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 responses
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
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
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
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.
Permissible transformations
This refers to what we may due 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
Coop/Condo– D
Barracks or other military – E
Prison – F
Shelter – G
Other – H
Or
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
Or
None – 1
Some – 2
Moderate – 3
Severe – 4
Excruciating – 5
Or even
None – 0
Some 4.2
Moderate 12
Severe 13
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:
Fahrenheit Celsius
32 0
212 100
392 200
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”.
Themocracy
great post as usual!
What a great resource!
I want to know why there is no meaningful zero in interval respones，thank you
Hi May
There is no meaningful zero because in interval scales (unlike ratio scales) zero doesn’t represent “none” of something. When it is 0 degrees F, there is nothing that is really at 0. Compare Fahrenheit temperature with height, for instance. If something is 0 inches tall, then it does not exist at all.
Thanks, but what kinda data percentages are?
Hi Suli
Percentages are one of the problematic types of data for the typology. Probably best treated as interval, but they have some odd properties.
Peter
Why are percentages not ratio?
For example, the percentage of unemployment for a country. There is a zero point –> 0% unemployment means there is no unemployment and % figures can be divided and multiplied.
in this case are percentages a ratio?
confused
Hi Jo,
Percentages are problematic, and don’t really fit into the scheme. It’s true that (e.g.) 5% is double 2.5%. But what if you double 80%? You get 160%, which often makes no sense, but sometimes does. Unemployment of over 100%, to use your example, makes no sense.
Peter
Hello,
Are “yes” or “no” responses nominal or ordinal?
Hi Effie
Either or both. When there are only 2 levels (such as yes and no) then ordinal and nominal are really the same. In fact, you can code “no” and “yes” as numbers and that’s OK too.
Peter
hi all,
i have comparing questions between XP and CMMI ,,,
the scales is
1- strongly support
2- support
3- partially support
4- not-addressed
5- conflict
please what is the suitable scales to be used here??
It depends on a couple things: If there are very few 4′s or if the 4 answer adds nothing, you could recode 4 as missing. Then the question is what 5 means in terms of this scale, and if a lot of people give that answer
Hello,
I’m really confused…
I’m trying to create a correlational relationship study and am looking at the ratio outcome of one variable and the standardized test score of another vaiable to see if there is any type of correlation.
I am having trouble understanding if these two mentioned variables are considered ratio or ordinal. Also, I once read that ratio shouldn’t be used within educational research, why is this?
Standardized test scores are usually interval level. You say the other variable is “ratio”. You can do a regular correlation between an interval level variable and a ratio level variable
I have no idea where that statement about ratio data in educational research came from..
Hello, I’m so confused after hours of research and looking in numerous text books.
I’m looking at a study which has used a paired t-test to test the significance of the sensitivity and specificity of radiographers within a red dot system before and after taking part in a training programme.
Sensitivity and specificity were calculated from false and true positives and negatives.
But what kind of data was used?… I’m getting the understanding it would be interval?
Thank you!
Sensitivity and specificity are both fractions. They are ratio level data. That’s why a t-test could be used.
Thank you for your quick response!!
sir,
can you provied more examples on nominal ordinal ration and interval
Another nominal variable would be “What sort of pet do you own?” Ordinal Mohs scale of hardness of rocks Interval – it’s hard to think of any that aren’t ratio, but these are treated very similarly in most analysis. Ratio – height, weight, salary etc.
What scale would the following fall under,
My adress on March 22, 1999
Would that be nominal?
Addresses would be nominal, in the vast majority of cases. But if you plotted address as latitude and longitude, it might be some thing other than the big four – a bivariate variable
Time is usually an example of a Ratio Scale. If there is an open ended issue such as Length of time on the job whereby
1- is less than 3 months 2- is 3 thru 8 months and 3- is 9 months or more, would this also be a Ratio Scale or would it be considered an Ordinal Scale?
Hi Ron – if this is all you have, then it would be ordinal. It’s not ration because (3 through
divided by (less than 3) is not meaningful, nor is (9 or more) divided by (less than 3).
It’s not interval because (3 through
minus (less than 3) is not meaningful, nor is (9 or more) minus (less than 3).
HOWEVER, if you are willing to make some assumptions, there may be sensible ways to look at it as interval or ratio. It would depend on your application. In your case, suppose that this is a survey among people on the job less than 1 year. If people are hired at a uniform rate throughout the year, then it would be reasonable to recode 1 as 1.5, 2 as 5.5 and 3 as 11.5, and then it would be ratio.
when the standard deviation of a distribuiton is 4.0 and if you mulitply each score in the distribution by 2 and then add 3 to each of the products, would the new standard deviation be 8? I get my answer as a function of when you add a constant the std deviation remains the same and when a mulitple is introduced it impacts the standard devilation of the old distribuiton by that factor or in this case 2 X 4.0 = 8 Am I correct in my logic? Thank-you.
yes
[...] Interval responses 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. Nominal, ordinal, interval, ratio: Stevens’ typology and some problems with it | Statistical Analy… [...]
Hi Peter
I’ve been thinking..if we are measuring the time spent on playing computers and it is sorted into “1- rarely, 2- seldom, 3- often, and 4- always”, is that ordinal or interval? It can be ordinal because there is a sequence to it (‘always’ is obviously more than ‘often’), but how much more? we dont know. so it cannot be interval in this case? however, we can also assume that a true zero exist (rarely is equivalent zero). pls help! thank you
Hi Estee
As measured, it’s ordinal; but if you can assume some things then it might be interval (e.g. if you have some notion of how much time is meant by 1, 2, 3 and 4). But here, 4 seems odd, “always” playing on computer? No sleep?
Like many things, it doesn’t fit neatly into Stevens scale
Peter
Hi Peter..
Can you give me a clear examples how we transform the interval into ordinal and nominal
It’s usually a bad idea to do this. But it can happen when a variable is categorized, for instance, if age is measured as < 18, 18-21, 22-30 and >30 then it is really ordinal. It’s very unusual to transform interval data into nominal data. I can’t think of a good example.
is international classification of disease a nominal variable
I think so, unless there is some method to how the numbers relate to some quality of the disease. It might be ordinal in certain parts of the classification (e.g. if, within a certain category, higher numbers were more severe).