What are the four levels of measurement — nominal, ordinal, interval and ratio?

## What a level of measurement is A data column is not just a set of values; it is a set of values plus the rules governing which operations on those values mean anything. The four-level taxonomy — nominal, ordinal, interval, ratio — proposed by S. S. Stevens in 1946 is the standard shorthand for those rules. It is the difference between a number as a *measurement* and a number as a *code*. ## Nominal Values are labels that distinguish categories with no inherent order: product category, browser, country, blood type. The only relation defined between two values is equality — they are the same category or they are not. - Legitimate: counts, proportions, a frequency table, the most common category. - Illegitimate: any arithmetic. Averaging seven product categories coded `1..7` to get `3.4` produces a number whose referent does not exist — there is no category three-point-four. Nominal columns are very often stored as numbers: jersey numbers, zip codes, store IDs, department codes. Storage type is a fact about the file; scale is a fact about the meaning. `12` on a jersey is a name, not a quantity, and sorting players by it conveys nothing. ## Ordinal Values are ranked, but the distance between adjacent values is unknown and not assumed equal: education level, injury severity, medal placement, T-shirt sizes, survey response options. - Legitimate: order comparisons, counts and proportions, percentiles, the median — all of which need only ordering. - Illegitimate: sums, means, and any statement about how far apart two values are. Second place beat third place, but the ranks do not say by how much: the top two may have finished a hair apart while third trailed by a mile. ## Interval Ordered, with equal and meaningful distances between values, but the zero point is a convention rather than the absence of the quantity: Celsius temperature, calendar year, clock time. - Legitimate: everything ordinal permits, plus differences, the arithmetic mean, and the standard deviation. - Illegitimate: ratios and percent change. The year 2000 is not twice the year 1000; the difference of 1000 years is real, the ratio is an artefact of when the count was started. ## Ratio Equal distances *and* a true zero that means none of the quantity: age, elapsed time, count of clicks, revenue, distance, mass, temperature in Kelvin. - Legitimate: everything, including ratios and percent change. Forty years really is twice twenty years, and it stays twice as long whether you measure in years, months or seconds. ## The levels are cumulative Each level permits everything the level below permits, plus more. So the scales form a ladder of information, and a legitimate move is always *downward*: a ratio column can always be ranked or grouped into categories; an ordinal column can never be promoted to interval by wishing. ## A cleaner test than memorising the list Rather than classifying by name, ask what transformations leave your conclusion intact: - Nominal: any relabelling of the categories changes nothing that matters. - Ordinal: any order-preserving relabelling changes nothing that matters. - Interval: any positive affine transformation `y = a*x + b` with `a > 0` preserves the meaningful facts (order and equal differences). - Ratio: only positive scaling `y = a*x` is allowed, which is exactly why ratios survive a unit change. If a statistic you computed changes under a transformation the scale permits, that statistic was not measuring a property of the data. ## Discrete versus continuous is a separate axis Discrete versus continuous is about how many values are possible between two points, not about how much structure the values carry. Counts of orders are discrete and ratio; height is continuous and ratio; nominal and ordinal columns are necessarily discrete. The two classifications answer different questions and you need both to know what a column supports. ## Frequency and proportion tables For nominal and ordinal columns the honest summary is usually a table: each category with its count and its share of the total. It preserves the whole distribution instead of collapsing it into a single number that the scale cannot support, and it is directly comparable across time periods or segments as long as the category set is stable. ## Where the taxonomy is fuzzy Stevens's scheme is a working rule, not a law of nature, and real columns sometimes resist it. Percentages are bounded ratio quantities; composite indices and standardised scores are treated as interval by convention rather than by proof; counts are ratio but discrete, which matters for modelling even though it does not change what summaries are allowed. Interviewers are not testing whether you can place every exotic variable perfectly — they are testing whether you check what a column means before you average it.