Understanding Data Distributions

Center, Spread, and Shape

In this lesson:

• Describe data using center, spread, and shape
• Read distributions from dot plots and histograms
• Connect distributional features to real-world context

What You Will Learn Today

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

1. Describe a distribution using center, spread, and overall shape
2. Identify center, spread, and shape from a dot plot or histogram
3. Explain what each feature reveals about the real-world context

What Can a Test Score Graph Tell Us?

Your class just took a math test. Here are the scores:

55, 60, 65, 68, 70, 72, 72, 75, 75, 75, 78, 80, 82, 85, 90

Look at this data. What can you say about how the class did?

Think for a moment before we continue...

Three Features Describe Every Distribution

Every distribution can be described using three features:

• Center — the typical value; where data tends to cluster
• Spread — how far apart the values are from each other
• Shape — the overall visual pattern of the data

Annotated Dot Plot: Test Scores

Center ≈ 75 | Spread: 55 to 90 | Shape: roughly symmetric

Center — The Typical Value

The center tells us the typical, or middle-of-the-road, data value.

• Ask: "Where do most values cluster?"
• Informal description: "Most scores were around 75"
• Center is the balance point — not the highest dot

Exact measures (mean, median) come in 6.SP.A.3 — for now, estimate visually

Quick Check: Finding the Center

Here is a dot plot showing hours of sleep for 12 students:

Data: 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10

Where is the center of this distribution?

Estimate from the dot plot — where do values tend to cluster?

Spread — Width and Concentration

The spread tells us how varied the data values are.

• Ask: "How far apart are the values? Are they packed or scattered?"
• Full description: "Scores ranged from 55 to 90, with most between 70 and 85"
• Just stating min and max misses where most values fall

Shape — The Visual Pattern

Shape Gallery: Four Distribution Patterns

Shape tells you about real-world structure — not just visual aesthetics

Reading a Histogram for Distribution

A histogram groups data into intervals (bins). Read it like a dot plot:

• Center ≈ peak bar or group of tallest bars
• Spread = range from leftmost to rightmost occupied bin
• Shape = overall pattern of bar heights

Formal histogram construction is covered in 6.SP.B.4

Histogram: Hours of Sleep per Night

• Center: approximately 7–8 hours
• Spread: from about 4 to 11 hours
• Shape: roughly symmetric, slight right skew

Two Classes, Same Mean — Different Spread

Quick Check: Shape and Real-World Context

A dot plot of household incomes in a neighborhood shows a right-skewed distribution.

What does this shape tell you about the neighborhood?

Think: what does a long right tail mean for incomes?

Your Turn: Describe Three Distributions

For each data set, describe center, spread, and shape.

1. Plant heights (cm): 12, 13, 14, 15, 15, 15, 16, 16, 17, 18
2. Commute times (min): 5, 8, 10, 12, 15, 20, 25, 30, 45, 60
3. Test scores histogram: peak 80–90, tail extending left to 40

Three Distribution Descriptions: Sample Answers

Display 1: Center ≈ 15 cm; spread 12–18 cm, clustered; shape symmetric.

Display 2: Center ≈ 12–15 min; spread 5–60 min, most under 25; shape right-skewed.

Display 3: Center ≈ 80–85; spread 40–100; shape left-skewed.

Three Features Work Together Always

✓ Center = typical value (balance point)
✓ Spread = how varied; note concentration, not just range
✓ Shape = overall pattern; reveals real-world structure
✓ All three together tell the full story

Center is the balance point — not the highest dot
Report where most values fall, not only min and max
Shape always has meaning — never skip it

Coming Up Next: Measuring Precisely

Next: 6.SP.A.3 — Measuring Center and Spread

• Today: described center informally ("around 75")
• Next: calculate mean and median for center
• Next: calculate range and mean absolute deviation for spread