Peter Flom’s Statistics 101: Nominal, ordinal, interval, ratio: Stevens’ typology and some problems with it
The nominal ordinal interval ratio scheme
Stevens (Stevens 1946) divided types of variables into four categories, and these have become entrenched in the literature. The levels are nominal, ordinal, interval and ratio. To fully understand these, you have to use the same methods that Stevens used, which involve permissible transformations. However, it will be clearer to first describe each level more casually.
Nominal comes from the Latin for ‘name’ and nominal variables are those that are simply names – they have no order. Examples are hair color or religion.
Ordinal responses are those that have a sensible order, but no fixed distances between the levels. Questions about subjective responses are often ordinal, for example, responses to a question such as “how much pain are you in?” with responses such as “none”, “a little”, “some”, “a lot”, “excruciating” would be ordinal, because, while it’s clear that they go from least to most pain, it’s not at all clear whether the difference between (e.g) “none” and “a little” is bigger, smaller, or the same as the difference between (e.g) “a lot” and “excruciating”.
Interval responses mean that, in addition to order, the scale has some sort of sensible spacing, so that the difference between two numbers is meaningful. Perhaps the best known example is temperature, in degrees Celsius or Fahrenheit. The difference between 10 degrees and 20 degrees is, in some sense, the same as the difference between 60 degrees and 70 degrees. In interval scales, addition and subtraction make sense, but multiplication and division do not. That is, 70 degrees is not “twice as hot” as 35 degrees. If this is confusing, think what a negative temperature would mean, or a 0 temperature! 30 degrees is -1 times as hot as -30 degrees? It doesn’t make sense!
Ratio responses mean that not only is there order and spacing, but that multiplication makes sense as well. Two common examples are height and weight. A person who weighs 200 pounds weighs double what a person who weighs 100 pounds weighs. Ratio scales have a meaningful zero.
This refers to what we may do to the responses without changing their meaning. For nominal responses, we can do anything at all, as long as it is 1-1, that is, as long as each unique level stays unique. For example, if we ask about residences, it does not matter if we label the responses as
Private house – A
Attached house– B
Rented apartment– C
Barracks or other military – E
Prison – F
Shelter – G
Other – H
Attached house – A
Private house– B
Barracks or other military– C
Prison – D
Shelter – E
Coop/Condo – F
Rented apartment – G
Other – H
But, if you combined any of the categories, you would change the meaning of the scale.
Ordinal responses may be transformed in any way that preserves their order. Thus, if we ask how much pain a person is in, and the choices are “none”, “some”, “moderate”, “severe”, and “excruciating” we could code
None – 0
Some – 1
Moderate – 2
Severe – 3
Excruciating – 4
None – 1
Some – 2
Moderate – 3
Severe – 4
Excruciating – 5
None – 0
Excruciating – 1,929,292
For interval data, we can transform in any way that preserves the relative size of the intervals. For example, it does not matter if we measure temperature in degrees Celsius or Fahrenheit. Although the size of the differences will vary, they will vary consistently. For example:
The gaps are 100 degrees on the C scale and 180 on the F scale, but they are consistent. This means that we can add and multiply by any numbers we like, as long as we do it consistently (e.g. to go from Celsius to Fahrenheit, multiply by 9 divide by 5 and add 32).
Finally, for ratio data, we may only multiply. We may go from pounds to kilograms, for example. But we cannot add or subtract constants.
Problems with the nominal-ordinal-interval-ratio categorization
Although Stevens’ scheme is useful, and is very commonly used, it is not without its problems. First, the categories are not exhaustive and alternate scale taxonomies are possible. For example, Mosteller and Tukey (Mosteller and Tukey 1977) proposed: Names, grades (e.g. freshman, sophomore, junior, senior), ranks, counted fractions bound by zero and one (such as percentages or proportions), counts (non negative integers), amounts (non-negative real numbers) and balances (any real number). So, are percentages nominal, ordinal, interval or ratio? Technically, they are not even ratio – you cannot double a percentage without distorting the meaning (Velleman and Wilkinson 1993); in addition, data transformations can be very useful, even if they are disallowed under Stevens’ rules – for example, taking the log or square root of a ratio variable would not be permitted by Stevens. Treating variables that are technically ordinal as if they were interval or ratio is often sensible (Abelson and Tukey 1963) and methods such as multidimensional scaling and item response theory turn ordinal level measures into ratio level ones. Also, although the transformations listed above for ordinal measures are both technically legitimate in Stevens’ typology, we sense that there is something wrong about the second transformation – although we may not know precisely how far apart “none”, “some”, “moderate” and excruciating are, we sense that the differences are at least somewhat similar. For more on these problems, see (Velleman and Wilkinson 1993), but, in short, any typology (whether that of Stevens or not) should be a guide, not a straitjacket. In words attributed to David Cox: “There are no routine statistical questions, only questionable statistical routines”.