**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 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

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”.

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Hi Josie…. Nearly all variables can be categorized this way. What variables do you have in mind?

Sir, plz let me know what scale is this

1. Do you have access to new media?

a. Yes

b. No

c. Sometimes

d. Often

You didn’t need to submit the same question twice.

As is, you would have to treat this as categorical. It is a very poor set of choices.

hello sir, how to transform ratio data to ordinal data?

Hi Alim – why would you want to do that? You could, I suppose, divide it at arbitrary cut points and then relabel them but it’s hard to see a reason to do so.

Dear Mr. Peter Flom,

Pls advise – Is it an ordinal or interval scale?

Q. How does the aggregate turnover of your firm during the 5-year period of current global economic slowdown compare with that of 5-year period immediately preceding the slowdown?

a. 100% decline (turnover is now zero)

b. 50% decline (half of what it was earlier)

c. No change

d. 50% increase

e. 100% increase (twice as much)

f. > 100% (pls. specify)

Hi Nitin

As is, it’s ordinal. Percentages are a bit tricky, the simplest is to treat this as ordinal, but you could make it interval since you asked them to specify the amount in F. The trouble would be whether to add noise to A through E to simulate the variation that is really there but not captured.

Peter

Dear Peter,

Many thanks! I’m a research scholar (pursuing PhD) based in India. I want my research to be technically sound, therefore I would like to avail your professional services (if I can afford it!).

1. Can you pls. tell me how much would you charge if I seek your help in my research, sampling, and instrument design and analysis of the data.

2. How would we communicate – over emails?

3. Where are you based?

Thanks,

Nitin

In response to your reply on June 19:

I’d prefer an interval or ratio scale. What scale will it be if I make a continuous scale (unbalanced) as below and ask the respondent to place a mark at an appropriate place on the continuous scale and mention the exact figure for the mark (say 71% increase):

100% decrease(that is, zero turnover)– 75% decrease — 50% decrease — 25% decrease — 0% (no increase or decrease; same turnover as earlier)– 25% increase — 50% increase — 75% increase — 100% increase (turnover is now double of what it was earlier) — more than 100% increase (pls provide exact %).

If I have percent statistics for where the “lost remote” can be. Would that be nominal, being that the places it can be found would have no paprticular order of importance. Or would you worry about the percentages and call it ratio?

Percentages of locations for lost remote. Nominal or Ratio?

Percents are one of the things that doesn’t fit well on this scheme

Hi Peter

Please can you assist? I am in the process of analysing data from my questionnaire for a dissertation for my master’s degree. I am struggling on how to correctly analyse data from the question below. I have coded the data as scale but I am not sure how to analyse the data meaningfully

Question:

How would you describe the ethnic composition of your congregation? (Please select all that apply and if other, please specify. Please can you also provide a rough estimate percentage of the composition?)

White British % __________________

East European % __________________

Other White % ______________________

Mixed % _________________________

Chinese/Japanese/Korean % __________________

Black Caribbean/Black African/Black Other % _________________

Gypsies and Travellers %_________________________

Indian/Pakistani/Bangladeshi % ________________

Asian Other % _______________

Many thanks

Tsitsi Nyahwo

This will be tricky, since some people may give % and some may just give check marks. It will also depend on what you have to do with the data.

If a high school test is given and results are given in percentages. Would this be considered a ratio scale? Wouldn’t these scores be arguably at the ratio level of measurement because there is an absolute zero value (0% correct) and differences between values can be compared meaningfully. There is a fixed order, fixed unity of measurement and fixed zeror for measuring the variable. The results (percentage) is indicating how many were answered correctly not the students ability in the subject.

Thanks for your help!!

Percentages do not fit well in Stevens’ scheme. How to deal with them depends on what you are trying to do

Peter

Dear Peter

What type of a variable is a companies’ R&D/sales, so percentage? The purpose is to perform a cluster analysis.

Kind regards,

Remco

Percentages don’t fit on Steven’s scale. However you ought to be able to do cluster analysis with percentage data.

what test can be done between nominal(2 variables) and ordinal(5 variables)

You could try Jonckheere Terpstra test