Scales of Measurement

Measurement Basics

• Definition: Assigning numbers/symbols to characteristics of objects according to rules (Stevens, 1946).

• Scale: A set of numbers/symbols that model properties of what’s being measured.

• Sample space: All possible values a variable can take.

  • Example: gender = {male, female, nonbinary}, age = {0, 1, 2, …}, height = [0, +∞].

Discrete scale

• Countable values (e.g., year in high school = {freshman, sophomore, junior, senior})

• No in-between values (e.g., cannot have 2.5 hospitalizations).

Continuous scale

• Any real number within range (fractions, decimals, even irrational numbers).

• Requires rounding → should match precision of instrument (e.g., grams vs kilograms)

Measurement Error

• Scientific “error”: Not necessarily a mistake → refers to all factors influencing a score beyond what’s being measured.

Examples of error sources

• Environmental distractions (e.g., thunderstorm)

• Test item selection

• Instrument precision limits (e.g., ruler showing 35.5 in vs true 35.484 in)

Continuous scales always involve error

because continuous scale scores are approximations, not exact.

• Example: anxiety test score of 25 may represent real score like 24.7 or 25.4.

Levels of Measurement (NOIR – mnemonic)

• Nominal (N): Categories only; no order (e.g., gender, blood type).

• Ordinal (O): Ordered categories; no equal intervals (e.g., ranking, class standing).

• Interval (I): Ordered with equal intervals; no true zero (e.g., temperature in °C).

• Ratio (R): Ordered, equal intervals, true zero → allows all math operations (e.g., height, weight, reaction time).

Nominal (N)

Categories only; no order (e.g., gender, blood type).

Ordinal (O)

Ordered categories; no equal intervals (e.g., ranking, class standing)

Interval (I)

Ordered with equal intervals; no true zero (e.g., temperature in °C)

Ratio (R)

Ordered, equal intervals, true zero → allows all math operations (e.g., height, weight, reaction time).

What is the simplest form of measurement scale?

Nominal scale is the simplest for of measurement scale

What is required of categories in a nominal scale?

The required categories must be mutually exclusive and exhaustive

Do nominal scales have an inherent order?

No, there is no inherent order among categories.

examples of nominal variables

College major, gender, race, place of birth, phone numbers, zip codes, DSM diagnostic codes

Why are DSM diagnostic numbers (e.g., 303.00 for alcohol intoxication) considered nominal?

Because they are classification codes, not quantities—cannot be added, subtracted, or averaged

What mathematical operations are possible with nominal data?

Checking equality (=) or inequality (≠), counting cases, and determining proportions/percentages.

Example of a nominal yes/no test item?

“Have you ever been convicted of a felony?” (yes/no → felon vs non-felon).

Difference of Ordinal from Nominal

• Categories have an order (Nominal has none).

• Examples:

  • Frequency responses: {Never, Sometimes, Often}

  • Agreement scales: {Strongly disagree → Strongly agree}

  • Job applicants ranked by desirability

  • Clients ranked by need for therapy

  • High school year: Senior > Junior > Sophomore > Freshman

  • Rokeach Value Survey: rank values (freedom, happiness, wisdom, etc.)

Properties of Ordinal

• Allows comparison with relational operators (<, ≤, >, ≥)

• Ranks, but no info about actual differences between ranks

• Distances between ranks are unequal/unknown (e.g., gap between 1st and 2nd ≠ gap between 2nd and 3rd)

• No absolute zero (e.g., nobody has “zero ability”).

Statistical Limitation of ordinal scale

Can rank order but cannot meaningfully average scores/ranks

Give an example of ordinal scale measurement.

Likert responses (e.g., strongly disagree → strongly agree), job applicant rankings, or high school year (Senior > Junior > Sophomore > Freshman).

What did Alfred Binet say about intelligence testing?

Intelligence tests provide classification and ranking(ordinal), not actual measurement of ability.

What operations are possible with ordinal data?

Relational operators like <, ≤, >, ≥ (to compare ranks).

Why can’t ordinal data be averaged?

Because the distances between ranks are unequal/unknown and there’s no true zero point.

Does an ordinal scale have an absolute zero?

No, zero has no meaning in ordinal scales (e.g., no one has “zero ability”).

What survey uses ordinal ranking of values?

The Rokeach Value Survey, where test-takers rank personal values from most to least important.

Interval Scales

• Has all features of nominal (categories) and ordinal (ordered ranks).

• Adds equal and meaningful distances between numbers.

• Allows addition and subtraction, calculation of means and standard deviations.

• No absolute zero (zero ≠ absence of the trait).

Key Example

Temperature (°C, °F)

• 0°C does not mean “no heat” → just the freezing point of water.

• Differences are meaningful (e.g., 30°C - 20°C = 10°C).

• Cannot say 200°C is “twice as hot” as 100°C.

What scale is Kelvin scale

(Ratio Scale)

• 0 K = absence of heat → has an absolute zero.

• 200 K is twice as hot as 100 K (true ratio comparison possible).

Interval scale in Psychological Testing

• Individual test items (e.g., Likert) are ordinal.

• Combined total scores (e.g., IQ) treated as interval.

• Example: IQ 80 → 100 is assumed same “distance” as 100 → 120.

• But no absolute zero IQ → cannot compare as ratios (IQ 100 ≠ “twice as intelligent” as IQ 50)

Statistical Limitations of Interval Scale

• Cannot use ratios, proportions, or percentages.

• Division is meaningless for interval scales.

• Example: IQ 110 is not “10% higher” than IQ 100.

What makes interval scales different from nominal and ordinal scales?

They have equal and meaningful distances between numbers, allowing addition and subtraction.

Do interval scales have an absolute zero?

No. Zero does not mean absence of the trait (e.g., 0°C ≠ no heat).

Why can’t 200°C be described as “twice as hot” as 100°C?

Because interval scales lack an absolute zero, so ratio comparisons are meaningless.

Give examples of interval scales besides temperature

Calendar years, piano notes, color hues

Why are IQ scores treated as interval scales?

Because the differences between scores are considered equal, allowing calculation of averages and standard deviations.

Why can’t IQ scores be expressed as ratios (e.g., IQ 100 is twice IQ 50)?

Because there’s no true zero point for intelligence.

Ratio Scales

• Includes all properties of nominal, ordinal, and interval scales.

• Has a true absolute zero, meaning the complete absence of the measured trait.

• Allows all mathematical operations (addition, subtraction, multiplication, division).

• Values represent true magnitudes → can compare with ratios and proportions.

Key features of interval scale

• Zero = absence of the trait (e.g., 0 siblings = no siblings).

• Ratios are meaningful (e.g., 20 kg is twice as heavy as 10 kg).

• Negative numbers:

  • Nonsense for countable things (e.g., −3 siblings is meaningless).

  • Possible for variables like money (e.g., −$10 = debt).

Examples of Interval Scale

• veryday Life: weight, height, income, age, distance, reaction time.

• Psychology:

  • Neurological tests: hand grip strength (measured in pressure units).

  • Timed tasks: puzzle completion time (0 seconds = absolute zero).

Limitation / Special Note of interval scale

• True zero exists in theory, but sometimes unattainable in practice.

• Example: Completing a puzzle in 0 seconds is impossible, even if zero is meaningful as a reference.

What makes ratio scales unique compared to interval scales?

They have a true absolute zero, allowing ratios and proportions to be meaningful

What does a “true zero” mean?

It represents the complete absence of the measured trait (e.g., 0 siblings = no siblings).

Can ratio scales use all mathematical operations?

Yes—addition, subtraction, multiplication, and division.

Give examples of ratio variables in everyday life.

Weight, height, age, income, distance, reaction time.

Give examples of ratio variables in psychology.

Hand grip strength tests, timed puzzle completion tasks.

Why are ratios meaningful in ratio scales but not in interval scales?

Because ratio scales have an absolute zero, so “twice as much” or “10% larger” comparisons are valid.

Can negative values exist in ratio scales?

Sometimes—nonsense for countable traits (e.g., −3 siblings), but possible for things like money (e.g., debt = negative balance).

Why is a score of “0 seconds” in a timed test theoretical?

Because no one can complete a task in literally 0 seconds, though the scale conceptually allows it.

Most frequently scale used in psychology

Ordinal measurement is most frequently used in psychology

What scales are Intelligence, aptitude, and personality test scores

Intelligence, aptitude, and personality test scores are basically ordinal (Kerlinger, 1973)

Psychological and educational scales

may approximate interval equality, but not perfectly

If ordinal data are treated as interval

users must be cautious of unequal intervals.

Psychologists prefer treating data as interval

for greater flexibility in statistical manipulation

Interval/ratio data allow for techniques like

techniques like computing averages

Ordinal/nominal data are more limited but

can still be used for graphs and tables