Summarizing Numerical Data Sets

Grade 6 Statistics and Probability

In this lesson:

• Describe what a data set measures and how it was collected
• Compute mean, median, range, IQR, and MAD
• Choose the right measure pair based on distribution shape

What You Will Learn Today

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

1. Report the number of observations and describe the attribute, units, and measurement method
2. Compute the mean and median; compute the range, IQR, and MAD
3. Describe overall patterns and striking deviations in context
4. Choose between mean/MAD and median/IQR based on distribution shape

What Story Does This Data Tell?

You already know that data describes real-world situations:

• A dot plot shows the shape of a distribution
• Center tells you where most values cluster
• Spread tells you how much values vary

Today: we go deeper — compute the statistics and interpret them in context.

Describing a Data Set: Four Questions

Before computing anything, answer these four questions:

• How many? — Number of observations ()
• What was measured? — The attribute (e.g., hours of sleep)
• In what units? — Hours, cm, dollars, books
• How was it collected? — Survey, measurement, school records

Describing the Books Data Set

27 sixth graders reported books read over summer.

Context report:

• observations
• Attribute: books read over summer
• Units: books
• Method: self-reported survey

Mean Measures Equal Distribution of Values

The mean is what each person would get if the total were divided equally:

• Add every value; divide by
• Interpretation: "the fair share" — equal portion for each
• Sensitive to outliers — extreme values pull the mean

Computing the Mean: Books Data

But wait — most students read 5 or fewer books. One student read 18.

The outlier (18) pulled the mean upward — 6.4 doesn't feel representative.

Finding the Median: Sort Then Locate

The median is the middle value when data is sorted:

• Odd : exact middle → position
• Even : average the two middle values

For our data: (odd) → 14th value

What the Gap Between Mean and Median Reveals

Key insight: When mean and median are close → symmetric distribution.
When mean > median → right-skewed (or high outlier).

Quick Check: Which Measure Fits?

Scenario A: Heights of 20 students — mean = 62 in, median = 62 in

Scenario B: Monthly allowances — one student gets $500, rest get $10–30

For each scenario: which measure better represents a "typical" value?

Think before the next slide...

Three Tools to Measure Data Spread

We know the center. Now: how spread out are the values?

Three measures of variation:

• Range — the total span
• IQR — the middle 50%
• MAD — average distance from the mean

Each tells a different story about variability.

Range: Maximum Minus Minimum Value

For books data: books

Pro: Simple to compute
Con: One outlier inflates it — range only reports extremes

IQR: Spread of the Middle 50%

The IQR (interquartile range) measures the span of the middle half of the data:

• = median of the lower half
• = median of the upper half
• Resistant to outliers — ignores the top and bottom 25%

Computing IQR Step by Step

Median = 14th value = 5. Split remaining 26 values into halves.

• Lower half (13 values):
• Upper half (13 values):

Middle 50% of students: between 3 and 8 books.

MAD Measures Typical Distance from Mean

The MAD is the average of how far each value is from the mean:

1. Find the mean
2. Compute for each value
3. Average those absolute deviations:

MAD Computation Using a Deviation Table

Data set: {2, 3, 5, 5, 7, 8, 12},

Values in this set typically vary by about 2.6 books from the mean of 6.

Outlier Effect: Comparing All Three Measures

Add an outlier — change 12 to 30 in our data set. What changes?

IQR is resistant. Range and MAD are sensitive to outliers.

Center and Spread: Choosing the Right Pair

We have our center and variation. Now: which pair should we report?

• Symmetric, no outliers → mean + MAD
• Skewed, or has outliers → median + IQR

The choice isn't arbitrary — it's about honest communication.

Two Paths for Choosing Summary Measures

Look at the display first. Then decide.

Writing a Summary for Symmetric Data

Quiz scores (25 students) — shape: roughly symmetric, no outliers → mean + MAD

• Mean = 74 points; MAD ≈ 2.1 points
• Summary: "25 quiz scores. Mean = 74 pts. Roughly symmetric. Values vary ~2 pts from the mean."

Writing a Summary for Skewed Data

House prices (20 homes): mostly $150K–$250K, one at $1.2M → right-skewed

→ Choose median + IQR

• Mean = $342K (pulled by outlier); Median = $198K; IQR = $62K
• Summary: "20 homes. Median = $198K. Right-skewed. Middle 50% within $62K range."

Mean of $342K overstates a typical home.

Your Turn: Complete the Summary

Given: 18 plant heights (cm) measured in a school garden.
Shape: right-skewed (a few very tall plants)

Which measure pair should you report? Write the summary statement.

Plant Heights: Correct Summary Statement

Right-skewed → median + IQR

• Median = 19.5 cm; IQR = 8.2 cm

Summary: "18 plant heights in cm. Right-skewed with tall outliers. Median = 19.5 cm. Middle 50% within 8.2 cm range."

Mean of 24.3 cm overstates a typical plant.

What's Wrong with This Summary?

Error to identify:

A student summarized skewed income data as:
"The mean household income is $87,000 with a MAD of $31,000."

The data has several households earning over $500,000.

What's wrong? What should they have reported instead?

Lesson Summary and Four Common Warnings

✓ Always describe context first: , attribute, units, method

✓ Mean = fair share; Median = middle value — compute both and compare

✓ Range, IQR, MAD each measure variation differently — IQR and MAD are more informative than range

✓ Symmetric/no outliers → mean + MAD; Skewed/outliers → median + IQR

Watch out: For even , average the two middle values — don't just take one

Watch out: MAD requires absolute values — signed deviations always sum to 0

Watch out: Skewed data or outliers → median, not mean

Watch out: Always interpret MAD with units — "values vary by ___ [units] from the mean"

Coming Up Next: Seventh Grade Statistics

You've mastered 6.SP.B.5 — summarizing data in context.

In 7th grade statistics, you'll use these skills to:

• Compare two populations using mean and MAD
• Draw inferences from random samples
• Explore overlapping distributions