1.3 Four levels of measurement

Four levels of measurement

Why do we care?

The level dictates what we can do with the data.

Nominal (or categorical) data

When numbers are not numbers

  • They act like names or labels.
  • Can not be put into order.

Examples

  • Country code: 1 (USA), 44 (UK), 81 (Japan), …
  • Social security numbers
  • Your UMID
  • Marital status: single, married, divorced, …

Tokyo Olympics men’s marathon

What information are nominal data?

Nominal (or categorical) data

Implications

  • They can not be compared (e.g., \(>, <\))
  • We can not carry out any calculations (\(+, −, ×, \div\))
  • The only math operations that can be done are \(=, \neq\)

Ordinal data

When numbers are almost numbers

  • They go into order.
  • But that’s about it.
  • They don’t tell us about the distance between two numbers.
  • Airline cabin classes
    • 1: first class, 2: business, 3: premium economy, 4: economy
  • Hotel ratings (1-5 stars)

Tokyo Olympics men’s marathon

What information are ordinal data?

Ordinal data

Implications

  • Ordinal numbers can be compared (e.g., \(>, <\))
  • However, we can not carry out calculations for differences or magnitudes (\(−, \div\))

Interval & ratio data

When numbers are really numbers

  • Not only can be put to order
  • The distance between them tell us something.
  • Tom drinks 3 cups of coffee. Kate drinks 1 cup.

Ratio data

Ratio data have a genuine1, non-arbitrary zero point.

  • I have zero dollars.
    • absence of money
  • I walk at a speed of 0 mph
    • no movement

Examples of ratio data

  • Length (100 meters)
  • Weight (50 lbs)
  • Size (800 square feet)
  • Speed (20 mph)
  • Time duration (5 hours)

The majority of scientific measures are ratio measures.

Interval data

Interval data have an arbitrary zero point.

Zero does not mean there is none of the things being measured.

  • What is the temperature for 0°C?
    • 0°C does not mean there is no warmth.
  • Calendar year 2025. What does year 0 mean?
    • Year 0 does not mean it’s the beginning of time.
  • What is longitude (geographic coordinate) of 0°?

Examples of interval data

  • Temperature in Fahrenheit (70°F) or Celsius (20°C)
  • Calendar year
  • GPS coordinates (42.3185, -83.2327)
  • GPS heading (90°)

Implications without a genuine zero

We can talk about ratios with ratio measures (duh!)

  • Tom drinks ☕☕☕. Kate drinks ☕.
    • Tom drink ☕☕ more coffee than Kate.
    • Tom drink 3 times as much coffee as Kate.
  • Today is 40°F. Yesterday was 20°F.
    • Today is 20°F warmer than yesterday.
    • The average temp of the two days is 30°F.
    • Today is twice as warm as yesterday.

Implications without a genuine zero

  • With interval data
    • we can calculate the difference (\(-\)).
    • we can not calculate the ratio (\(\div\)).
  • With ratio data we can carry out most calculations.

Tokyo Olympics men’s marathon

What information are ratio data?

Qualitative and quantitative data