You have run your survey, you want to report a mean, and your course notes call the scale “cardinal”. You look the term up in English and it is barely there. Meanwhile the textbook says “metric”, and nobody tells you what you are actually allowed to calculate.
A cardinal scale is a level of measurement whose values sit at equal, meaningful distances from one another, so you can say by how much two values differ and not just which one is larger. In English this covers the interval and ratio scales.
📌 Key takeaways
- Cardinal scale and metric scale mean the same level of measurement.
- In English, say interval scale and ratio scale instead.
- Equal gaps allow differences, means and standard deviations.
- Ratios and percentages need a true zero point.
- Stevens named four scales in 1946; cardinal was not one.
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What is a cardinal scale?
A cardinal scale is a level of measurement whose values are spaced at equal intervals, so that two numbers tell you a genuine difference and not merely an order. The second common name for exactly the same thing is metric scale.
The name comes from the cardinal numbers, the counting numbers one, two, three, which answer the question “how many”. Ordinal numbers, first, second, third, only say who comes ahead of whom. That distinction runs straight through the levels of measurement: the ordinal scale puts values in order, and a cardinal scale measures the distance between them as well.
It therefore sits above the two non-metric levels. The nominal scale only tells values apart, by blood group or by town. The ordinal scale sorts them too, by GCSE grade or finishing position. On a cardinal scale, “more” finally becomes a number you can work with.
Nominal names, ordinal orders, cardinal measures.
The levels stack on top of one another: each higher one does everything the level below does and adds one more statement. Which level you have decides which analysis is permissible, and our guide to levels of measurement works through that decision variable by variable.
Why English textbooks do not use the term
The levels of measurement go back to the psychologist S. S. Stevens, who set them out in the journal Science in 1946. He distinguishes four scales there: nominal, ordinal, interval and ratio (Stevens 1946). No fifth term for the upper levels appears in his paper. The word “cardinal” does turn up once, in a different sense: he calls counting itself, the cardinal number, the foremost of the ratio scales. A separate level of measurement is not what he means by it.
So “cardinal scale” is not the standard term in English-language statistics. It is a translation of the German Kardinalskala, a collective term that took hold in German-speaking teaching for the levels on which real distances are measured. You will meet it in English now and then, but British statistics guides do not use it as a heading. Oxford Brookes University, for instance, lists the four Stevens levels and groups interval and ratio data under the label “scale data” (Oxford Brookes, Types of data).
Two readings in German-language sources
German sources also disagree about what the term covers, which is why looking it up gives you two different answers. Nicola Döring, in the glossary to Forschungsmethoden und Evaluation (6th edition 2023), counts the interval and the ratio scale (Döring, glossary to the 6th edition). A University of Graz handout on working out the level of measurement lists three, adding the absolute scale (Graz handout on levels of measurement). Both readings are common, and for your analysis the count changes nothing.
What to say and search for in English
To find the right literature, or to write a methods section a British marker will recognise, swap the collective term for the individual scales. Everything on this page still applies; only the label changes.
| If your notes say | In English, use |
|---|---|
| Kardinalskala (cardinal scale) | interval scale and ratio scale |
| Metrische Skala (metric scale) | scale, continuous or quantitative data |
| Intervallskala | interval scale |
| Verhältnisskala | ratio scale |
| Absolutskala | absolute scale, rarely listed separately |
One thing is the same in both languages, and it is the only question that decides your analysis: does the variable have a true zero? That settles whether you may report ratios and percentages.
The three types of cardinal scale
Within the cardinal scale there is a further split, and it turns on the zero point and the unit. All three types have equal gaps between their values. They differ in how many statements those values will support.
| Type | How you recognise it | Example |
|---|---|---|
| Interval scale | equal gaps, zero point set by convention | degrees Celsius, year of birth |
| Ratio scale | plus a true zero point | height, weight, income |
| Absolute scale | plus a natural unit | population, number of replies |
On the interval scale the gaps are equal, but the zero has been placed there by agreement. There are as many degrees between 15 and 20 Celsius as between 25 and 30. Zero stands for the freezing point of water, not for “no temperature”, which is why minus figures exist and why “20 degrees is twice as warm as 10 degrees” does not work. A year of birth is the same construction, a point on an agreed timeline. The age you calculate from it is a duration measured from zero, and therefore ratio scaled.
The ratio scale, by contrast, has a zero that means a genuine nothing. Nobody is shorter than zero centimetres or weighs less than zero grams. That is why you may say here that one value is twice another.
The absolute scale is a special case of the ratio scale with a unit that is fixed for you, usually a count. Temperature can be measured in Celsius or in Fahrenheit, whereas people can only be counted in people. English-language guides rarely give it a heading of its own: Oxford Brookes and the University of Nottingham both stop at the four Stevens levels.
Is my variable cardinal? The test in four questions
A checklist settles this faster than a definition does. The University of Graz handout sets out a chain you work through from top to bottom. The first two questions decide cardinal or not, and the last two sort out which type you have.
- Can the values be put in order? If not, the variable is nominal and the test stops here.
- Is the gap between two values the same everywhere? If not, it is ordinal. If yes, it is cardinal.
- Is there a natural zero point? If not, you have an interval scale. If yes, carry on to the last question.
- Is there a natural unit? If not, it is a ratio scale. If yes, it is an absolute scale.
Question two carries the real decision, and it is the only one where you have to be honest with yourself. “The same everywhere” means demonstrably the same, not “looks about right”. With centimetres, pounds and seconds nobody argues. With answer options such as “fairly satisfied” and “very satisfied” it is an assumption.
Numbers in your dataset prove nothing
The commonest error is to treat a variable as cardinal because the column is full of digits. A postcode carries digits and is still nominal: SW1A 1AA is not “more” than EC1A 1BB, and the gap between the two measures nothing at all. GCSE grades make the same point one level up. The grades run 9 to 1 and they are genuinely ordered, but nobody can show that the step from 9 to 8 is the same size as the step from 4 to 3. They are ordinal, digits and all. Shirt numbers in football, and coded answers where 1 stands for renting and 2 for owning, belong in the same box.
So check the meaning, not the format. The question is not “is there a number here” but “does the difference between these two numbers say anything”.
The zero point only matters at the second step
A variable is already cardinal as soon as the gaps hold. Whether the zero is agreed or natural then decides only which type of cardinal scale you have, and whether ratios are allowed. Pull the two questions together and you will write off interval variables such as the year of birth as non-metric, giving away analyses you could have run.
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What you can calculate with cardinal data
On a cardinal scale every method that works with distances is open to you. Anything beyond that depends on the zero point, so it does not apply to all three types equally. The table keeps the two apart.
| Method | Cardinal scale | Condition |
|---|---|---|
| Mode and median | Yes | available from nominal or ordinal upwards |
| Difference between two values | Yes | equal gaps are enough |
| Mean and standard deviation | Yes | equal gaps are enough |
| Pearson correlation | Yes | works with deviations from the mean |
| Ratios, “twice as much” | Only sometimes | needs a true zero point |
| Percentage change | Only sometimes | a ratio written another way |
| Coefficient of variation | Only sometimes | divides the spread by the mean |
The “only sometimes” rows hold on the ratio and absolute scales, and not on the interval scale. One income may be called twice another, a temperature in degrees Celsius may not. In practice that means checking, before every percentage figure, whether zero on your variable stands for a genuine nothing. The University of Nottingham puts the same rule plainly for interval data, noting that multiplying and dividing are out because there is no absolute zero (University of Nottingham, levels of measurement).
Methods that rest on the mean and the spread assume interval level at least, by common convention. That includes the t-test, analysis of variance and linear regression, all of them part of inferential statistics, which reasons from your sample to the population. For ordinal variables you are left with the median, quartiles and rank correlations. Like the mean and the standard deviation, those figures describe your own data first of all.
Cardinal scales in a questionnaire: the question decides
Whether you end up with cardinal data is settled when you write the questionnaire, not when you analyse it. The same variable gives you a different level depending on the answer format, and afterwards it can only be converted downwards, from fine to coarse.
Age is the clearest case. Ask “How old are you?” with a free number field and you get ratio scaled data, from which you can form a mean, a range and age groups. Ask instead for “18 to 29”, “30 to 49” and “50 and over” and you get ordinal data, with the individual years gone for good. You can always turn numbers into groups. You can never turn groups back into numbers.
Tip
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Is a rating scale a cardinal scale?
A rating scale in a questionnaire is, strictly speaking, ordinal, and therefore not cardinal. There is no way to show that the step from “tend to agree” to “agree” is the same size as the step from “agree” to “strongly agree”. The same holds for a single item of a Likert scale. It looks different once you add several items into a sum score, as the classic Likert design intends. Averaged over many items, the total moves closer to a metric level, which is exactly why it is usually analysed as one. It stays an assumption all the same.
This leniency is not a modern invention. In 1946 Stevens called means on ordinal scales “illegal statisticizing”, against which, he wrote, “a kind of pragmatic sanction” can be invoked, because the results are often useful in practice. In the same passage he names the price: on an ordinal scale the mean and the standard deviation are wrong to exactly the degree that the successive intervals are unequal. Stevens wrote nothing about questionnaires, but the direction follows anyway. The more carefully your steps are spaced and labelled, the smaller the error.
If you do average agreement ratings, the assumption belongs in your methods section: “The response options were treated as equidistant.” Without that sentence, your analysis claims a level of measurement your data cannot supply.
Common mistakes with cardinal scales
Most errors happen one step before the arithmetic. A variable is filed under the wrong level, and everything after that follows the same wrong turn. It rarely shows, because a badly founded mean looks exactly like a sound one.
Two of the three cases below start in the questionnaire, while the third only appears in the analysis software. All three are avoided by a single habit, which is to assign every variable its level of measurement before you apply the first number to it.
Asking for counts in bands
“How many surveys did you analyse last year?” with the options “none”, “1 to 5” and “more than 5” returns ordinal data, even though the variable itself is absolute scaled. The question lowers the level of measurement, and it does so irreversibly. A free number field costs your respondents barely any extra time and keeps your mean.
Percentages on interval data
A sentence such as “the temperature rose by 50 per cent” sounds harmless and is not. A percentage change divides the difference by the starting value, which makes it a ratio. From 10 to 15 degrees Celsius that is 50 per cent, the same warming comes to 18 per cent in Fahrenheit and to 1.8 per cent in Kelvin. Three numbers for one warming is a reliable sign that the percentage is measuring nothing here. Report the difference in degrees instead.
Treating coded answers as measurements
Statistics packages will happily calculate with anything stored as a number. A mean of 1.4 for a variable where 1 means renting and 2 means owning is formally correct and substantively meaningless. So set the level of measurement for every variable straight after the export, before you run the first analysis.
Careful
“Don’t know” and “prefer not to say” are not scale points, they are opt-out categories. They belong at the end of the answer list, visually separated, and they should be coded as missing values in the analysis. Sitting in the middle of the scale with the middle number, they pull your mean towards the centre and fake a balanced result. Do not confuse them with a genuine midpoint such as “neither agree nor disagree”, which is a substantive answer, rightly sits in the middle and is counted.
Conclusion
The cardinal scale is the level at which statistics becomes comfortable. Differences, means and standard deviations are all permitted, and for most analyses that is plenty. Whether your course counts two types or three, you can safely leave open. The question that matters is the true zero, because it decides whether you may report ratios and percentages. And if you are writing in English, name the interval and ratio scales directly, so your reader knows exactly what you mean. The most important decision comes earlier anyway, in how you word the question.
Where to go next
- Want to see all the levels together? Determining the level of measurement
- No true zero in your variable? Interval scale
- Want to report ratios? Ratio scale
- Counting whole units? Absolute scale
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In empirio.ai you can set up a number field or a graded scale in a few minutes, and after a handful of test responses you will see whether the analysis gives you what you want to report.