# Difference between "interval" and "ratio" variable: How to explain?

I teach a basic statistics class, and I have trouble explaining how to tell if a continuous variable is an interval or a ratio variable. Further, there is some disagreement between when an interval variable stops being interval and starts being ordinal. I'm an expert at data, but splitting hairs in definitions for a class is not my expertise.

I am grateful to anyone who can give me practical, non-theoretical examples to help guide me and my class as to how to split hairs between ordinal, interval, and ratio data classification.

UPDATE 5/24/2013

Thank you all for your helpful answers. Of course, everything you are saying is correct, and as you know, I am looking for a way that really resonates (at least with me) so I can use that to explain to my students. Taking into account your thoughtful answers, here are my challenges on the periphery:

In interval vs. ratio, imagine a person has $0. Is this interval or ratio? $0 is having no money. However, people can have bank accounts that are -$100. You could say it is interval, like Celcius temperature, or you could say it is ratio, like height - a person with no height is at 0, and a person with no money has $0.

Another challenge comes to bear with units. Let's say I have schools A, B, and C, and they have 100, 600, and 1,000 students respectively. A new school, school D, has 0 students until they open, so these units could be ratio. However, students are not equal because people are all different. If school A is a high school and B is a pre-school, then adding a unit to A is not the same kind of unit as adding it to B. Therefore, the intervals are not equal. This can also be said with age, or years of education. Yes, you can have 0 years and 0 years of education, but is your first year anything like your 16th year? Can they be counted on the same scale?

In ordinal vs. interval (which I admittedly did not ask but unfortunately is relevant), imagine I give a test worth 100 points. The nuance is that questions 1-50 are difficult essays and 51-100 are simple multiple choice. Many would call this interval, because it does not matter which 90 points a student gets, if s/he gets 90 points, I will give him/her an A. On the other hand, getting points on the first part vs. the second part is different amounts of effort, suggesting ordinal. This is why I do not like to use rank examples in ordinal - anyone in high school knows/feels that if s/he is ranked 12 and his/her friend is ranked 6 that s/he is twice the distance from her friend from the top. The fact that the intervals between rank 1 (valedictorian) and rank 2 (salutatorian) and all the other ranks are technically different is lost on any high school student because they do not see it that way, and that is how most people encounter rank in daily life.

I especially like Jochen's example of degrees in terms of color - or even, a circle or an arc. We all know that an angle or arc of 0 degrees does not mean there is no angle or arc, just that it is at 0 degrees. This one I will use as a good difference between "interval" and "ratio". However, the class will challenge me with the above and I want to have some better answers.

Thanks again, and I will keep reading as I'm working on my materials for my stats class in fall.

UPDATE 5/29/2013

Clerot, Venkata, and Mohammad: You have convinced me NOT to use temperature as an example! I think it tries to be a good example, but has too many grey areas around the edges.

Charles: I love No Oil In Rivers! I will add that!

I am still really liking Jochen's example of degrees in terms of a color or an arc. Are there any more similar to that that can really illustrate when something is interval but NOT ratio?

I am grateful to anyone who can give me practical, non-theoretical examples to help guide me and my class as to how to split hairs between ordinal, interval, and ratio data classification.

UPDATE 5/24/2013

Thank you all for your helpful answers. Of course, everything you are saying is correct, and as you know, I am looking for a way that really resonates (at least with me) so I can use that to explain to my students. Taking into account your thoughtful answers, here are my challenges on the periphery:

In interval vs. ratio, imagine a person has $0. Is this interval or ratio? $0 is having no money. However, people can have bank accounts that are -$100. You could say it is interval, like Celcius temperature, or you could say it is ratio, like height - a person with no height is at 0, and a person with no money has $0.

Another challenge comes to bear with units. Let's say I have schools A, B, and C, and they have 100, 600, and 1,000 students respectively. A new school, school D, has 0 students until they open, so these units could be ratio. However, students are not equal because people are all different. If school A is a high school and B is a pre-school, then adding a unit to A is not the same kind of unit as adding it to B. Therefore, the intervals are not equal. This can also be said with age, or years of education. Yes, you can have 0 years and 0 years of education, but is your first year anything like your 16th year? Can they be counted on the same scale?

In ordinal vs. interval (which I admittedly did not ask but unfortunately is relevant), imagine I give a test worth 100 points. The nuance is that questions 1-50 are difficult essays and 51-100 are simple multiple choice. Many would call this interval, because it does not matter which 90 points a student gets, if s/he gets 90 points, I will give him/her an A. On the other hand, getting points on the first part vs. the second part is different amounts of effort, suggesting ordinal. This is why I do not like to use rank examples in ordinal - anyone in high school knows/feels that if s/he is ranked 12 and his/her friend is ranked 6 that s/he is twice the distance from her friend from the top. The fact that the intervals between rank 1 (valedictorian) and rank 2 (salutatorian) and all the other ranks are technically different is lost on any high school student because they do not see it that way, and that is how most people encounter rank in daily life.

I especially like Jochen's example of degrees in terms of color - or even, a circle or an arc. We all know that an angle or arc of 0 degrees does not mean there is no angle or arc, just that it is at 0 degrees. This one I will use as a good difference between "interval" and "ratio". However, the class will challenge me with the above and I want to have some better answers.

Thanks again, and I will keep reading as I'm working on my materials for my stats class in fall.

UPDATE 5/29/2013

Clerot, Venkata, and Mohammad: You have convinced me NOT to use temperature as an example! I think it tries to be a good example, but has too many grey areas around the edges.

Charles: I love No Oil In Rivers! I will add that!

I am still really liking Jochen's example of degrees in terms of a color or an arc. Are there any more similar to that that can really illustrate when something is interval but NOT ratio?

## All Answers (16)

Masoumeh Khosravi· Shiraz University of Medical SciencesI think this refer to the scale of measurement not the variable.We shouldn't say our variable is ordinal or interval,instead we speak about the measurement scale that we use to calculate data for our variables,so for one variable we can report different numbers which can be calculated by different scales.

For example think about length of sticks, our variable is continuous and can be measured with interval or ordinal scale,we can give numbers in inches(interval) or only give them number 1 if it is shorter than 1 inch and 2 if it is longer than 1 and shorter than 3 and so on(ordinal).

Although any of these scales have their own characteristics that make them different of each other ( you can find in statistics books) and should use for proper variables.

Fabrice Clerot· Orange Labs"Interval and ratio level data are classified as quantitative or numerical data.

Interval data are measured and have constant, equal distances between values, but the zero point is arbitrary. The zero isn’t meaningful, it doesn’t mean a true absence of something. An example of interval level data is intelligence, as measured on some IQ test. We know that the scoring difference between a 100 and a 110 is equal to the scoring distance between 120 and 130, but there is no true zero on this test and an IQ of 140 is not twice as high as an IQ of 70.

Ratio level data are those which have equal intervals between values, and the zero is meaningful. Some laboratory tests are good examples of ratio level data. A person who weighs 100 kilos is twice as heavy as a person who weighs 50 kilos, and a measure of zero kilos is meaningful. Other examples include pulse rate and respiratory rate."

taken from this presentation :

www.med.wright.edu/sites/default/files/psych/.../6TypesofVariables.pptx

(it even provides a small test at the end !)

.

Jochen Wilhelm· Justus-Liebig-Universität GießenWhen a ratio between two values of a quantitative/metric variable is meaningful, it's a ratio scaled variable, otherwise it's an interval scaled variable.

Not that the "variate" is some property one likes to analyse/measure. If one applies a defined measuring procedure to get some kind of values (names or numbers), then the result is the "variable". The way how a variate is measured defines the scale of the variable. For example, the variate "color" could be measured by naming the colors (the variable would be nominal with values "red", "green" ...). Also, the variate could be measured as "wavelength" in nm, what would give a quantitative variable. This variable is ratio scaled, since 0nm is a physically meaningful value and in fact the wavelength 200nm is twice as long as 100nm. As a third way, one could take a colored pallette system like the hue color system to measure the color (http://dba.med.sc.edu/price/irf/Adobe_tg/models/hsb.html). This gives values between 0° and 360°, again a quantitative variable. But here the 0° point is arbitrary, and a hue of 40° does not mean twice the hue of 20°. So it's an interval scale, but not a ratio scale.

Dr. Mukesh Kumar· University of BahrainLet us take an example of “100 meter race” in a tournament where three runners are participating from three different states of Malaysia. Each runner is assigned a number (displayed in uniform) to differentiate from each other. The number displayed in the uniform to identify runners is an example of nominal scale. Once the race is over, the winner is declared along with the declaration of first runner up and second runner up based on the criteria that who reaches the destination first, second and last. The rank order of runners such as “second runner up as 3”, “first runner up as 2” and the “winner as 1” is an example of ordinal scale. During the tournament, judge is asked to rate each runner on the scale of 1–10 based on certain criteria. The rating given by the judge is an example of interval scale. The time spent by each runner in completing the race is an example of ratio scale.

Mohammad Firoz Khan· Jamia Millia Islamia1. Ratio measurement assumes a zero point where there is no measurement. Suppose you want to know straight line distance between your house and your college or university department, centre of your house will be taken as zero (0.0) and say distance between your house and your destination is measured as 10.523 km, it is ratio measurement (Keep in mind that measurement re seldom perfect even measured at nano-scale, rounding is always done). Values at this scale can be added, subtracted, multiplied and divided.

2. Interval scale never assumes as absolute zero (0,0). For example, temperature measured in degree C or F. Even in the condition of zero when some liquids or fluids solidified or condensed to solid as ice., we cannot say there is no heat (temperature) in them. Therefore, if at your place day temperature is 40 degree C and at nearby hill station it is only 20 degree C, one cannot say that your place is twice hot than that of the hill station and similarly one cannot say that hill station is 1/2 time less hot than your place. In the absence of absolute zero, we cannot multiply or divide interval values with each other. However, to arrive at a mean value, these values can be added and subtracted from each other.

Syed Shahadat Hossain· University of Dhaka1. Does ‘zero’ mean ‘none’?

2. Does ‘multiplied by 2’ have the exact meaning of ‘twice as much as’?

If the answers to the both questions are ‘yes’, the scale of measurement used is a ratio scale.

The following example may be helpful:

Suppose a group individual is investigated and their heights in centimeter are recorded. The scale in this case is obviously a ratio one. Because:

0 cm height means ‘no height’

A person with 180cm is twice as tall as a person with 90cm height.

Now assume that the investigator could not find an exact measurement instrument for some reason and instead he chalks two lines on the wall to construct an interval. Then the interval is divided into ten equal divisions (units). The measurements are now made by the units marked on the wall. This in reality will give an interval scale of measurement. Because

0 unit height will not mean ‘no height’

A person with 8 unit height is NOT twice as tall as a person with 4 unit height.

Let a further assumption be made that marking the wall in equal units is also not feasible, and the investigator only draw one line on the wall and call the people having height above the line as Tall and people having height below the line as Short. This measurement only can be utilized to order the two groups in terms of height and hence will be an ordinal scale of measurement.

Finally consider that the two groups Tall and Short are just named as Height Group 1 and Height Group 2, the information contained by these two group names would not be helpful even for ordering the observations in terms of height. This is nominal scale.

Fabrice Clerot· Orange Labsnot sure that the following will make things clearer (a little background in physics may be needed) : consider the temperature

- when T is given in Kelvin, it has an absolute zero and kT (k the Boltzmann constant) has the dimension of an energy so that 400 K represents twice the energy coresponding to 200 K : ratio variable !

- the same temperature given in Celsius has no absolute zero (the melting point of ice at atmospheric pressure is just a useful but otherwise arbitrary point in the scale) but 10 more degrees correspond to the same energy difference whatever the point on the scale : interval variable !

the funny thing here is that it is the same quantity ; depending on its coding, it is either represented by a ratio or interval variable and the codings do not differ much : C = K - 273 (C in Celsius, K in Kelvin ; the absolute zero is about -273°C).

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Fabrice Clerot· Orange LabsVenkata Prasad Palakiti· Indian Institute of Technology MadrasMohammad Firoz Khan· Jamia Millia Islamiayou are right in terms of physics. It is possible at Boltzmann constant as when K=0, all molecular activity stops. But we don't know more what happens at -K. It is quite possible at K=0, the universe will become a cold waste.

Fabrice Clerot· Orange LabsDear Mohammad,

my answer was casted in the frame of "classical" thermodynamics, the one where the "temperature" is considered as the "real" variable of interest.

This "classical" thermodynamics is quite sufficient to interpret macroscopic properties and most microscopic ones ... but not all and, indeed, systems with negative absolute temperatures do exist, although their interpretation is more that these "negative" temperatures are "above infinity" (such systems have a decreasing entropy as their energy is increased, because they have a bounded number of high energy states ; in a sense, for such systems, the more you add energy, the more you are bound to fill in the low enregy states (as they are the only available) therefore reducing the entropy).

So well, well, well ... temperature may not be such a brilliant example (a scale that starts again beyond infinity ... definitely not interval nor ratio !!!) ... at least at the quantum level but this should not be a such a surprise as the relevant parameter for such systems is the conjugate of the temperature, the infamous "thermodynamic beta" !

for more details :

http://en.wikipedia.org/wiki/Negative_temperature

However, for macroscopic systems (from galaxies to superfluid helium,say), the three laws of thermodynamics provide an excellent framework and within this framework, the notion of temperature is reasonably well understood as a ratio variable (if expressed in Kelvin).

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Charles Vincent· Pontifical Catholic University of PeruFabrice Clerot· Orange Labs@ Monika

regarding the different examples you list in your update, it seems that the temperature above is a good warning : measurements (aka "data") are not "variables" by themselves ; they only become "variables" (and variables of a given type) through a clear specification of the purpose of the analysis.

For meteorologists, temperature coded in °C is appropriate and therefore temperature measurements are conveniently considered as occurrences of an interval variable.

For solid state physicists, temperature is quite often considered as a proxy for the energy kT (with T in Kelvin) and the zero temperature has its classical interpretation as the absolute zero of the energy scale ; temperature measurements are therefore recoded in Kelvin and conveniently considered as occurrences of a ratio variable.

For quantum physicists, temperature is sometimes not the right parameter of interest and using it in the discussion of the results (because of tradition or just for the fun of writing about "negative absolute temperature" !) leads to more questions than answers !

In summary, it is not the nature of the measured quantity (the "data") which determines univocally the nature of the type of variables these mesurements will be recoded into. You have to consider both the nature of the quantity _and_ the way they enter the analysis.

In your example about units, it is the role of the analyst to decide whether s/he wants to use units or not. Once this choice is made and agreed upon, the type of variables is fixed and the analysis can proceed ... and no "ex post whining" about units if the choice has been made not to use them ! Instead, it is possible to start another analysis with units and compare results.

.

Geoffrey Boynton· University of Washington SeattleDory Fisher· Australian College of Applied PsychologyYa I refer to the question and keep in mind that only ratio has a true zero. So if I am measuring hours of study, and I can have zero hours, then my data is ratio

Personally I think of age as interval because it is arbitrary that the West considers birth year 0; some south east Asian countries don't consider a child 'alive' until it reaches it's second birthday.

Jochen Wilhelm· Justus-Liebig-Universität GießenNice example, Dory.

"Time from birth" is, by definition, ratio scale, because, for instance, 6 years from birth are just twice the time of 3 years from birth. But the intersting point you brought up is about the

rationaleof this variable (in scientific context). I do not see any biological, psychological or sociological variable that would be (even only vaguely) linearily associated with this "time from birth". (Nevertheless the measure is useful in the juristical context, and there the scale is not relevant)Can you help by adding an answer?