Scales of Measurement

Overview

The previous lecture classified variables broadly as categorical or numerical. A scale of measurement goes one step further: it describes what the recorded values mean and which comparisons remain scientifically meaningful.

The classical framework distinguishes four scales: nominal, ordinal, interval and ratio [1]. These labels provide a useful language for thinking about medical data, but they should not be treated as a mechanical rule that selects a statistical test. The scientific meaning of the measurement, the research question, the study design, the distribution of the data and the intended effect estimate must all be considered [2], [3].

This lecture develops the four scales using one simulated postoperative-recovery study. It also addresses common sources of confusion, including numerical category codes, temperature scales, pain ratings and multi-item questionnaire scores.

Why this matters

A column of numbers may represent very different kinds of information. The code 3 could mean blood group AB, severe nausea, a pain rating of 3, three previous admissions or a body temperature recorded to one decimal place. Although all appear numerical in a spreadsheet, the permitted interpretations are different.

Incorrect assumptions about measurement can lead to meaningless statements. Examples include:

• calculating an average blood-group code;
• claiming that severe nausea is exactly twice moderate nausea;
• saying that a temperature of 40 °C is twice as hot as 20 °C; or
• assuming that a pain score of 8 represents exactly twice the pain of a score of 4.

Understanding the scale helps a researcher preserve the clinical meaning of the data, choose defensible summaries and avoid interpretations that the measurement does not support.

What a scale of measurement describes

A measurement scale describes the relationships represented by a variable’s values. Four questions are useful:

1. Do the values identify categories?
2. Can the values be placed in a meaningful order?
3. Do equal numerical differences represent equal differences in the measured quantity?
4. Does zero represent the absence of the measured quantity, making ratio statements meaningful?

The answers lead to the classical nominal, ordinal, interval and ratio scales.

A variable’s scale is different from its statistical role. Time to readiness for discharge is a ratio-scale variable and may be the outcome in one study but an explanatory variable in another. Its role changes with the research question; its measurement meaning does not.

A scale is also different from a software storage type. A statistical program may store blood group as text or as integer codes. Neither storage choice changes blood group from nominal data into a measured numerical quantity.

Nominal scale

A nominal scale distinguishes categories without assigning a meaningful order. The valid comparison is whether two observations belong to the same or different categories.

Examples include:

• blood group: A, B, AB or O;
• hospital site;
• type of surgical procedure; and
• culture organism.

For nominal data, suitable initial summaries are the number and percentage in each category. A bar chart may display the category distribution. The mode, meaning the most frequent category, can be identified, although it is often less informative than reporting all relevant counts and percentages.

The categories should be clearly defined and mutually exclusive where the clinical concept permits. When percentages are reported, the denominator and any missing values must be clear.

Ordinal scale

An ordinal scale has categories with a meaningful order, but the distance between adjacent categories is not established as equal.

Postoperative nausea severity may be recorded as:

• none;
• mild;
• moderate; or
• severe.

Severe indicates more nausea than moderate, but the difference between mild and moderate is not necessarily equal to the difference between moderate and severe. Coding the categories as 0, 1, 2 and 3 preserves their order for data management but does not prove equal spacing.

Other ordinal examples include:

• disease stage;
• functional class;
• pressure-injury category;
• a single agreement item from strongly disagree to strongly agree; and
• ordered levels of clinical urgency.

Ordinal data are commonly described using category counts and percentages. The median and percentiles may be meaningful when the ordering and number of categories make them interpretable, but the full category distribution should not be concealed by a single summary.

Interval scale

An interval scale has ordered values with equal intervals, but its zero point is arbitrary rather than a representation of complete absence of the measured quantity.

Temperature in degrees Celsius is the standard medical example. The difference between 36 °C and 37 °C represents the same temperature interval as the difference between 38 °C and 39 °C. Differences are therefore meaningful.

However, 0 °C does not indicate the absence of temperature. A temperature of 40 °C is not twice 20 °C. Adding a constant when converting Celsius to kelvin also changes such a ratio while preserving temperature differences. The International System of Units defines the kelvin as the unit of thermodynamic temperature [4].

Calendar years provide another interval-like example. The difference between 2010 and 2020 is ten years, but the calendar origin does not represent an absence of time. In contrast, a duration measured from a defined start point, such as postoperative recovery time in hours, can have a meaningful zero and is usually treated as ratio-scale data.

Interval-scale measurements can often be summarised using means, standard deviations and differences when these summaries suit the distribution and research question. Measurement scale alone does not guarantee that the mean is the best clinical summary.

Ratio scale

A ratio scale has ordered values, equal intervals and a non-arbitrary zero that represents the absence of the measured quantity within the measurement system. Differences and ratios can therefore be meaningful.

Examples include:

• body weight in kilograms;
• elapsed time in hours;
• number of previous admissions;
• drug dose in milligrams; and
• serum concentration in mg/L, when zero represents none of the measured substance in the defined system.

A recovery time of 48 hours is twice a recovery time of 24 hours. A participant with four previous admissions has twice the count of a participant with two, although the clinical importance of that ratio remains a separate question.

Ratio-scale variables may be discrete counts or continuous measurements. Their scale does not determine their distribution. Recovery time may be right-skewed, and admission counts may contain many zeros. Suitable summaries and models must account for those features.

Comparing the four scales

Table 1. Meaning supported by the classical measurement scales

Note: The table summarises the classical framework. Whether a particular statistical method is appropriate also depends on the study design, research objective, distribution, sample size and model assumptions.

The scales are sometimes presented as a strict hierarchy in which each level permits every operation available at lower levels. This is useful as an introduction, but real measurement is more nuanced. Velleman and Wilkinson argued that the four labels can become misleading when used as rigid rules for statistical analysis [2].

Data structure

Simulated postoperative-recovery variables

simulated data

Table 2. Measurement scales in the postoperative-recovery study

Variable	Example value	Broad data type	Measurement scale
participant_id	ER041	Identifier	Not a clinical measurement scale
blood_group	AB	Categorical	Nominal
nausea_severity	Moderate	Categorical	Ordinal
pain_nrs_24h	6	Ordered rating	Ordinal for a single item
temperature_c	37.8 °C	Numerical measurement	Interval
weight_kg	72.4 kg	Numerical measurement	Ratio
recovery_time_h	46 hours	Numerical measurement	Ratio
satisfaction_item	Agree	Categorical rating	Ordinal
care_group	Enhanced recovery	Categorical	Nominal

Note: NRS = numerical rating scale. The measurement interpretation should be documented in the data dictionary rather than inferred from the column’s software type.

Table 3. Four simulated participant records

participant_id	care_group	nausea_severity	pain_nrs_24h	temperature_c	recovery_time_h
ER001	Enhanced recovery	Mild	3	37.1	34
ER002	Standard care	Severe	8	38.0	71
ER003	Enhanced recovery	None	1	36.8	29
ER004	Standard care	Moderate	5	37.4	52

Note: Each row represents one individually randomised participant.

Rating scales require particular care

A single 0–10 pain item has ordered responses: 8 represents more reported pain than 4. However, equal numerical gaps and ratio interpretations are not automatically established. It is not generally defensible to say that pain rated 8 is exactly twice pain rated 4.

A single Likert-type item, such as strongly disagree to strongly agree, is ordinarily treated as ordinal [5]. A multi-item score combines responses from several items according to a prespecified scoring rule. Such a total score is not automatically interval-scale merely because it has many possible numerical values.

In practice, researchers sometimes analyse a validated multi-item score as approximately continuous. This may be reasonable when the score’s development, distribution, reliability, interpretation and statistical model support that decision. Research has also shown that some common parametric procedures can be robust to departures from ideal measurement and distributional assumptions [6]. Robustness is not permission to ignore the measurement process: the estimand, model assumptions and sensitivity of conclusions should still be considered.

Why scale alone does not select the analysis MMed extension

Rules such as “ordinal data always require a non-parametric test” or “interval data always require a t-test” are too rigid.

A suitable method depends on several connected features:

• the research question and estimand;
• whether observations are independent, paired, repeated or clustered;
• the number and distribution of categories;
• whether the analysis concerns a single item or a validated composite score;
• the plausibility of the model assumptions; and
• the interpretation required from the result.

An ordinal outcome may be analysed using an ordinal regression model that preserves category order. A composite score may sometimes be analysed using a numerical-outcome model, supported by diagnostic assessment and a sensitivity analysis. The decision should be justified rather than based only on the label attached to the scale.

How scale guides descriptive summaries

Table 4. Starting points for describing variables at each scale

Note: Interquartile range describes the middle 50% of ordered observations. The choice of summary should reflect the distribution and clinical purpose, not only the scale label or a formal normality-test p-value.

Binary variables illustrate why the framework should not be applied mechanically. A binary outcome coded 0 and 1 remains categorical, yet its arithmetic mean equals the observed proportion coded 1. This is a valid mathematical property, not evidence that the underlying clinical outcome has become a continuous measurement.

Worked example

Suggested reporting

Measurement definitions should be reported with categories, units, time points and scoring rules. Readers should not have to infer meaning from abbreviated variable names.

The reporting wording states the measurement origin, unit, treatment of ordered ratings, comparison direction and statistical method.

Common mistakes

Knowledge check

Blood group is coded as 1 = A, 2 = B, 3 = AB and 4 = O. What is its measurement scale?

Answer and reasoning

It is nominal. The numbers are labels for distinct categories and do not create a clinical order or meaningful numerical distance. Counts and percentages should be used rather than arithmetic summaries of the codes.

Why is postoperative nausea graded as none, mild, moderate or severe considered ordinal?

Answer and reasoning

The categories have a meaningful order, but the differences between adjacent categories are not established as equal. Severe represents more nausea than moderate, but it is not valid to assume that each category step represents the same increase.

Why is temperature in degrees Celsius interval rather than ratio scale?

Answer and reasoning

Equal differences in degrees Celsius are meaningful, but 0 °C is an arbitrary reference point rather than an absence of temperature. Consequently, 40 °C cannot be interpreted as twice 20 °C.

A researcher records recovery time in hours and finds values of 24 and 48 hours. Is a ratio statement meaningful?

Answer and reasoning

Yes. Elapsed time has a meaningful zero, so 48 hours is twice 24 hours as a duration. Whether that ratio is the most clinically useful effect measure is a separate question; a difference in hours may be easier to interpret for the study objective.

Does an ordinal outcome automatically require the same statistical method in every study?

Answer and reasoning

No. The method also depends on the research question, number of categories, study design, dependence among observations, target effect and model assumptions. The ordinal nature should be respected, but it does not identify one universal test.

Key take-home messages

References

1. Stevens SS. On the theory of scales of measurement. Science. 1946;103(2684):677–680. https://doi.org/10.1126/science.103.2684.677
2. Velleman PF, Wilkinson L. Nominal, ordinal, interval, and ratio typologies are misleading. Am Stat. 1993;47(1):65–72. https://doi.org/10.1080/00031305.1993.10475938
3. Altman DG. Practical statistics for medical research. London: Chapman and Hall; 1991.
4. Bureau International des Poids et Mesures. The International System of Units (SI). 9th ed. Sèvres: BIPM; 2019. https://doi.org/10.59161/AUEZ1291
5. Sullivan GM, Artino AR Jr. Analyzing and interpreting data from Likert-type scales. J Grad Med Educ. 2013;5(4):541–542. https://doi.org/10.4300/JGME-5-4-18
6. Norman G. Likert scales, levels of measurement and the “laws” of statistics. Adv Health Sci Educ Theory Pract. 2010;15(5):625–632. https://doi.org/10.1007/s10459-010-9222-y