A Data Set Has a Distribution

6.SP.A.2

In this lesson:

• See why a dot plot reveals what a list hides
• Describe any distribution by center, spread, and shape
• Compare two distributions qualitatively

What You Will Learn Today

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

1. Recognize that answers to a statistical question form a distribution — data plus how often each value occurs
2. Describe a distribution using center (where values cluster), spread (how stretched), and shape (the visual pattern)
3. Identify named shapes: symmetric, skewed-right, and distributions with clusters or gaps
4. Compare two distributions informally on center, spread, and shape without computing

Can You Read This List?

Heights (inches) of 25 sixth-graders:

52, 54, 54, 55, 55, 55, 56, 56, 56, 56, 57, 57, 57, 58, 58, 58, 59, 59, 60, 60, 61, 62, 63, 63, 65

• What is the typical height?
• What is the range of heights?
• Does anything stand out?

The Same Heights Data, Now Plotted

Defining Distribution: Data Plus Pattern

A distribution is the data plus how often each value occurs.

• The list gives you the values
• The dot plot shows how often each one appears
• Together: that pattern is the distribution

Key idea: plotted answers form a structured shape — that shape is what we describe.

Which One Is the Distribution?

Which of these is the distribution of 25 heights?

• A. The sorted list: 52, 54, 54, 55, …
• B. The dot plot showing how many students have each height
• C. The number 57 (the most common value)

Think, then advance for the answer.

Three Features Describe Any Distribution

Any distribution can be described with three features:

1. Center — where do most values pile up?
2. Spread — how far do values stretch?
3. Shape — what does the outline look like?

Three features, three gestures, before any calculation.

Center: Finding the Densest Region

Center is where most values pile up.

• Heights: most cluster around 56–58 inches — that's the center
• Not the axis midpoint — data clusters at 56–58, not at 58

Cover the axis and ask: where are the dots?

Predict: Center of This Data Set

A new dot plot shows test scores.

Axis runs from 0 to 30. Most dots cluster between 6 and 10.

Where is the center?

• A. Around 15 — the middle of the axis
• B. Around 8 — where most dots pile up

Pick A or B before advancing.

Spread: How Far Values Range

Spread is how stretched out the values are.

• Heights range from 52 to 65 inches — sweep left to right
• That span is the spread: wide or narrow

Not a number yet — say "spans 52 to 65" or "moderately wide."

Describe Center and Spread: Check-In

Describe the heights distribution so far:

1. Center: most values cluster around ___
2. Spread: heights range from about ___ to ___; this is a ___ spread

Fill in each blank, then advance.

Shape — Symmetric: Mirror Image Shape

• The left half mirrors the right half
• One peak near the center
• Tapers gradually on both sides

Shape — Skewed: One Side Has a Tail

A skewed distribution has a long tail on one side.

Sibling counts: 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 5, 8

• Most students have 0–1 siblings
• Values 3, 5, 8 form a tail

Which side?

Predict Before Seeing the Answer

Sibling count distribution:

Bulk of data: near 0–1. Long tail: extends to 8 (the right side).

Before seeing the answer — predict:

• A. Right-skewed (tail points right)
• B. Left-skewed (tail points left)

Skew Names the Direction of the Tail

Answer: right-skewed — tail points right.

Skew = direction the tail points (bulk and name are opposite)

Shape — Clusters and Gaps: Two Humps

Some distributions have clusters separated by a gap.

Bus arrivals (minutes after 7:00): 5, 6, 7, 8 … then 18, 19, 20, 21

Putting All Three Features Together

Center: most values cluster around 57 inches

Spread: heights range from 52 to 65 inches — moderately wide

Shape: roughly symmetric, single peak, gradual taper both sides

Three sentences — this is a complete description.

Describe a Distribution: Guided Practice

Math club ages: mostly 11–12, with a few as young as 9 and one aged 15.

1. Center: most ages cluster around ___
2. Spread: ages range from ___ to ___
3. Shape: the distribution appears ___ because ___

Fill in each line, then advance.

Describe This Library Distribution Independently

Books checked out at the library last week:

Values: 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 7

Write three sentences — center, spread, shape.

Write your description, then advance.

Three Features Enable Comparing Two Distributions

Why did we learn center, spread, and shape?

So we can compare any two distributions with three questions:

1. Where is each centered?
2. How spread out is each?
3. What shape does each have?

Comparison is the purpose.

Comparing Distributions: Ask Three Questions

To compare two distributions:

1. Center first: which is centered higher? Lower?
2. Spread second: which is more spread out?
3. Shape third: do they share a shape, or differ?

Two distributions, three questions, three short statements.

Comparing Sixth-Graders and Basketball Players

Sixth-graders (n=25): heights 52–65 in, clustered near 57

Varsity basketball (n=9): heights 72, 73, 74, 75, 76, 76, 77, 78, 80

Sibling Counts vs. Books Read Summer

Comparing sibling counts and books-read distributions:

• Center: siblings peak near 1; books peak near 5–6 — books higher
• Spread: siblings span 0–8; books span 1–12 — similar
• Shape: both right-skewed — same shape, different centers

Which Distribution Is More Spread Out?

Two dot plots:

• Plot A spans from 10 to 40
• Plot B spans from 15 to 25

Which is more spread out?

Think, then advance.

Find and Fix the Skew Error

A student wrote: "The sibling count distribution is left-skewed because most of the data is on the left side."

What's wrong? Write the correct statement.

• Hint: where does the long tail point?

Compare Two Quiz-Score Classes Guided

Class A: scores 60–95, most around 80, roughly symmetric

Class B: scores 50–95, most around 60, tail extending right

Complete the comparison:

1. Center: Class ___ is higher
2. Spread: Class ___ is wider
3. Shape: A is ___; B is ___

Fill in each blank, then advance.

Write Your Own Comparison Sentences

Set 1: 3, 3, 4, 4, 5, 5, 5, 6, 6, 7

Set 2: 1, 1, 2, 2, 2, 3, 3, 4, 8, 12

Write three sentences — center, spread, shape.

Your own words, then advance.

Describe and Compare: Unscaffolded Practice

Books read this summer: 1, 2, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9

1. Write a complete 3-feature description
2. Compare to the heights distribution (center ≈ 57, spread 52–65, symmetric)

No fill-in-the-blank — your own sentences.

What You Learned: Key Takeaways

✓ Distribution = data + how often each value occurs

✓ Three features: center (where), spread (wide?), shape (outline)

✓ Compare two distributions: ask center → spread → shape

Center ≠ axis midpoint; right-skewed = tail right

Spread is qualitative now — numbers come next lesson

Next Lesson Puts Numbers on These Features

You can now describe any distribution qualitatively.

Next lesson: attach numbers to those features.

• Center → mean and median
• Spread → range (and later, IQR and MAD)

The three features you named today are exactly what those numbers measure.