Inventing and Defending Quantities

Lesson 2 of 2: Designing and Evaluating Measures

In this lesson:

• Design a measure for an attribute that has no standard unit
• Defend a quantity choice using a structured four-part framework

Deck 2 Learning Objectives Today

By the end of this lesson:

1. Design a novel measure for an attribute without a standard unit, checking it for dimensional sense and appropriate behavior
2. Defend a quantity choice by stating what it measures, its units, its scope, and its blind spots

When No Standard Measure Exists

Problem: A scientist suspects surface area affects how quickly a liquid evaporates.

She wants to test this experimentally.

What should she measure?

Surface area exists as a unit (cm²) — but is that the right quantity for this study?

Step 1: Name the Relevant Attribute

What exactly are we trying to capture?

Not "surface area" in general — but surface area exposure: how much of the liquid's surface is exposed to air relative to the liquid's volume.

The attribute is more specific than "surface area alone."

Naming the attribute precisely is Step 1.

Step 2: Express the Attribute as a Ratio

The units follow from the design — they don't need to be a pre-existing named unit.

Step 3: Test the Measure at Boundaries

Double the surface area, keep volume constant — the measure should double:

✓

If volume increases while area stays fixed → ratio decreases ✓

A good measure varies proportionally with the attribute it claims to measure.

Three Proposals for School Building Use

Which is most fit for purpose — and for which question?

What Makes a Measure Well-Designed?

A well-designed measure is:

1. Dimensionally sensible — the units make mathematical sense
2. Proportional — doubling the attribute roughly doubles the measure
3. Practically computable — can be calculated from data you can actually collect

A measure that fails any of these three should be revised or rejected.

Novel Units Are Invented — Not Found

Wrong: "A quantity is only valid if it uses a recognized unit name."

True: Widely-used units were all invented to fill a need:

• Kilowatt-hours, person-hours, parts per million, points per game

Define a new unit freely — then check it behaves correctly.

Quick Check: Does This Measure Pass?

Measure: books checked out per week

1. If library visits double, does this measure double?
2. What does it miss?
3. How would you modify it to fix the blind spot?

Answer all three before advancing.

Design a Measure for Room Usage Efficiency

Available data: floor plan (room areas), weekly schedule, enrollment by period

Propose one quantity:

• State the formula and units
• Check it against the three-question test

Combine two or three of the available data sources.

Design a Gym Membership Measure

A gym wants to measure how effectively members use their membership.

Design a measure from scratch:

1. Name the attribute precisely
2. Express it as a ratio with units
3. Test it at one boundary condition

No example provided — build the complete measure yourself.

Designing vs. Defending: The Next Step

You can now design a quantity from scratch.

The next step: when multiple quantities exist for the same attribute, how do you defend your choice?

Design answers "what should we measure?" — defense answers "why this one and not the others?"

Housing Affordability: Three Competing Measures

Which quantity should a household use to evaluate a city?

Four Questions for Evaluating Any Quantity

1. What attribute does it measure?
2. What are its units (and how is it computed)?
3. What scope — which population, which time window?
4. What blind spots — what does it not capture?

Apply this to any quantity you use or encounter.

Apply Four Questions to Median Home Price

Apply the framework to: Median Home Price as a housing affordability measure

1. What attribute does it measure?
2. What are its units?
3. What population and time window does it cover?
4. What does it not capture?

Write all four responses before advancing.

A Quantity Is a Position Worth Defending

Defining a quantity is taking a position:

"This is what matters about this situation, measured this way."

A well-defined quantity is:

• Transparent about what it captures
• Honest about what it misses

Choosing deliberately and justifying the choice is what rigorous quantitative reasoning looks like.

Coming Up: Accuracy and Precision

You can now choose, design, and defend appropriate quantities.

The next question (HSN.Q.A.3): once you've defined your quantity, how precisely should you report it?

The right level of precision depends on the measurement context — not on the number of digits available.