Define Appropriate Quantities

Lesson 1 of 2: Quantification and Standard Measures

In this lesson:

• Understand what quantification means and why quantity choices matter
• Evaluate standard derived quantities for fitness of purpose

Deck 1 Learning Objectives Today

By the end of this lesson:

1. Explain what quantification means — the deliberate act of choosing what to measure
2. Evaluate competing measures for the same attribute, noting what each captures and misses
3. Select an appropriate standard quantity (rate, per-capita, compound) for a given situation

Recall: Rates and Per-Unit Expressions

• A rate compares two quantities with different units: miles per hour, deaths per year
• "Per" means divided by: fatalities per driver = fatalities ÷ drivers
• A unit rate reports the amount per one unit of the denominator
• Rates can use any denominator: per person, per mile, per year

Can you read a rate and name its numerator and denominator?

Three Analysts, One Safety Question

Question: Are U.S. highways getting safer?

• Analyst A looks at total fatalities per year
• Analyst B looks at fatalities per licensed driver per year
• Analyst C looks at fatalities per 100 million vehicle-miles traveled

Same data. Three different answers about whether safety is improving.

One Dataset Produces Three Different Stories

Which line is telling the truth?

Attribute vs. Quantity: A Key Distinction

Attribute: a quality we care about — safety, efficiency, well-being

Quantity: a specific measurable expression — a number with units computable from data

Quantification: choosing which aspect to measure, what units to use, and what the result means

Quantification: Bridging Attribute to Measure

Each quantity answers a different question about highway safety.

Matching Safety Quantity to Purpose

Each quantity is right for one question. None is universally correct.

From Using Units to Choosing Them

Lesson 1 (HSN.Q.A.1): Units were given — your job was to use them correctly.

This lesson (HSN.Q.A.2): Quantities are not given — your job is to choose them.

Choosing what to measure is the first act of any data analysis.

Quick Check: Attribute or Quantity?

Classify each:

1. "Productivity of a factory floor"
2. "35 units produced per worker per day"
3. "The quality of a city's schools"
4. "71% of students scoring proficient on state math tests"

Which items are attributes? Which are quantities? What makes the difference?

Standard Derived Quantities Around Us

These quantities were invented — then adopted because they proved useful.

Person-Hours: Captures Labor, Misses Organization

• 2 workers × 20 hr = 40 person-hours
• 4 workers × 10 hr = 40 person-hours

Same total labor — different organization.

Captures: total labor input
Misses: how work is organized, quality, efficiency per worker

Per-Capita Income: Not the Typical Person

9 residents earn 20,000/yr; 1 earns 1,000,000:

• Total income: 1,180,000
• Per-capita: 1,180,000 / 10 = 118,000

Nobody in this town earns 118,000.

Warning: per-capita = total ÷ count — not a typical value when distributions are skewed.

Misconception: One Right Quantity Exists

Wrong: "There is a correct quantity for highway safety."

True: Different questions require different quantities.

• Physical danger per mile? → fatalities per vehicle-mile
• My annual risk? → fatalities per driver
• Absolute death toll? → total fatalities

"Appropriate" means fit for a specific purpose.

MPG vs. L/100km: Orientation Matters

Same efficiency, opposite orientation. Car B is twice as efficient in both scales — but improvement savings are more linear in L/100km.

Evaluate a Standard Measure for a Purpose

Scenario: A school district wants to compare schools by how well students learn.

Proposed quantity: percent of students scoring "proficient" on state tests

1. What attribute does this measure?
2. What does it capture well?
3. What does it miss?

Apply the capture/miss framework before advancing.

Your Turn: School Library Use Measure

A school wants to track how effectively students are using the library.

Your task: Propose one quantity to measure library use.

• State the quantity name and units
• Explain what it captures
• Name one thing it does not capture

No setup provided — build the full measure yourself.

A Quantity Is a Lens with Blind Spots

Every quantity:

• Shows some things clearly
• Hides other things completely

Choosing the right quantity = choosing the right lens for the question you're asking.

The goal isn't to find the perfect measure — it's to choose deliberately and know what you're missing.

Coming Up: Inventing New Quantities

You can now evaluate and select standard measures.

Next lesson: what happens when no standard measure fits your attribute?

• Design a measure from scratch using a three-step process
• Test whether the measure behaves correctly
• Defend a quantity choice using a four-part framework