Levels of Measurement

In social science and criminology, how you measure a concept dictates
- which numerical operations are sensible
- which graphs/descriptives are legitimate
- which statistical tests/regression models are valid
Psychologist S. S. Stevens (1946) formalised four scale types – nominal, ordinal, interval, ratio – still the dominant convention
Error in classifying level of measurement can lead to
- biased estimates (e.g., calculating $\bar{x}$ on categorical data)
- invalid inferences (e.g., running Pearson’s $r$ on Likert items without checking assumptions)
Ethical responsibility: choose measures/analyses that represent participants’ data faithfully; mis-categorisation can inflate Type I/II error rates and misinform policy

Also called qualitative variables
Function: place each observation into mutually exclusive and collectively exhaustive groups
Numerical codes are arbitrary labels – operations such as $+,-,\times,\div$ are meaningless
Central tendency: report frequencies, percentages, mode; never compute mean/SD
Graphs: bar chart, pie chart
Typical statistical analyses (two variables)
- Nominal × Nominal ⇒ $\chi^2$ test of independence, Cramer’s V, lambda
- Nominal × Interval/Ratio ⇒ $t$ -test (2 groups), one-way ANOVA (>2), logistic regression
Examples from slides
- Gender: (1) Male, (2) Female, (3) Non-binary, (4) Other
- Religious affiliation, favourite chocolate, preferred streaming service
Additional examples
- Blood type, licence class, prison security level (minimum/medium/maximum as categories rather than ranks)

Order matters; distance between categories unknown/unequal
Permitted statistics
- Median, percentile, sometimes mean when distances are assumed equal (e.g., 5-point Likert)
- Non-parametric tests: Mann–Whitney U, Kruskal–Wallis, Wilcoxon signed-rank, Spearman’s $\rho$
Visuals: bar chart (ordered), boxplot (if treated as numeric)
Examples from slides
- Educational attainment (Primary → Postgraduate)
- Likert agreement (Strongly disagree → Strongly agree)
Extra examples
- Socio-economic status quintile, pain severity scale, prison misconduct seriousness (minor → severe)
Beware: treating ordinal data as interval without justification may violate assumptions of parametric tests

Both have equal intervals; only ratio possesses a true zero allowing meaningful ratios
Continuous vs. discrete
- Continuous: can take any value in range (e.g., height $=172.3\,\text{cm}$ )
- Discrete: whole counts (e.g., number of arrests $=5$ )
Permissible operations: all arithmetic + logarithms/exponentials (especially for ratio)
Descriptives: mean $\bar{x}$ , variance $s^2$ , standard deviation $s$ , median, mode, range, IQR
- $\bar{x}=\frac{\sum<em>{i=1}^{N}x</em>i}{N}$
- $s^2=\frac{\sum<em>{i=1}^{N}(x</em>i-\bar{x})^2}{N-1}$
Graphs: histogram, density plot, boxplot, scatterplot
Common parametric analyses: Pearson’s $r$ , simple/multiple linear regression, ANOVA, t-tests
Slide examples
- Income (\$), age (years), height (cm), crime counts/rates, days to recidivism
Further examples
- Blood alcohol concentration, IQ score (interval), distance to nearest police station (ratio)

Ask: “Does $0$ mean ‘none of the quantity’?”
- Temperature in $^{\circ}C$ ⇒ interval (0 ≠ no heat)
- Number of crimes ⇒ ratio (0 = crime-free)
Consequence: with ratio variables, statements like “area A has twice the crime of area B” are meaningful; not so with interval

Identify nature of categories: labels only? ✔ Nominal
Can you rank them? ✔ Ordinal
Are distances equal? ✔ Interval
Is there a true zero? ✔ Ratio
When in doubt, code conservatively (lower level) or collect additional metric information

Type of case disposition (Dismissed / Acquitted / Diverted / Convicted) ⇒ Nominal
Assault injury severity (None → Major + hospitalisation) ⇒ Ordinal (clear ordered progression)
Number of self-reported delinquent acts (1–12) ⇒ Interval/Ratio (count, true zero)

Choice of test/model depends on:
- Number of variables (bivariate vs. multivariate)
- Measurement level (nominal, ordinal, interval, ratio)
- Type of comparison (difference vs. association)
- Research design (experimental, quasi-experimental, correlational)
Example mapping (not exhaustive)
- 2 nominal groups, interval DV ⇒ independent-samples $t$ -test
- >2 nominal groups, interval DV ⇒ one-way ANOVA
- Ordinal IV & DV ⇒ Spearman’s $\rho$ or Kendall’s $\tau$
- Interval predictors, nominal DV ⇒ logistic regression
Always check assumptions: normality, homoscedasticity, independence, linearity, etc.

Correct identification of measurement level is foundational to valid empirical research.
Nominal: categories only; summarise with mode/frequency.
Ordinal: ordered categories; median & non-parametric tests.
Interval: equal units, arbitrary zero; full arithmetic except ratio statements.
Ratio: interval + true zero; allows meaningful ratios (twice, half, etc.).
Statistical method selection hinges on level of measurement, number of variables, comparison type, and research design.
Develop habit: on encountering a new variable, walk through the four-question heuristic (label? order? equal distance? true zero?) before coding or analysing.