Levels of Measurement: Nominal, Ordinal, Interval and Ratio Explained

What is a level of measurement?

The level of measurement (also called scale of measurement or measurement scale) describes how much information the numbers of a variable carry. The idea goes back to the psychologist S. S. Stevens, who in 1946 distinguished four levels: nominal, ordinal, interval and ratio.

Why does this matter? Numbers in a data set do not always behave like numbers. If you code gender as 1, 2 and 3, the “mean gender” of 1.8 is meaningless. If you code satisfaction as 1 = very dissatisfied to 5 = very satisfied, you know that 4 is more than 3, but not whether the step from 3 to 4 is as large as the step from 4 to 5. The level of measurement tells you which calculations are legitimate, which charts make sense and which statistical test you can run. That is why it is the first thing to check before any analysis and the first row in most test-selection charts.

The four levels at a glance

Each level includes all properties of the levels below it. A ratio variable can always be treated as interval, ordinal or nominal, but never the other way around.

Nominal and ordinal: categorical data

Nominal variables sort cases into categories without any order. The numbers you assign are only labels: coding psychology as 1 and law as 2 does not make law “more” than psychology. A special case is the dichotomous (binary) variable with exactly two categories, such as passed/failed. You describe nominal data with frequencies, percentages and the mode, and you test relationships between two nominal variables with the chi-square test.

Ordinal variables have a meaningful order, but the distances between categories are not necessarily equal. In a race, the gap between first and second place can be one second or one minute. Letter grades (A, B, C) and single questionnaire items with answer options from “strongly disagree” to “strongly agree” are classic ordinal variables. Here the median is the right measure of central tendency, and rank-based tests such as the Mann-Whitney U test or Spearman’s rank correlation are the safe choice.

Interval and ratio: metric data

Interval variables have equal distances between values, so differences are meaningful: the step from 10 °C to 20 °C is as large as the step from 20 °C to 30 °C. But the zero point is arbitrary. 0 °C is not “no temperature”, so 20 °C is not twice as warm as 10 °C. In kelvin, which has a true zero, the ratio is only 293.15 / 283.15 = 1.035.

Ratio variables add a true, meaningful zero. Zero hours of study really means no studying, so 10 hours is twice as much as 5 hours. Most physical and count variables are ratio: age, income, distance, reaction time, number of errors.

For almost all statistical tests the difference between interval and ratio does not matter. Both allow means, standard deviations and parametric tests. That is why textbooks and statistics software often merge them into one category called metric, continuous or simply scale.

• Continuous variables can take any value in a range (height, time).
• Discrete variables take countable values (number of siblings, number of exam attempts). Both can be ratio.
• Only for ratio data can you compute the coefficient of variation (CV = SD / M) or say “twice as much”.

The Likert scale debate: ordinal or interval?

Hardly any question causes more confusion in student theses. The short answer: a single Likert item is ordinal; a scale score made from several items is usually treated as interval.

When you average or sum, say, five items measuring exam anxiety, the resulting score takes many values, its distribution is often close to normal, and simulation studies show that parametric tests work well with such scores. In psychology, education and the social sciences this is standard practice. Strict measurement theorists disagree, so you should state your decision openly in your methods section.

• Single item with 4 to 5 options: report median and frequencies; compare groups with the Mann-Whitney U test.
• Scale score from several items with acceptable reliability (for example Cronbach’s alpha ≥ .70): report M and SD; use t-tests, ANOVA, correlation, regression.
• Item with 7 or more options and a roughly symmetric distribution: many researchers treat it as approximately interval; justify this choice.

Which statistics and tests fit which level?

Use the level of your dependent variable (the outcome) to narrow down your choice of test. You can always move down to a lower level, for example by analyzing ranks instead of raw scores, but you lose information and usually some statistical power.

Common mistakes and how to avoid them

Most errors happen when numeric codes are confused with real quantities. Statistics software will happily compute a mean of any column of numbers, so the responsibility is yours.

• Computing a mean for nominal codes such as “1 = biology, 2 = chemistry, 3 = physics”.
• Turning a ratio variable into categories (for example age into “young/old”) without a good reason. You throw away information and reduce power.
• Calling a single 5-point item “metric” without discussion.
• Forgetting that percentages of nominal categories need a chi-square test, not a t-test.
• Reporting “twice as satisfied” for a rating scale: ratio statements need a true zero.