A ratio variable is a quantitative variable measured with ordered values, equal intervals, and a meaningful zero point. Because zero represents the absence of the measured quantity, both differences and ratios are interpretable. For example, 20 kilograms is twice 10 kilograms, whereas 20°C is not meaningfully twice as warm as 10°C.
Key Takeaways
- A ratio variable has ordered numerical values, equal intervals, and a non-arbitrary zero.
- Differences, sums, products, divisions, and proportional comparisons can be meaningful.
- Ratio variables may be continuous, such as height, or discrete, such as number of children.
- Exact ratio measurements become ordinal when they are replaced by ordered response bands.
- Measurement level alone does not determine the correct statistical test.
- A numeric column is not necessarily ratio-level; its construct, zero point, unit, and coding must be examined.
Introduction
Ratio variables appear throughout scientific, social, educational, health, and business research. Researchers use them to record quantities such as duration, distance, mass, revenue, reaction time, number of visits, and units produced.
Recognizing a ratio variable matters because its measurement properties affect how results can be interpreted. With a valid ratio scale, a researcher can compare not only differences but also proportions. A duration of 20 minutes is 10 minutes longer than a duration of 10 minutes and is also twice as long.
However, the classification is frequently oversimplified. A variable is not ratio-level merely because it contains numbers or includes a value coded as zero. The zero must represent a theoretically meaningful absence of the measured quantity, and equal numerical differences must represent equal differences in the underlying attribute.
This article explains how to identify ratio variables, distinguish them from interval and ordinal variables, collect and analyze ratio data, recognize difficult cases, and report the results appropriately.
What Is a Ratio Variable?
A ratio variable is a numerical variable whose values have a meaningful order, equal units, and a true or non-arbitrary zero point. The zero establishes an origin from which proportional comparisons can be made.
Suppose researchers measure task-completion time in seconds:
- 0 seconds represents no elapsed time.
- The difference between 10 and 20 seconds is the same duration as the difference between 30 and 40 seconds.
- A task lasting 20 seconds takes twice as long as one lasting 10 seconds.
These properties make task duration a ratio variable.
The term ratio scale refers to the measurement system. The term ratio variable refers to a variable measured using that system. “Ratio data” describes the resulting observations.
Characteristics of Ratio Variables
1. The values are numerical
Ratio variables represent measurable or countable quantities. Their values are numbers with quantitative meaning, not numbers used merely as labels.
For example:
25 kgrepresents a measured mass.25 visitsrepresents a count.- Student ID
25is only an identifier and is therefore nominal, not ratio-level.
2. Values have a meaningful order
Higher and lower values represent more and less of the measured quantity.
A distance of 12 metres is greater than a distance of 8 metres. A person who completed 15 transactions completed more transactions than a person who completed 6.
Order alone is not sufficient, however. Ordinal variables also have an order. A ratio variable must additionally have equal intervals and a meaningful zero.
3. Intervals between values are equal
An increase of one unit has the same quantitative meaning throughout the scale.
For example, the difference between 5 and 10 kilograms is the same amount of mass as the difference between 20 and 25 kilograms. Both differences equal 5 kilograms.
This property allows subtraction and difference-based statistics to be interpreted.
4. The scale has a true zero
A true zero represents the absence of the quantity defined by the variable.
Examples include:
- Zero purchases: no purchases occurred.
- Zero minutes: no time elapsed.
- Zero metres: no distance was travelled.
- Zero children: no children are present in the defined count.
- Zero kelvin: the origin of the thermodynamic temperature scale.
A true zero does not mean that zero must actually appear in a particular dataset. A study of adult heights may contain no observations close to zero, but the measurement system still has a meaningful origin.
5. Ratios are meaningful
Because the origin is fixed, proportional statements can be interpreted.
If one package weighs 4 kilograms and another weighs 2 kilograms, the first package has twice the mass of the second.
The following ratio is meaningful:
[
\text{Ratio}=\frac{x_1}{x_2}
]
For example:
[
\frac{4\text{ kg}}{2\text{ kg}}=2
]
The value 2 indicates that the first mass is twice the second.
6. Unit conversions preserve ratios
A rigorous property of ratio scales is that a valid change of unit multiplies every value by the same positive constant:
[
x’=bx,\qquad b>0
]
For example:
[
1\text{ metre}=100\text{ centimetres}
]
Changing from metres to centimetres multiplies each value by 100. It does not change the origin, order, or ratios.
By contrast, converting Celsius to Fahrenheit requires multiplication and addition:
[
F=\frac{9}{5}C+32
]
The added constant changes the location of zero. This is one reason Celsius and Fahrenheit are classified as interval rather than ratio scales.
Examples of Ratio Variables
| Variable | Discrete or continuous? | Why it can be ratio-level | Valid proportional interpretation |
|---|---|---|---|
| Height measured from a defined baseline | Continuous | Equal units and zero height as the origin | 180 cm is 1.5 times 120 cm |
| Body mass | Continuous | Zero represents no mass | 80 kg is twice 40 kg |
| Distance travelled | Continuous | Zero means no distance travelled | 10 km is twice 5 km |
| Task duration | Continuous | Zero means no elapsed duration | 30 minutes is twice 15 minutes |
| Reaction time | Continuous | Measured from the onset of a stimulus | 600 ms is twice 300 ms |
| Number of children | Discrete | Zero means none in the defined count | Four is twice two |
| Number of purchases | Discrete | Zero means no purchases | Ten is twice five |
| Units produced | Discrete | Zero means no units produced | 200 units is twice 100 |
| Gross revenue during a period | Usually continuous | Zero means no revenue received | $2,000 is twice $1,000 |
| File size in bytes | Discrete in storage units | Zero means no stored bytes | 20 MB is twice 10 MB |
| Area | Continuous | Zero represents no area | 40 m² is twice 20 m² |
| Volume | Continuous | Zero represents no volume | 6 L is twice 3 L |
| Thermodynamic temperature in kelvin | Continuous | Kelvin has a theoretically defined origin | 200 K is twice 100 K in thermodynamic temperature |
| Frequency count | Discrete | Zero means the event did not occur | Eight events is twice four |
| Energy consumption | Continuous | Zero means none consumed during the defined period | 20 kWh is twice 10 kWh |
The operational definition remains important. “Distance” may be a ratio quantity, but a geographical coordinate such as “10 kilometres east of a reference point” can involve direction and an arbitrarily chosen origin.
Variables That Are Not Ratio-Level
| Variable | Likely level | Why it is not ratio-level |
|---|---|---|
| Temperature in Celsius | Interval | Zero does not mean the absence of thermodynamic temperature |
| Temperature in Fahrenheit | Interval | The zero point is conventional |
| Calendar year | Interval | Year zero is not the absence of time |
| Time of day | Interval or circular | Midnight is a reference point, not no time |
| IQ score | Commonly treated as interval | Zero does not mean no intelligence |
| Standardized z-score | Interval | Zero represents the sample or reference mean |
| Satisfaction from 1 to 5 | Ordinal | Ordered categories do not necessarily have equal intervals |
| Class rank | Ordinal | The difference between ranks is not a fixed quantity |
| Postal code | Nominal | Digits are identifiers rather than measured amounts |
| Student ID | Nominal | Numbers label individuals |
| pH | Logarithmic | A pH of zero does not mean no acidity |
| Grouped age band | Ordinal | Exact ages have been replaced by ordered categories |
| Letter grade | Ordinal | Grades have order but unequal or undefined distances |
Discrete and Continuous Ratio Variables
A ratio variable can be either discrete or continuous.
Discrete ratio variables
A discrete ratio variable takes countable values, commonly whole numbers.
Examples include:
- Number of hospital admissions
- Number of publications
- Number of errors
- Number of customers
- Number of website visits
- Number of defects
A count of 0 represents no observed events in the defined period or setting. A count of 10 is twice a count of 5.
However, the statistical distribution of count data often differs from that of continuous measurements. Poisson, negative-binomial, zero-inflated, or hurdle models may be more suitable than ordinary linear regression.
Continuous ratio variables
A continuous ratio variable can theoretically take any value within an interval, subject to measurement precision.
Examples include:
- Height
- Weight
- Duration
- Distance
- Volume
- Reaction time
- Concentration measured on an appropriate linear scale
A device may round a continuous quantity to a fixed number of decimal places, but rounding does not necessarily make the underlying construct discrete.
Ratio Variable Versus Interval Variable
A ratio variable has all the usual properties attributed to an interval variable plus a meaningful origin.
| Property | Interval variable | Ratio variable |
|---|---|---|
| Values can be classified | Yes | Yes |
| Values can be ordered | Yes | Yes |
| Equal differences are meaningful | Yes | Yes |
| Zero represents absence | No | Yes |
| Addition and subtraction are interpretable | Yes | Yes |
| Multiplication and division of measurements are interpretable | Generally no | Yes |
| “Twice as much” is meaningful | No | Yes |
| Unit change may add a constant | Yes | No; a valid ratio-scale conversion multiplies by a positive constant |
| Examples | Celsius temperature, calendar year | mass, duration, distance, counts |
Celsius and Kelvin example
The difference between 10°C and 20°C equals the difference between 20°C and 30°C. Celsius therefore has equal intervals.
However, 20°C is not twice the thermodynamic temperature of 10°C because Celsius zero does not represent the absence of thermodynamic temperature.
Kelvin has a theoretically defined zero. Ratios of thermodynamic temperatures expressed in kelvin can therefore be interpreted mathematically.
It is preferable to say “twice the thermodynamic temperature” rather than “twice as hot,” because subjective hotness is a different construct.
Ratio Variables and the Four Levels of Measurement
The traditional classification introduced by Stevens (1946) distinguishes four levels.
| Level | Categories | Order | Equal intervals | True zero | Example |
|---|---|---|---|---|---|
| Nominal | Yes | No | No | No | Research discipline |
| Ordinal | Yes | Yes | No | No | Satisfaction category |
| Interval | Yes | Yes | Yes | No | Celsius temperature |
| Ratio | Yes | Yes | Yes | Yes | Task duration |
This framework remains useful for understanding what numerical statements mean. However, it should not be treated as a complete automatic test-selection system.
Later methodological discussions have shown that statistical validity also depends on the model, research question, design, sampling process, distribution, and robustness of the method (Hand, 1996; Velleman & Wilkinson, 1993).
How to Identify a Ratio Variable
Use the following five-step process.
Step 1: Define the construct
State exactly what is being measured.
“Time” is too vague. It could mean:
- Duration since an event
- Clock time
- Calendar date
- Time remaining
- Change in duration
Duration can be ratio-level, whereas clock time and calendar dates generally are not.
Step 2: Determine whether the values represent quantities
Ask whether the numbers measure or count an amount.
A postal code contains numbers but does not measure postal quantity. It is nominal.
Step 3: Test whether equal differences have equal meanings
Ask whether a one-unit increase represents the same amount throughout the scale.
If the difference between adjacent values is unknown or inconsistent, the variable is not interval- or ratio-level.
Step 4: Examine zero
Ask:
Does zero represent a complete absence of the specifically defined quantity, rather than a conventional reference point, average, threshold, missing value, or arbitrary code?
If not, the variable is unlikely to be ratio-level.
Step 5: Test proportional statements
Choose two possible values and ask whether a statement such as “twice as much” remains meaningful.
For instance:
- 20 minutes is twice 10 minutes: meaningful.
- 20°C is twice 10°C: not meaningful.
- Satisfaction score 4 is twice score 2: not established.
- Rank 2 is twice rank 1: meaningless.
Quick classification checklist
A variable is a strong candidate for ratio measurement when all the following are true:
- The values measure or count a quantity.
- The values have a meaningful order.
- Equal numerical differences represent equal differences in the construct.
- Zero is non-arbitrary and represents absence of the defined quantity.
- Ratios remain meaningful after valid unit conversions.
How Ratio Data Are Collected
Direct physical measurement
Researchers may use:
- Scales
- Rulers
- Timers
- Sensors
- Laboratory instruments
- Wearable devices
- Environmental monitors
Instrument calibration, precision, detection limits, and measurement error should be documented.
Counting
Researchers may count:
- Events
- People
- Errors
- Purchases
- Visits
- Symptoms
- Publications
- Correct responses
The counting period and unit of observation must be defined. “Number of visits” is incomplete without specifying visits by whom, to where, and during which period.
Administrative or digital records
Ratio variables may come from:
- Financial systems
- Web analytics
- Hospital records
- Learning-management systems
- Transaction logs
- Scientific databases
- Machine sensors
Researchers should verify how zeros, missing values, duplicates, and automatically generated records are coded.
Questionnaires and interviews
Ratio data can be collected by requesting an exact quantity:
How many hours did you study during the last seven days?
___ hours
This preserves substantially more information than response categories such as:
- None
- 1–5 hours
- 6–10 hours
- More than 10 hours
The second format records an ordinal variable, even though the underlying number of hours is ratio-level.
Exact entry is not always preferable. Respondents may not remember precise quantities, may consider the question sensitive, or may make unit-entry errors. The instrument should balance analytical precision with response quality.
How to Analyze Ratio Data
Ratio measurement permits a wide range of numerical summaries. Nevertheless, the appropriate analysis depends on the variable’s distribution and the study design.
Descriptive statistics
Common summaries include:
- Count and percentage of valid observations
- Minimum and maximum
- Mean
- Median
- Mode
- Range
- Interquartile range
- Variance
- Standard deviation
- Coefficient of variation
- Geometric mean for suitable positive data
Arithmetic mean
[
\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}
]
The arithmetic mean uses every observation but can be strongly affected by outliers and skewness.
Sample standard deviation
[
s=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}
]
The standard deviation describes dispersion in the original measurement unit.
Coefficient of variation
[
CV=\frac{s}{\bar{x}}\times100%
]
The coefficient of variation expresses standard deviation relative to the mean. It is most interpretable for ratio variables with a meaningful zero and a mean sufficiently far from zero.
Geometric mean
For positive values:
[
G=\left(\prod_{i=1}^{n}x_i\right)^{1/n}
]
The geometric mean may be helpful for multiplicative processes, growth factors, concentrations, or positively skewed data. It cannot be calculated directly when values include zero or negative numbers.
Visualizations
Useful graphs include:
- Histogram
- Density plot
- Box plot
- Dot plot
- Violin plot
- Scatterplot
- Time-series plot
- Error-bar plot
- ECDF or cumulative distribution plot
The graph should match the variable and question. A histogram may describe a continuous distribution, while a bar chart may be more suitable for small count values.
Inferential methods
| Research question | Possible method | Important additional considerations |
|---|---|---|
| Compare two independent group means | Independent-samples t-test | Independence, outliers, variance, sampling distribution |
| Compare paired measurements | Paired t-test | Distribution of within-pair differences |
| Compare three or more means | ANOVA | Independence, residual behavior, variance structure |
| Estimate association between continuous variables | Pearson correlation | Linearity, influential observations, dependence |
| Predict a continuous outcome | Linear regression | Functional form, residuals, heteroscedasticity, dependence |
| Analyze positive skewed outcomes | Gamma regression or lognormal model | Link function, zeros, model fit |
| Analyze counts | Poisson or negative-binomial regression | Exposure time, overdispersion, excess zeros |
| Analyze time until an event | Survival analysis | Censoring, time origin, proportional-hazards assumptions |
| Use a robust or rank-based comparison | Mann–Whitney, Wilcoxon, Kruskal–Wallis, permutation methods | Interpretation differs from a simple mean comparison |
| Model repeated measurements | Mixed-effects model or GEE | Within-subject dependence and time structure |
A ratio measurement does not guarantee normality. Income, reaction time, length of hospital stay, publication counts, and transaction values are often skewed.
Worked Example
Suppose a researcher records the time, in minutes, required by five participants to complete a task:
[
12,\ 15,\ 18,\ 20,\ 25
]
Step 1: Confirm the level
Completion time has:
- Numerical values
- Meaningful order
- Equal units
- A meaningful zero representing no elapsed time
- Meaningful proportional comparisons
It is therefore a continuous ratio variable.
Step 2: Calculate the mean
[
\bar{x}=\frac{12+15+18+20+25}{5}
]
[
\bar{x}=\frac{90}{5}=18
]
The mean completion time is 18 minutes.
Step 3: Calculate the sample standard deviation
The squared deviations from the mean are:
[
(12-18)^2=36
]
[
(15-18)^2=9
]
[
(18-18)^2=0
]
[
(20-18)^2=4
]
[
(25-18)^2=49
]
Their sum is 98.
[
s=\sqrt{\frac{98}{5-1}}
]
[
s=\sqrt{24.5}\approx4.95
]
The sample standard deviation is approximately 4.95 minutes.
Step 4: Calculate the coefficient of variation
[
CV=\frac{4.95}{18}\times100%\approx27.5%
]
Completion times vary by approximately 27.5% of the mean.
This estimate should be interpreted cautiously because the sample contains only five observations. The example demonstrates the calculation rather than supporting a population conclusion.
Advantages of Ratio Variables
Rich interpretation
Researchers can discuss differences and proportions. This makes statements such as “twice the duration” or “25% lower consumption” potentially meaningful.
Broad descriptive possibilities
Means, standard deviations, ranges, percentiles, geometric means, and coefficients of variation may be available when their individual assumptions are satisfied.
Flexible modeling
Ratio variables can appear as outcomes, predictors, mediators, moderators, controls, offsets, or exposure variables in statistical models.
Unit conversion
Researchers can convert compatible units without altering substantive ratios. For example, a length can be reported in metres or centimetres.
Practical relevance
Many research outcomes are naturally measured as amounts, durations, counts, distances, costs, or physical quantities.
Limitations and Practical Considerations
Measurement level does not guarantee validity
A precisely measured variable may still fail to represent the intended construct. Time spent on a learning platform, for example, may not accurately measure attention or learning.
Ratio variables may be skewed
Financial values, duration data, and counts often have long right tails. The mean and standard deviation may not adequately summarize them.
Zeros can have different meanings
A zero may represent:
- Genuine absence
- No event during the observation window
- A value below detection
- Not applicable
- Missing information
- A system default
- Rounding
Researchers must distinguish these possibilities in the codebook.
Measurement error remains possible
Ratio scales do not eliminate:
- Instrument error
- Recall error
- Data-entry errors
- Calibration problems
- Rounding
- Digit preference
- Unit inconsistencies
Some constructs cannot be forced onto ratio scales
Attitudes, preferences, perceived quality, and many psychological constructs do not automatically acquire ratio properties merely because researchers assign numbers to response categories.
Ratios involving zero need care
A value may validly equal zero, but division by zero is undefined. Percentage changes are also unstable when the baseline is zero or extremely small.
Can Ratio Variables Have Negative Values?
In elementary explanations, ratio variables are usually described as nonnegative because zero represents absence and values below absence are impossible. This description works well for mass, duration, length, and counts.
However, the presence of a negative sign should not be used as the only classification rule.
Negative values may appear when a variable measures:
- Directional displacement
- Net profit or loss
- Change from baseline
- Net cash flow
- Signed velocity
- A difference between two ratio quantities
In such cases, the researcher must define what the variable represents. Gross revenue may have a natural zero, while net profit can be negative. Distance is nonnegative, while directional displacement can be signed.
The most defensible question is:
Do the origin, units, and proportional statements have a coherent meaning for the defined construct?
Common Mistakes
Mistake 1: Assuming every numeric variable is ratio-level
Identifiers, rankings, rating categories, and postal codes can all contain numbers without measuring amounts.
Mistake 2: Treating a coded zero as a true zero
Coding “male = 0” or “control group = 0” does not create a ratio scale. The number is only a category label.
Mistake 3: Calling grouped ranges ratio data
Exact age may be ratio-level. An age category such as 25–34 is ordinal because the exact value is no longer known.
Mistake 4: Assuming a ratio variable must be continuous
Counts are commonly discrete ratio variables.
Mistake 5: Choosing a test solely from the measurement level
Distribution, dependence, design, exposure, missingness, and the research hypothesis also affect test selection.
Mistake 6: Treating zero as missing
Replacing missing income or study time with zero changes the substantive meaning of the data and can bias estimates.
Mistake 7: Believing percentages are automatically ratio-level
A percentage may represent a proportion, percentage change, standardized score, index, or transformed measure. Its measurement properties depend on how it was constructed.
Mistake 8: Making ratios across incompatible units
A meaningful ratio requires compatible quantities measured using compatible units.
Mistake 9: Ignoring the observation period
Ten visits per week and ten visits per year are not equivalent quantities. Counts often need an exposure period or offset.
Mistake 10: Using the coefficient of variation near a zero mean
Because the mean appears in the denominator, the coefficient of variation becomes unstable when the mean is close to zero.
Ratio Variables in Modern Research
Experimental research
Common ratio outcomes include:
- Reaction time
- Task duration
- Number of errors
- Distance travelled
- Physiological measurements
- Quantity consumed
Health and clinical research
Examples include:
- Body mass
- Drug dosage
- Length of hospital stay
- Number of admissions
- Concentration measured on a suitable scale
- Time to an event
Researchers must account for censoring, detection limits, repeated measurements, and clinical interpretation.
Educational research
Possible ratio variables include:
- Time on task
- Number of correct responses
- Number of absences
- Number of completed assignments
- Pages read
- Login frequency
A score of zero correct answers means no items were answered correctly, but it does not necessarily mean the learner possesses no knowledge. The observed count can be ratio-level while the underlying latent ability is not.
Business and economic research
Examples include:
- Units sold
- Transaction value
- Gross revenue
- Advertising expenditure
- Production time
- Customer purchases
- Website conversions
Net income and account balance require special care because they may be negative.
Digital-behaviour research
Digital platforms generate ratio-like records such as:
- Session duration
- Click count
- Scroll distance
- File size
- Download count
- Response latency
- Number of interactions
Researchers must distinguish genuine user behaviour from automated traffic, duplicate events, idle browser time, and tracking failures.
Digital Research Tools and Artificial Intelligence
SPSS
SPSS generally groups interval and ratio variables under the measurement label Scale. Selecting Scale does not independently prove that a variable has a meaningful zero. The researcher must make that methodological judgment.
jamovi
jamovi commonly uses Continuous for variables that would traditionally be called interval or ratio. As with SPSS, this is a software classification that facilitates analysis rather than a theoretical verification.
R
R distinguishes storage structures such as numeric vectors and factors. A numeric vector can contain ratio measurements, interval measurements, codes, or identifiers. R does not infer the measurement theory from the values.
Python and pandas
A pandas column may have an integer or floating-point data type. That indicates how values are stored, not what they mean. A numeric data type does not establish a true zero or equal intervals.
Spreadsheets
Spreadsheet applications may detect numbers, dates, percentages, and currencies automatically. Researchers should verify:
- Units
- Decimal symbols
- Date conversion
- Missing-value codes
- Percentage formatting
- Hidden rounding
- Mixed-unit columns
Artificial intelligence
AI tools can assist with:
- Drafting a data dictionary
- Suggesting possible visualizations
- Generating analysis code
- Checking unit consistency
- Identifying suspicious zeros
- Explaining model output
AI should not classify a variable from its name alone. A column named time, score, income, or temperature is ambiguous without additional context.
A useful AI prompt should provide:
- The construct being measured
- The operational definition
- The unit
- The possible range
- The meaning of zero
- Whether negative values are possible
- How missing values are coded
- Whether the values are exact or grouped
- The study design
- The intended analysis
All AI-generated classifications and code should be checked against the research protocol and statistical assumptions.
How to Report a Ratio Variable
A methods section should identify more than the measurement level.
A strong description states:
- What was measured
- How it was operationalized
- The unit
- The instrument or data source
- The observation period
- The meaning of zero
- Precision or rounding
- Missing-data coding
- Any transformation
- The statistical summary and model used
Example methods statement
Task-completion duration was recorded automatically in seconds from presentation of the first item until submission of the final response. A value of zero indicated that no elapsed duration was recorded. Durations were treated as continuous ratio measurements. Because the distribution was right-skewed, results were summarized using the median and interquartile range and analyzed with a prespecified log-linked model.
Example results statement
Median completion time was 18.4 minutes (IQR = 14.2–25.7). The adjusted model estimated that the intervention group required 16% less time than the comparison group, with the uncertainty interval reported alongside the estimate.
The reporting language should match the model. A difference in means, ratio of geometric means, rate ratio, hazard ratio, and incidence-rate ratio are not interchangeable.
Conclusion
A ratio variable measures a numerical quantity using ordered values, equal intervals, and a meaningful zero. These properties support valid difference and proportional interpretations and distinguish ratio variables from interval variables.
Correct classification requires more than checking whether a column contains numbers or zero. Researchers must examine the construct, operational definition, unit, origin, collection method, and intended interpretation. They must also select statistical procedures based on the research design, distribution, dependence structure, and model assumptions—not the measurement label alone.
References
- Hand, D. J. (1996). Statistics and the theory of measurement. Journal of the Royal Statistical Society: Series A, 159(3), 445–473. https://doi.org/10.2307/2983326
- Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103(2684), 677–680. https://doi.org/10.1126/science.103.2684.677
- Velleman, P. F., & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), 65–72. https://doi.org/10.1080/00031305.1993.10475938
