Quick test. Two sentences about the same thermometer: “Yesterday the high was 40 °F, today it is 80 °F, so the day got 40 degrees warmer.” And: “Today is twice as warm as yesterday.” The first sentence is correct, the second one is nonsense.
The reason is called the interval scale. An interval scale is a level of measurement where the distances between the values are equally wide everywhere, while the zero point was set by agreement and marks no real absence. Differences and averages are fair game, ratios such as “twice as much” are not.
📌 The key points at a glance
- An interval scale has equal distances but no natural zero point.
- Degrees Fahrenheit, calendar years and IQ scores are interval scaled.
- Allowed: difference, mean, standard deviation and Pearson correlation.
- Not allowed: ratios, percentage change and the coefficient of variation.
- Rating scales in a questionnaire are strictly speaking ordinal.
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What is an interval scale?
An interval scale is a metric level of measurement where the distances between neighboring values are equally wide everywhere, while the zero point was fixed by convention and therefore does not mark a real absence. The name comes from the Latin intervallum, meaning gap or distance.
Compared with the two weaker levels, an interval scale does one thing more. The scale tells categories apart like the nominal scale and puts them in order like the ordinal scale, and it also states how far apart two values are.
The four levels of measurement go back to the psychologist S. S. Stevens, who described them in 1946 in the journal Science while he directed the Psycho-Acoustic Laboratory at Harvard. About the zero point of an interval scale Stevens notes expressly that it is “a matter of convention or convenience” (Stevens 1946). Everything else follows from that.
On an interval scale you can compare distances. Ratios you cannot.
Where the interval scale sits among the other three levels is covered in our guide to levels of measurement.
Examples of an interval scale
Interval scaled variables have a fixed unit but a zero point that people agreed on: degrees Fahrenheit, calendar years and test scores such as the IQ. The overview shows where each set zero comes from and which sentence falls apart.
| Example | Where the zero comes from | What does not follow |
|---|---|---|
| Degrees Fahrenheit | Set by definition, well below freezing | “80 °F is twice as warm as 40 °F” |
| Calendar year | Starting point of the calendar | “The year 2000 is twice as much as 1000” |
| IQ score | Population mean, set at 100 | “IQ 140 is twice as smart as IQ 70” |
A thermometer makes the point fastest. The gap between 50 °F and 60 °F is ten degrees, and the gap between 80 °F and 90 °F is ten degrees as well. The zero, however, does not mean “no temperature”: water freezes at 32 °F, which puts the Fahrenheit zero a full 32 degrees below freezing.
The missing zero shows up the moment somebody doubles a number. Converted into Celsius, 80 °F and 40 °F become roughly 26.7 and 4.4 degrees, a ratio of six to one instead of two to one, and a ratio that changes with the unit was never a property of the weather. Calendar years fail the same test, and IQ scores center on 100 by construction, which is why 140 is not “twice as intelligent” as 70.
Strictly speaking, the IQ carries the same caveat as a rating scale: the equal steps come from the way the scale was standardized, not from the measurement itself. Stevens still counted intelligence tests among the ordinal scales that aim at interval level.
Interval scale vs. ratio scale: how to tell them apart
Interval scale and ratio scale differ in exactly one place, namely at the zero point. A ratio scale has a natural zero that marks a real absence, an interval scale does not. Equal distances and a fixed order both of them share.
The same phenomenon can therefore sit on two levels depending on the unit. Temperature in degrees Fahrenheit is interval scaled, and so is temperature in degrees Celsius, while the same temperature in kelvin is ratio scaled, because 0 K is the lowest value that exists. The full comparison is in our guide to the ratio scale.
The zero point decides, not the minus sign
A popular shortcut says you recognize an interval scale by the fact that negative values are possible. Temperature happens to fit, but the rule fails as a test: the IQ is interval scaled and never goes negative, because the scale is built so that scores cluster around 100.
A date and a duration sit on different levels
A date is interval scaled, while a duration is ratio scaled. Stevens describes both side by side in 1946, and about periods of time he writes that “one period may be correctly defined as double another”.
For your own questionnaire the wording of a question fixes the level of measurement. Ask for the birth year and you collect interval data. Ask for the age in years and you collect ratio data, because zero years of lived time is a real absence.
What you can calculate with interval data
Interval data allows every procedure that works with distances and rules out every procedure that needs the zero point. Stevens summarized that assignment in a table in 1946 which still appears in textbooks today.
| Calculation | Interval scale | Reason |
|---|---|---|
| Mode and median | ✓ | Mode from nominal, median from ordinal |
| Difference between two values | ✓ | distances are equally wide everywhere |
| Mean and standard deviation | ✓ | need equal distances, not a zero point |
| Pearson correlation | ✓ | works with deviations from the mean |
| Ratio, meaning “twice as much” | ✕ | meaningless without a true zero |
| Percentage change | ✕ | a ratio in different notation |
| Coefficient of variation | ✕ | divides the spread by the mean |
| Geometric mean | ✕ | multiplies the values with each other |
Procedures built on the mean and the spread therefore require at least an interval scale, among them the t test, the analysis of variance and linear regression. For ordinal variables, the median, quartiles and rank correlations remain. Stevens himself still placed rank correlation on the interval scale in 1946; today it counts as the standard method for ordinal data.
Conversions have to stay linear
An interval scale stays an interval scale as long as you multiply every value by the same positive number and add the same number. Converting Fahrenheit into Celsius works exactly like that: subtract 32, then divide by 1.8, and the other direction multiplies by 1.8 and adds 32. Squaring or taking a logarithm destroys the equal distances.
One trap follows from this. A difference of 18 °F is not a difference of 18 °C but a difference of 10 °C, because the factor 1.8 acts on the distances while the added 32 only shifts the zero.
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Is a rating scale in a questionnaire an interval scale?
A rating scale in a questionnaire is strictly speaking ordinal, because nobody can show that the step from “somewhat agree” to “agree” is as wide as the step from “agree” to “strongly agree”. In practice such scales are nevertheless analyzed like interval scales all the time.
The UCLA Office of Advanced Research Computing, Statistical Methods and Data Analytics puts it in one sentence: an interval variable differs from an ordinal one in that “the intervals between the values of the numerical variable are equally spaced”. A five point Likert item is classified there as ordinal, and the same page names the stake: “an average requires a variable to be numerical” (UCLA, Statistical Methods and Data Analytics).
The distance between theory and practice is no modern invention. Stevens noted back in 1946 that a “kind of pragmatic sanction” can be invoked for computing means on ordinal data, but he also named the condition: the error grows to the degree that the distances are unequal.
The GPA makes the same assumption every semester
Grade point averages rest on that assumption too, and every graduate student carries one around. Letter grades from each course are translated into grade points and then averaged, although the letters underneath express a rank order rather than measured distances. Nobody has shown that the step from one letter to the next is equally wide.
The point is not that a GPA is wrong. A GPA is useful, comparable and thoroughly established. Your rating scale simply sits in the identical situation: a defensible convention, not a measured property.
Five choices that make equal distances more plausible
Scale design decides how well the assumption holds, and none of the choices below turns an ordinal scale into an interval scale.
- Five to seven levels, fine enough but still easy to grasp.
- Equally many positive and negative levels.
- Every level labeled, not only the two ends.
- No negative numbers as markers, they push response behavior upward.
- Equal visual spacing between all levels.
The hardest part remains the choice of words, because two labels that look neighboring on paper can sit far apart in a reader’s head. Scale building is covered in our guides to the rating scale and the Likert scale.
⚠️ Careful
Whenever you compute a mean from agreement levels, write the assumption into your methods section: “The response levels were treated as equidistant.” Without that sentence your analysis claims a level of measurement that the data does not support.
Using an interval scale in a questionnaire: five steps
Whether you end up with interval data is decided while you build the questionnaire, not while you analyze it. Afterwards a variable can only be converted downward, from fine to coarse.
- Define the variable. Write down in one sentence what is being measured, for example satisfaction with the dining hall.
- Choose the response format. An open number field, a graded scale or categories, each option delivers a different level.
- Build the levels evenly. Five to seven levels, symmetrical, without negative numbers, with equal visual spacing.
- Test the scale. Ask five people from your target group to put the levels into their own words.
- Settle the analysis in advance. Decide now whether you will report means or frequencies.
Step four gets skipped most often and pays off most. When two out of five test readers sort “slightly agree” differently than you do, the assumption of equal distances is gone before the first real response arrives. With empirio.ai, an online survey tool from Germany, you change the levels after a pretest in minutes.
💡 Tip
Collect age, income and duration as an open number instead of in brackets. From an open number you can always build brackets later, from brackets you never get the number back. A respondent who skips the dollar amount costs one empty field, a bracketed income costs the variable.
Common mistakes with the interval scale
Most mistakes around the interval scale happen during the classification rather than the calculation, and all three below are easier to catch yourself than to hear from your advisor.
Treating minus signs as the giveaway
Negative values are a hint, not a proof. The IQ is interval scaled and never turns negative, because the scale is built around 100. The other way round, temperature in degrees Fahrenheit stays interval scaled even when a summer data set holds no reading below zero at all. Check the zero itself: does it stand for “nothing present” or for an agreed starting point?
Mixing up a date and a duration
A birth year is interval scaled, an age in years is ratio scaled. Anybody who treats both variables as the same thing ends up with sentences like “the participants are on average twice as old as their birth year suggests”. Ask for the age from the start when you want to report ratios.
Computing percentages on interval values
A sentence such as “the temperature rose by 50 percent” sounds harmless and is not. A percentage change divides the difference by the starting value and is therefore a ratio. Going from 50 °F to 68 °F is a rise of 36 percent, while the identical warming written in Celsius, from 10 °C to 20 °C, is a rise of 100 percent. Report the difference in degrees instead.
Conclusion
The interval scale is the level at which statistics gets comfortable: mean, standard deviation and correlation are all allowed, and for most analyses in a thesis that is plenty. The price is the set zero point, and it costs exactly one kind of statement, namely the ratio.
Where to go next
- Want to see how the four levels fit together? Levels of measurement
- Missing the boundary at the top? Ratio scale
- Building a response scale right now? Rating scale
- Working with agreement levels? Likert scale
Want to try your response scales right away?
With empirio.ai you build a scale in a few minutes, send it to five test readers and adjust the levels before the real data collection starts.