Nominal, Ordinal, Interval and Ratio Scales

The four levels of measurement: nominal, ordinal, interval, ratio

Every answer you collect sits at one of four levels. The level is not about the numbers in your spreadsheet; it is about which comparisons between them mean something.

The scheme comes from a single 1946 paper. The Harvard psychologist S. S. Stevens coined the terms nominal, ordinal, interval and ratio, and classified statistical procedures by the scales for which they were, in his word, permissible. Nearly every textbook since has used his four names.

One question separates them: what survives if you change the numbers you happened to assign? If only the identities survive, the level is nominal. If the order survives, ordinal. If the size of the differences survives, interval. If the ratios survive, ratio. Each level keeps everything below it.

Nominal and ratio are the easy ends. Nominal is a list of labels, so you count them and report the mode: the average of Chrome and Firefox is not Safari. Ratio answers are counts and amounts, and because zero means none, every statistic is available.

Ordinal adds order and nothing else. Education level, seniority, medal position, a satisfaction rating, a bronze to platinum tier. The trap is that ranks look like measurements. Coding very dissatisfied as 1 and very satisfied as 5 does not create equal steps; it records the order. The median and the frequency distribution are the honest summaries.

Interval is the level people get wrong, because the definition sounds like it is about the numbers when it is really about the zero. An interval scale has equal, meaningful gaps, so subtraction works. What it lacks is a true zero, a point meaning none of the quantity. Celsius has a zero, but it is the freezing point of water. As the University of Nottingham puts it, the arbitrary 0C does not mean no temperature.

Is a Likert scale ordinal or interval?

This is the argument the topic is actually about, and most pages dodge it. The answer turns on a distinction almost nobody makes: a single question is not a scale.

One question is ordinal. A single item running from strongly disagree to strongly agree has a clear order and unknown gaps. Nothing guarantees that agree to strongly agree is the same distance as neutral to agree. Susan Jamieson, writing in Medical Education, puts the objection in a form that is hard to argue with: the average of fair and good is not fair and a half, and that stays true after you assign the integers 3 and 4.

Her reading of the methods textbooks is blunt: the categories have a rank order, but the intervals between them cannot be presumed equal, so the median or mode is the measure of central tendency, frequencies describe the spread, and the tests are non-parametric ones such as chi-square, Spearman rho or Mann-Whitney U.

Several items combined into one score are usually treated as interval. That is what Rensis Likert built in 1932. As Boone and Boone describe it, he used a series of questions and analyzed the composite score representing the attitude; he did not analyze individual questions. Their working rule: four or more items averaged into one variable make a Likert scale, and that composite is analyzed at the interval level, with means, standard deviations, Pearson correlation, t-tests, ANOVA and regression.

So the two words people use interchangeably do different jobs. A Likert-type item is one question and is ordinal. A Likert scale is several items about one construct, averaged, and is conventionally analyzed as interval.

What follows for a real poll is short. Medians and frequency distributions are always safe. A mean of one rating question is shaky, and publishing 3.7 out of 5 from a single question is the most common mistake in survey reporting. It asserts an equal spacing the question never established, and it hides the shape of the answers: a room split between people who love the thing and people who hate it returns the same 3.0 as a room full of shrugs.

The other side is real. Sullivan and Artino, reviewing the evidence in the Journal of Graduate Medical Education, report that parametric tests can be used with ordinal data such as data from Likert scales and are generally more robust than non-parametric tests, while still recommending the median for Likert data. Treating a composite scale as interval is ordinary practice; treating one question as interval is where the case gets thin.

Which level of measurement is this variable? Forty examples and a quick quiz

Most people arrive here holding one specific variable and a question. Find it below, or test yourself on a random one.

The four level model is contested, and that is worth knowing

Stevens's taxonomy is a teaching tool with a serious critique attached. Knowing the critique stops you applying the rules mechanically.

In 1993 the statisticians Paul Velleman and Leland Wilkinson published a commentary in The American Statistician titled 'Nominal, Ordinal, Interval, and Ratio Typologies Are Misleading'. Their charge: using Stevens's categories to select or recommend a statistical method is inappropriate and can often be wrong, because the categories do not describe the attributes of real data that are essential to good statistical analysis.

The sharpest part is that the level is not purely a property of the data. In their words, the scale type of data may be determined in part by the questions we ask of the data or the purposes for which we intend it. Their example is a 1953 note by Frederic Lord about football jersey numbers: nominal labels by any reading, until somebody asks whether the freshmen were issued lower numbers than the sophomores, at which point arithmetic on them answers a real question.

Their conclusion inverts how the four levels are usually taught. Software built on the typology, they write, asserts that the scale type is evident from the data independent of the questions asked of the data. They thus restrict the questions that may be asked of the data. Such restrictions lead to bad data analysis and bad science.

The position that survives is practical. Use the four levels as a checklist for what a number means, not a gate that forbids an analysis. Nobody should average browser choices. But if your question makes an ordinal mean meaningful, and you say so and show the distribution, you are ahead of somebody who followed the rule blindly.

Every claim above is tied to a checked source in the list below. Where good sources disagree, as they do about single item means, both positions are set out.