Measurement Scales

foundations

scales

measurement

nominal

ordinal

interval

ratio

Nominal, ordinal, interval, and ratio scales – and which statistical operations each permits

Published

April 17, 2026

Research question

Before any test is chosen, the analyst must know the scale of each variable. Two concrete examples: (1) when summarising tumour grade in a retrospective breast-cancer cohort, is it appropriate to report the mean grade, or should the median or the mode be used? (2) In a multicentre trial, should site_id be treated as a predictor in a linear model, or only as a random grouping?

Assumptions

The taxonomy is descriptive rather than inferential. It assumes each variable has been coded faithfully: a variable intended to be ordinal has its levels ordered; a ratio variable genuinely has a meaningful zero.

Assumption	How to verify in R
Ordinal factor levels are ordered	`is.ordered(x)`; set with `factor(..., ordered = TRUE)`
No numeric code stored as `character`	`sapply(df, class)`
Zero is meaningful for a ratio variable	Inspect units: mass, counts, time elapsed are ratio; temperature in 0B0C is interval only

Hypotheses

Measurement-scale tutorials do not test hypotheses directly. They determine which downstream hypotheses are well-posed. A t-test on jersey numbers tests \(H_0: \mu_1 = \mu_2\) for a quantity that has no meaningful mean; the test is technically computable but substantively meaningless.

R code

library(tidyverse)
library(gtsummary)

set.seed(42)

clinical_cohort <- tibble(
  patient_id    = sprintf("P%03d", 1:120),
  sex           = factor(sample(c("Female", "Male"), 120, replace = TRUE)),
  tumour_grade  = factor(
    sample(c("G1", "G2", "G3"), 120, replace = TRUE, prob = c(0.3, 0.45, 0.25)),
    levels = c("G1", "G2", "G3"), ordered = TRUE
  ),
  temp_celsius  = round(rnorm(120, mean = 36.8, sd = 0.4), 1),
  wbc_count_giga_per_l = round(rlnorm(120, meanlog = log(7), sdlog = 0.3), 2)
)

# Inspect each scale level
sapply(clinical_cohort, function(x)
  if (is.ordered(x))      "ordinal"
  else if (is.factor(x))  "nominal"
  else if (is.numeric(x)) "metric"
  else "other"
)

# Permissible summaries by scale
clinical_cohort |>
  tbl_summary(
    include    = c(sex, tumour_grade, temp_celsius, wbc_count_giga_per_l),
    statistic  = list(all_categorical() ~ "{n} ({p}%)",
                      all_continuous()  ~ "{mean} ({sd})")
  )

The rstatix and gtsummary packages respect scale levels automatically: a factor is summarised by counts and percentages; a numeric variable by mean and standard deviation. An ordinal factor with ordered = TRUE stays ordered in downstream tests, so that Spearman or Jonckheere-Terpstra procedures pick up the ordering.

Interpreting the output

The inspection step confirms that sex is nominal, tumour_grade ordinal, and the two numeric columns metric. The tbl_summary() output reports the appropriate statistic for each column without manual intervention.

Effect size

Scale-level diagnostics are descriptive; no effect size applies. When a downstream analysis is chosen based on scale, the appropriate effect size for that test (Cohen’s \(d\), Cramer’s \(V\), etc.) is used.

Reporting (APA 7)

Typical methods-section phrasing:

Tumour grade was treated as an ordered factor (G1 < G2 < G3), sex as an unordered factor (female vs. male), and continuous variables as metric (interval / ratio).

Common pitfalls

Storing an ordinal variable as an unordered factor defeats downstream tests that depend on the ordering.
Treating a code (site identifier) as numeric invites regression models that fit slopes through arbitrary labels.
Reporting a mean of Likert-item scores for a single item is statistically dubious; a median or a composite scale of multiple items is safer.

Parametric vs. non-parametric alternative

Nominal outcomes route to chi-squared tests, Fisher’s exact test, logistic regression.
Ordinal outcomes route to rank-based tests (Mann-Whitney, Kruskal-Wallis, ordinal logistic regression).
Metric outcomes with approximately normal residuals support t-tests, ANOVA, linear regression; otherwise their rank-based analogues.