Assumptions of Parametric and Non-Parametric Statistics & Scales of Measurement

Introduction

Every statistical test rests on a set of assumptions — conditions that must be satisfied for the test results to be valid and interpretable. Choosing the wrong test, or using a test when its assumptions are violated, can lead to incorrect conclusions — even with perfectly collected data.

Two interrelated decisions guide statistical choice: (1) Do your data meet the assumptions of parametric tests? And (2) What scale of measurement were your variables measured on? These questions are inseparable: the scale of measurement often directly determines which assumptions can be met.

📄 Source PDF: Block-1/Unit-1.pdf
📚 MPC-006 Statistics in Psychology — Block 1: Introduction to Statistics

The Four Scales of Measurement

S.S. Stevens (1946) proposed the foundational classification of measurement scales — a framework so influential it remains standard in psychology and statistics textbooks worldwide.

1. Nominal Scale

The nominal scale is the most basic level of measurement. It assigns numbers or labels to categories purely for identification — not for any mathematical purpose.

Nature: Classifies into mutually exclusive, exhaustive categories
Operations: Can only determine equality or inequality
Example: Gender (male/female), blood type (A/B/AB/O), religion, caste
Labelling: Categories may be labelled 0/1, m/f, 1/2/3 — but these numbers have no mathematical meaning
Statistical use: Only frequency counts and percentages; modal category
Non-parametric test appropriate: Chi-square test

Why it's called "nominal": From Latin nomen (name) — it's a naming scale.

2. Ordinal Scale

The ordinal scale ranks data in a meaningful order, but the intervals between ranks are not equal.

Nature: Implies rank order but not the magnitude of differences
Operations: Can determine greater than/less than relationships
Example: Satisfaction scale (1=very dissatisfied to 5=very satisfied); academic rank (1st, 2nd, 3rd); pain severity (mild, moderate, severe)
Key limitation: The difference between rank 1 and rank 2 may not equal the difference between rank 4 and rank 5
Statistical use: Median, percentiles, rank correlations
Non-parametric tests appropriate: Mann-Whitney U, Wilcoxon, Spearman's ρ

3. Interval Scale

The interval scale has equal intervals between values — the distance between any two consecutive values is the same — but has no true zero point.

Nature: Equal intervals, but zero does not mean absence of the property
Operations: Addition and subtraction meaningful; ratios are not
Example: Temperature in Celsius or Fahrenheit (0°C does not mean no temperature); calendar years; IQ scores (an IQ of 0 doesn't mean zero intelligence)
Properties: Two of three key requirements — magnitude and equal intervals — but lacks the real/absolute zero
Statistical use: Mean, standard deviation, correlation
Parametric tests appropriate: t-test, ANOVA, Pearson's r

4. Ratio Scale

The ratio scale has all the properties of the interval scale plus a true (absolute) zero, meaning zero represents complete absence of the quantity.

Nature: Equal intervals + true zero; ratios between values are meaningful
Operations: All arithmetic operations are valid
Example: Height, weight, age, length, reaction time, income, number of correct answers
Key property: A person who is 180cm is genuinely twice as tall as a 90cm person
Statistical use: All parametric statistics
Parametric tests appropriate: t-test, ANOVA, regression, Pearson's r

Comparison Summary

Scale	Order	Equal Intervals	True Zero	Example	Test Type
Nominal	❌	❌	❌	Gender, Religion	Non-parametric
Ordinal	✅	❌	❌	Likert scale, Rank	Non-parametric
Interval	✅	✅	❌	Temperature (°C), IQ	Parametric
Ratio	✅	✅	✅	Height, Reaction time	Parametric

Assumptions of Parametric Statistics

Parametric tests (t-test, F-test, Pearson's r) operate under specific assumptions. Violating these can invalidate results.

Assumption 1: Normal Distribution

The population from which the sample is drawn should be normally distributed — the classic bell-shaped frequency distribution that is:

Symmetrical about the mean
Unimodal (one peak)
Infinite at both ends (theoretical)
Mean = Median = Mode

However, parametric tests (especially t and F) are considered robust — they remain valid even when the normality assumption is mildly violated, particularly with larger samples.

Assumption 2: Interval or Ratio Scale

Variables must be measured on at least an interval scale. This ensures that arithmetic operations (computing means, standard deviations) are mathematically meaningful.

Assumption 3: Independence of Observations

The inclusion or exclusion of any case should not unduly affect the results. Each observation must be independent of all others. Violations occur when:

The same participants provide multiple responses that are treated as independent
Participants influence each other (e.g., in group settings)
Time-series data are treated as independent

Assumption 4: Homoscedasticity (Equality of Variances)

The populations from which samples are drawn must have the same variance (or a known ratio of variances). This is also called homogeneity of variances and is particularly important with small samples.

The F-max test is used to check this:

F-max = Largest Variance ÷ Smallest Variance

If the computed F-max does not meet or exceed the critical value at p = .05, variances are considered homogeneous.

Assumptions of Non-Parametric Statistics

Non-parametric tests make fewer and weaker assumptions:

Situation 1: Small Sample Size

When N is as small as 5 or 6, parametric tests cannot be applied because the normal distribution of sample means (central limit theorem) does not hold. Non-parametric tests provide the only valid alternative.

Situation 2: Non-Normal Distribution

When the population distribution is unknown or known to be non-normal, parametric tests are inappropriate regardless of sample size.

Situation 3: Ordinal or Nominal Data

When measurements are expressed as:

Rankings or ordinal categories
Plus/minus signs (direction of difference only)
Classifications like "good-bad", "improved-not improved"

Situation 4: Unknown Population Distribution

When the nature of the population from which samples are drawn is simply not known.

Situation 5: Nominal Form of Variables

When variables are expressed purely as categories (e.g., religion, caste, type of therapy).

Variables: Types and Definitions

Understanding variables is inseparable from understanding scales and assumptions:

Variable Type	Definition	Example
Independent Variable (IV)	The variable considered to be a cause; manipulated by the researcher	Teaching method (lecture vs. activity-based)
Dependent Variable (DV)	The variable considered to be an effect; what is measured	Test score, anxiety level
Characteristic	Any variable that can take different values across cases	Age, IQ, gender

Homoscedasticity — a key parametric assumption — requires that the DV's variance is similar across levels of the IV. If anxiety scores have much greater variance in the control group than the experimental group, homoscedasticity is violated.

Practical Guidance: Selecting the Right Test

Real-world complication: Likert scales (1–5, 1–7) used extensively in psychology are technically ordinal — but many researchers treat them as interval. This is a longstanding controversy. In India, where attitude and satisfaction scales are commonly used in health and educational research, this decision has practical consequences for statistical analysis.

🌐 External Resources

Wikipedia

Level of Measurement — Wikipedia — Stevens's four scales explained with examples
Statistical Assumption — Wikipedia — What assumptions are and why they matter

Educational Videos

Research Papers

📄 Stevens, S.S. (1946). "On the Theory of Scales of Measurement." Science, 103(2684), 677–680. — The original paper proposing the four scales — one of the most cited papers in measurement theory
📄 Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage. — Standard comprehensive statistics reference
📄 Norman, G. (2010). "Likert scales, levels of measurement and the 'laws' of statistics." Advances in Health Sciences Education, 15(5). — Influential debate on treating Likert scales as interval

Additional Resources

🧠 Memory Aid: NOIR Scales

N — Nominal: Names/Categories only (gender, religion)
O — Ordinal: Order matters (1st, 2nd, 3rd) but intervals don't
I — Interval: Equal intervals, no true zero (temperature °C, IQ)
R — Ratio: Real zero + all operations valid (height, weight, age)

Parametric assumptions — HINDI:

H — Homoscedasticity (equal variances)
I — Independence of observations
N — Normal distribution
D — Data on interval/ratio scale
I — Inferential goal: estimating population parameters

✅ Self-Assessment Questions

Level 1 — Recall

Name the four scales of measurement in order from lowest to highest level.
What is homoscedasticity and why is it important for parametric tests?

Level 2 — Understanding

A researcher measures "years of education" (0–25 years). What scale of measurement is this? Is it appropriate for parametric tests? Explain.
Why does an interval scale lack a true zero? Give two examples to illustrate this limitation.

Level 3 — Application & Analysis

A study in India measures participants on the following variables: (a) Caste category (SC/ST/OBC/General), (b) Self-reported stress level on a 1–10 scale, (c) Systolic blood pressure in mmHg, (d) Rank in school class (1st to 40th). For each variable, identify the scale of measurement and state whether parametric or non-parametric analysis is appropriate, with justification.

Summary

The four scales of measurement — nominal, ordinal, interval, and ratio — directly determine which statistical analyses are appropriate. Parametric tests require at least interval-scale data, normally distributed populations, independent observations, and homogeneous variances. Non-parametric tests apply when these conditions cannot be met: with small samples, ordinal or nominal data, non-normal distributions, or unknown population characteristics. Understanding these assumptions is not bureaucratic box-ticking — it is the foundation of statistically sound and ethically responsible research.

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Next → Parametric Statistical Tests: t-test, F-test, and Pearson's r

Assumptions of Parametric and Non-Parametric Statistics & Scales of Measurement

Introduction​

The Four Scales of Measurement​

1. Nominal Scale​

2. Ordinal Scale​

3. Interval Scale​

4. Ratio Scale​

Comparison Summary​

Assumptions of Parametric Statistics​

Assumption 1: Normal Distribution​

Assumption 2: Interval or Ratio Scale​

Assumption 3: Independence of Observations​

Assumption 4: Homoscedasticity (Equality of Variances)​

Assumptions of Non-Parametric Statistics​

Situation 1: Small Sample Size​

Situation 2: Non-Normal Distribution​

Situation 3: Ordinal or Nominal Data​

Situation 4: Unknown Population Distribution​

Situation 5: Nominal Form of Variables​

Variables: Types and Definitions​

Practical Guidance: Selecting the Right Test​

🌐 External Resources​

Wikipedia​

Educational Videos​

Research Papers​

Additional Resources​

🧠 Memory Aid: NOIR Scales​

✅ Self-Assessment Questions​

Level 1 — Recall​

Level 2 — Understanding​

Level 3 — Application & Analysis​

Summary​