts every value
• Set B: mean is the center, but values differ widely

Real World: Delivery Service Comparison

Two pizza services both average 30 minutes delivery time.

• Service A: typical variation is 2 minutes (28–32 min)
• Service B: typical variation is 22 minutes (10–55 min)
• Same center — very different reliability. Which would you choose?

Neither Measure Alone Is Enough

The complete summary: Center + Variation together

"The data is centered around X, and values typically vary by Y."

• Center alone: "What's typical" — but hides consistency
• Variation alone: "How spread out" — but no anchor point
• Together: a complete picture of the distribution

Practice: Write a Two-Sentence Summary

Scores: 12, 14, 15, 15, 16, 17, 18 — Mean ≈ 15.3, Range = 6

Complete: "The scores are centered around ___, and values vary by about ___."

• Does the range suggest consistent or spread-out performance?
• What does each number add to your description?

Same Mean, Different Variation — Practice

Both classes: mean score = 14

• Class 1: 11, 12, 14, 14, 15, 16, 18 → range = 7, MAD = 1.6
• Class 2: 7, 10, 14, 14, 14, 19, 20 → range = 13, MAD = 3.4

Which class was more consistent? Write a two-sentence summary.

Answers: Center and Variation Together

• Both classes: mean = 14 → same typical score
• Class 1 (range 7, MAD 1.6): scores near 14 → consistent performance
• Class 2 (range 13, MAD 3.4): scores spread widely → uneven performance

✓ Same center — very different stories. Variation reveals the difference.

Common Mistake: The Mean Is the Data

What students sometimes think:

"The mean is 15, so everyone scored 15."

What's actually true:

"The mean is 15" means 15 is the typical value — a summary.

• Individual scores can be very different from the mean
• The mean describes the group, not any specific person

Center and Variation Answer Different Questions

These are different questions — they need different measures.

Which Measure Answers This Question?

"The daily high temperatures last week ranged from 58°F to 82°F. Which measure describes how consistent the temperatures were?"

• A) A measure of center
• B) A measure of variation

Think: which question is being asked — "what is typical?" or "how spread out?"

Choosing the Right Pair of Measures

Distribution shape guides the choice (6.SP.B.5 covers this fully):

Always pair a center measure with a variation measure.

Practice: Identify the Best Summary

30 students tracked books read in a month.
Most read 2–4 books; two students read 18 and 22.

Which pair best summarizes this data?

• A) Mean + MAD
• B) Median + IQR
• C) Either — it doesn't matter

Answers: Median and IQR Win Here

Best choice: Median + IQR (Option B)

• Extreme values (18, 22) pull the mean away from typical students
• Median: not affected by extreme scores
• IQR: focuses on the middle 50%, ignoring extremes

✓ Symmetric → mean + MAD ✓ Skewed / outliers → median + IQR

Summary: Center and Variation Together

✓ Center: one number represents the typical value

✓ Variation: one number describes the spread around the center

✓ Both together give a complete statistical summary

The mean describes the group — not any individual's value

Center and variation answer different questions

Symmetric → mean + MAD; outliers present → median + IQR

Up Next: Computing Measures Formally

Next lesson — 6.SP.B.5: Calculating and choosing measures

• Calculate mean, median, range, MAD, and IQR
• Decide which center fits the distribution shape
• Connect numerical summaries to graphical displays