Your course notes say “cardinal scale.” Your American statistics textbook has never heard of it. You search for the term and land on interval and ratio scales, or on a German glossary.
A cardinal scale is a level of measurement whose numbers carry real distances, so you can say not only which value is larger but by how much, and in English-language statistics you meet it split into two separate scales, the interval scale and the ratio scale. Cardinal scale and metric scale mean the same thing. This article shows you what to search for in English and how four questions settle the level.
📌 Key takeaways
- Cardinal scale and metric scale name the same level of measurement.
- In English that means the interval scale plus the ratio scale.
- Equal distances allow differences, means and standard deviations.
- Stevens named four scales in 1946, and cardinal was not one.
- A rating scale in a questionnaire is not automatically cardinal.
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What is a cardinal scale?
A cardinal scale is a level of measurement whose values sit at equally large distances from one another, so two numbers give you more than a running order: they give you a concrete difference. The second common name for it is metric scale.
The name comes from the cardinal number, the counting number one, two, three, which tells you how many there are. The ordinal number is first, second, third, and it only tells you who is ahead. That same distinction runs through the levels of measurement: the ordinal scale puts cases in order, and the cardinal scale measures the gap between them as well.
That places it above the two non-metric levels. The nominal scale only tells cases apart, by blood type or home state. The ordinal scale also ranks them, by letter grade or finishing position. Only on a cardinal scale does “more” turn into a number you can work with.
Nominal tells apart, ordinal ranks, cardinal measures.
The levels stack: each higher one does everything the level below it does, plus one more statement. Which level you are on decides which analysis is allowed, and our guide to levels of measurement makes that call variable by variable.
Why your statistics textbook never mentions it
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 two appears in his paper. The word “cardinal” does turn up once, in a different sense: he calls counting itself, the cardinal number we use for eggs, coins and apples, the foremost of the ratio scales. He does not mean a level of measurement by it.
So “cardinal scale” is not a measurement level of its own but a collective term, and it is not the standard term in English. It comes out of the German-language teaching tradition, where Kardinalskala covers the levels on which real distances are measured. American course material takes a different route. The statistical consulting group at UCLA works through categorical, ordinal and interval variables (UCLA OARC), and the statistics guide at National University sorts the four levels into categorical and continuous, or quantitative, variables (National University, Levels of Measurement). Neither one uses the word cardinal.
What to say and search for in English
If you need the idea in an English methods section or a search box, name the two scales: interval and ratio. If you want a single word for both, the usual choices are quantitative, continuous or metric. SPSS goes one step further and files interval and ratio together under one label, “Scale,” which is the closest thing to a cardinal scale you will meet in a US analysis package.
You will occasionally find “cardinal” used this way in English, mostly in translated material and in economics, where cardinal and ordinal utility are standard terms. The word is not wrong. It is simply not what an American reader, reviewer or search engine expects. Write “interval and ratio scales” and nobody has to guess.
Why sources disagree on what it covers
German-language sources also disagree among themselves, which is why looking the term up gives you two different answers. Nicola Döring, in the glossary to Forschungsmethoden und Evaluation (6th edition, 2023), puts the interval and ratio scales under the cardinal scale, with a practical reason: social science research statistics has no important procedure tailored specifically to ratio scales (Döring, glossary to the 6th edition, 2023). A handout on determining the level of measurement from the University of Graz lists three subtypes under “cardinal/metric” and adds the absolute scale (University of Graz, handout on levels of measurement).
Both counts are common, and for your analysis the count changes nothing. What matters is not whether your textbook names two subtypes or three, but whether your variable has a natural zero point. That is what decides whether you may report ratios and percentages.
The three types of cardinal scale
Inside the cardinal scale there is a further split, and it turns on the zero point and the unit. All three types have equally large distances, but they support different numbers of statements.
| Type | How you recognize it | Example |
|---|---|---|
| Interval scale | equal distances, zero set by convention | degrees Fahrenheit, year of birth |
| Ratio scale | plus a natural zero point | height, weight, income |
| Absolute scale | plus a natural unit | population count, number of responses |
On the interval scale the distances are equal, but the zero was put there by agreement. There are as many degrees between 60 and 65 °F as between 75 and 80 °F. Zero on that scale is a fixed point someone picked, not “no temperature,” which is why readings below zero exist and why “80 °F is twice as warm as 40 °F” does not work. Celsius is put together the same way, with its zero at the freezing point of water. Year of birth follows the same logic: a year is a position on an agreed timeline. The age you calculate from it is a duration measured from zero, and that makes it ratio scaled.
The ratio scale has a zero that means a genuine nothing. Nobody is shorter than zero centimeters or weighs less than zero ounces. That is why you are allowed to say here that one value is twice another.
The absolute scale is a special case of the ratio scale with the unit already fixed, usually a head count. You can measure temperature in Fahrenheit or Celsius, but residents only in people. Many textbooks do not list it separately, and nothing changes in your analysis if yours does not.
Is my variable cardinal? The test in four questions
Whether a variable is cardinal is settled faster by a list of questions than by a definition. The handout from the University of Graz sets out a chain you work through from the top down. The first two questions decide cardinal or not, and the last two assign the type.
- Can the values be put in order? If no, the variable is nominal and the test ends here.
- Is the distance between two values the same everywhere? If no, it is ordinal. If yes, it is cardinal.
- Is there a natural zero point? If no, it is an interval scale. If yes, go on to the last question.
- Is there a natural unit? If no, it is a ratio scale. If yes, an absolute scale.
Question two carries the actual decision, and it is the only one where you have to check yourself honestly. “The same everywhere” means demonstrably the same, not “looks about right.” With inches, dollars and seconds nobody argues. With answer options like “somewhat satisfied” and “very satisfied” it is an assumption.
Numbers in your data are not proof
The most common error is treating a variable as cardinal because the column happens to contain numbers. A ZIP code is a number and still nominal: 90210 is not “more” than 10001, and the distance between them means nothing. The same goes for jersey numbers, which were Stevens’ own example of numbers used as labels, and for coded answers where 1 stands for rent and 2 for own.
So check the meaning, not the format. The question is not “is there a number in this cell,” but “does the difference between these two numbers tell me anything.” If it does not, no software setting will make the mean meaningful.
The zero point only matters at step three
A variable is already cardinal as soon as the distances hold. Whether its zero is set by convention or natural only decides afterward which type of cardinal scale you have and whether ratios are allowed. Collapse the two questions into one and you will file interval variables like year of birth as non-metric and give away analyses you could have run.
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What you can calculate with cardinal data
On a cardinal scale every calculation that works with distances is allowed. Anything beyond that depends on the zero point and therefore does not apply to all three types equally. The table keeps the two groups apart.
| Calculation | Cardinal scale | Condition |
|---|---|---|
| Mode and median | Yes | already fine from nominal or ordinal up |
| Difference between two values | Yes | equal distances are enough |
| Mean and standard deviation | Yes | equal distances are enough |
| Pearson correlation | Yes | works with deviations from the mean |
| Ratios, so “twice as much” | Only sometimes | needs a natural zero point |
| Percentage change | Only sometimes | a ratio in different clothing |
| Coefficient of variation | Only sometimes | divides the spread by the mean |
The rows marked “only sometimes” hold on the ratio and absolute scales and not on the interval scale. One income may be called twice another, a temperature in degrees Fahrenheit may not. In practice that means checking before every percentage figure whether the zero of your variable stands for a genuine nothing.
Procedures that rest on the mean and the spread assume interval level at minimum, by common convention. That covers the t test, analysis of variance and linear regression, all of them tools of inferential statistics, which reasons from your sample to the population. For ordinal variables you keep the median, the quartiles and rank correlations. Like the mean and the standard deviation, those figures describe your own data first of all.
Cardinal scale in a questionnaire: the question decides
Whether you end up with cardinal data is settled when you write the questionnaire, not when you analyze it. The same variable comes back at a different level depending on the answer format, and afterward it can only be converted downward, from fine to coarse.
Age is the clearest case. Ask “How old are you?” with an open number field and you get ratio scaled data, from which you can form a mean, a range and any age groups you like. Ask instead for “18 to 29,” “30 to 49” and “50 or older” and you get ordinal data, and the individual years are gone for good. You can turn a number into groups at any time. You can never turn groups back into the number.
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Is a rating scale cardinal?
A rating scale in a questionnaire is strictly speaking ordinal and therefore not cardinal. There is no way to demonstrate that the step from “somewhat 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 combine several items into a sum score, as the classic Likert scale intends. Averaged over many items, the total moves closer to a metric level, which is exactly why it is usually analyzed as one. It stays an assumption all the same.
Put the assumption in your methods section
This leniency is nothing new. Stevens called averaging on ordinal scales “illegal statisticizing” in 1946, against which, he wrote, a kind of pragmatic sanction can be invoked, because it often leads to usable results. In the same passage he names the price: on an ordinal scale, the mean and the standard deviation are in error to the degree that the successive intervals are unequal. He wrote nothing at all about questionnaires, but the direction follows anyway. The more carefully the steps are designed, fully labeled and evenly spaced, the smaller the error.
So if you average agreement ratings, the assumption belongs in the methods section: “The response options were treated as equidistant.” Without that sentence, your analysis claims a level of measurement your data does not supply.
Common mistakes around the cardinal scale
Most errors do not happen during the calculation but one step earlier: a variable is filed at the wrong level, and everything after that follows the decision. It rarely shows, because a wrongly calculated mean looks exactly like a correct one.
Two of the three cases below start when the questionnaire is written, the third only 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 brackets
“How many surveys did you analyze 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. An open number field costs respondents almost no extra time and keeps your mean.
Reporting percentages on interval values
A sentence like “the temperature rose by 18 percent” sounds harmless and is not. A percentage change divides the difference by the starting value, which makes it a ratio. Going from 50 to 59 °F is 18 percent, the same warming in Celsius, 10 to 15 degrees, is 50 percent, and in Kelvin it is 1.8 percent. Three numbers for one and the same warming is a reliable sign that the percentage measures nothing here. Report the difference in degrees instead.
Treating coded answers like measurements
Statistics packages will happily compute with anything stored as a number. A mean of 1.4 for a variable where 1 stands for “rent” and 2 for “own” is formally calculated and substantively meaningless. So assign the level of measurement for every variable right after the export, before you start the first analysis.
Watch out
“Don’t know” and “Prefer not to answer” are not scale points but opt-out categories. They belong at the end of the answer list, set apart visually, and coded as missing values in the analysis. Sitting in the middle of the scale with the middle number, they pull your mean toward the center and fake balance. Not to be confused with a genuine midpoint such as “neither agree nor disagree,” which is a substantive answer, rightly sits in the middle and counts.
Conclusion
The cardinal scale is the level where statistics gets comfortable: differences, means and standard deviations are all allowed, and for most analyses that is plenty. Whether your textbook counts two subtypes or three you can safely leave open. The more useful question is the natural zero point, because that is what decides whether you may report ratios and percentages. In English, say interval and ratio scale and everyone will know what you mean. And the decision that matters most comes earlier anyway, when you word the question.
Where to go next
- Want to see all the levels together? Determine the level of measurement
- Your variable has no real zero point? Interval scale
- Want to report ratios? Ratio scale
- Counting units? Absolute scale
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