I have never heard of a Likert scale as being considered interval data. To be considered interval data, a scale or level of measurement must possess distance (i.e., the magnitude of the differences between successive numbers must be equal). That definitely does not hold for Likert scale data. Also, with interval data, you must be able to perform addition and subtraction with the values (and the resulting numbers must make sense). Again, you can't do that with Likert scale data.

well, most researchers consider such data as interval data :-). you can't even compute an arithmetical mean value with ordinal data. so, if you read any paper in psychology, sociology, marketing, whatever... you will find such scales (by the way: indeed we are talking about rating scales not likert scales - a likert scale is different from the answer categories for an item, and that's what we are talking about) and they do every statistical analyses with them...

but it's indeed a difficult issue. to my opionion one can do something by not labeling every answer category verbally but to just label the endpoints (e.g. 1, 5 with strongly agree, strongly disagree), like here: http://qualitysafety.bmj.com/content/10/4/238/F1.large.jpg (that's just an example, I prefer 6- or 7-point rating scales.

I agree with Gordon. In fact, depending on the possible responses, a Likert scale could be considered nominal. That said, most investigators treat Likert scale responses as ordinal, and evaluate the responses as such (even though it is common to see analyses of Likert scale data using ttests)

a likert scale could never be considered as nominal. it's in the definition of any scale in the social sciences that their purpose is to measure and order something. so any scale has to be at least ordinal.

but to my opionion we are not talking about a likert scale (which consists of several items for measuring a certain phenomenon or whatever) here. we are talking about rating scales or answer categories for items. that said i refer back to my first posting.

I agree with Manfred: any scale that is either continuous or nominal is not a Likert scale.

I agree with Marfred.

In dealing with Likert "scales" you have to distinguish between the way that individual items are scored (e.g., 1 to 5) which is ordinal, versus the scoring that you get when you create a scale by combining the full set of items (which is usually treated as interval).

My suggestion: http://www.researchgate.net/publication/278403191_Numerical_Coding_of_Nominal_Data

Article Numerical Coding of Nominal Data

Likert scale should be an ordinal scale variable because it has no meaningful difference

When standardised Likert scale instruments are used, the overwhelming majority of these papers treat the data as interval and use parametric methods. Technically, attitude data is ordinal because the intervals cannot be guaranteed to have equal distance. But, in scales that are aggregated to produce sum total scores, it is clear the data is being treated as interval. Either a large proportion of the research community using composite scales have it wrong, or it is the case that Likert scaled data can be acceptably used as interval. This is Manfred's earlier point.

I found 5 levels of Likert scale data as an ordinal scale. We can compute the composite mean score. Later on, if it is non-normality distributed, we can apply a Non-parametric test. Among the NP, the Mean Whitney u test to compare the mean between two different groups and spearman correlation to show the relationships among different variables.

Mohan Sharma , from the discussion above:

An individual Likert-type item is often considered ordinal in nature. If it were treated as ordinal, you would not be able to take a composite mean score. That is, if we are treating the data as ordinal, there is no way to know that a "neutral" and "strongly agree" average out the same as two "agree"s.

However, once we construct a composite score (by addition or average), we are treating the individual item scores as interval in nature. That's the only way we can do the addition.

its ordinal

Good post, Sal Mangiafico. It has prompted me to think more carefully about the common practice of summing or averaging several Likert-type items to obtain a scale score. On reflection, I think you are right in saying that we in essence treat the items as if they have interval scale properties when we add (or average) them.

Thinking about this reminded me of another situation in which we treat ordinal data as if it had interval scale properties--i.e., the ordinal Chi-square (or test of linear-by-linear association) we recently talked about in another thread. I know that I sometimes fail to remember that part of using that procedure is generating scores for the ordered categories, as described in this excerpt from p. 97 of Agresti's Categorical Data Analysis (2nd Ed.):

--- Start of excerpt ---

A test statistic that is sensitive to positive or negative linear trends utilizes correlation information. Let u1 LE u2 ... LE uI denote scores for the rows, and let v1 LE v2 ... LE vJ denote column scores. The scores have the same ordering as the categories. They assign distances between categories and actually treat the measurement scale as interval [emphasis added], with greater distances between categories that are farther apart. [NOTE: LE means less than or equal to.]

--- End of excerpt ---

I am now curious to find out if SPSS's test of "linear-by-linear" association (in CROSSTABS) is sensitive to such scoring. As we are getting a bit off the original topic now, I might start a new thread on that issue.

Cheers,

Bruce

Alireza Ghasemizad , please define what you mean by interval and ordinal. If you mean interval that for someone somewhere that the difference between 2 and 3 is the same as between 3 and 4, then it may be (we don't know everyone's thoughts). If you mean for everyone, then no. If you mean some mathematical ideal in some mathematical universe, then say why this matters to you. ALSO, and this is important and a lot of the commentators above seem to have read this into the question, why do you want to know since the validity of any of our answers may relate to the consequences (of these answers). Say also if you a believe a procedure requires one of these over the other, why it is. Also, see @article{Lord1953,

title = {On the Statistical Treatment of Football Numbers},

author = {Lord, Frederic M.},

year = {1953},

journal = {American Psychologist},

volume = {8},

pages = {750--751},

}

Sal Mangiafico , you state: "An individual Likert-type item is often considered ordinal in nature. If it were treated as ordinal, you would not be able to take a composite mean score." And @Manfred , you say "you can't even compute an arithmetical mean value with ordinal data."

Suppose I have two ordinal (or categorical) variables and I label their three values as 1, 2, and 3. Suppose I get "1" on first and "3" on the second. You say I "would not be able to take a composite mean score" and "can't compute an arithmetical mean." Maybe I am just some mathematical whiz kid (I am neither a whiz nor a kid!), but I was able to do (1+3)/2 and was ABLE to get the value 2. Can you careful what you mean, or am I real that special? This isn't just me being picky, but why you may claim I SHOULD not do this is relevant to this question.

The standard for assuming that you can add individual Likert-items into a scale score is to compute coefficient alpha, which is based on correlation coefficients. So, in order to treat a set of Likert-scored items as interval, you have to begin by seeing if they behave as if they were linear (i.e., are highly correlated).

There is a distinct irony here, but there is no way around it.

I guess from David L Morgan response, I should add into my options for Alireza Ghasemizad whether you are defining "interval" to mean the same "linear" and then with respect to what? Of course this would mean one of those terms is redundant (and other issues).

Regarding Daniel Wright's question in his mathematical whiz-kid post, we should also bear in mind that rank-based methods such as the Mann-Whitney U test (aka., Wilcoxon-Mann-Whitney test) do entail computing sums of ranks. AFAICT, that is done without any assumption of interval scale measurement.

That's the trick: ranks are interval scaled. The difference between rank r and rank r+1 is one step, for any value of r. So the intervals are well defined and always have the same meaning. Hence, rank based methods work on (disrete) interval-scaled data.

Let's say I have a variable that everyone agrees is interval (some how). And the values that I observed are 1, 3, 8, 9. Jochen Wilhelm says that transforming these into 1, 2, 3, 4 this new variable is also interval (and I'll assume people agree interval is not something about the distribution or its relationship with other variables). Or is it that the original variable wasn't? Suppose the sample was 1,2,3,4,5,6,7,8, so Jochen says these ranks are also interval. Is the difference in the original scale the same between 1 and 3 and between 3 and 8, as implied by the first sample (if ranks are automatically interval) or do we look at the larger sample? My point on these is simply that using the interval-ordinal distinction is usually not that useful, and I am certainly not the first person to say any of this. The issue for most of us is that it adds confusion when we teach it as if it matters a lot.

btw, just noticed this is a really old question.

I think David Stockberger's notes on measurement may be helpful:

• http://faculty.cbu.ca/~erudiuk/IntroBook/sbk06m.htm

I also think that things tend to get (a lot?) more complicated when the things we are interested in cannot be measured directly (as is often the case in psychology). For example, early in my career, I worked for an attention researcher who suspected that one mechanism of selective attention was inhibition of the representations of distracting items. The inhibition he was interested in could not be measured directly. Rather, we measured response time (RT) to displays on a computer screen under (at least) two conditions, and inferred the presence of the inhibitory mechanism if RT was higher in one condition than the other.

Q. What level of measurement did we have?

A. It depends, I think. RT = elapsed time from the onset of the computer display, and elapsed time is a ratio scale variable. But one might argue that humans cannot react faster than some minimum, so the actual 0-point might be arbitrary. But even so, I think most folks would agree that RT would still have interval scale properties in that case.

But remember that for us, RT, or actually, the difference in RT between two conditions, was an indirect measure of inhibition, the thing that really interested us. So one might argue that in order to truly claim interval scale measurement, a 30 ms difference in RT at any point along the RT scale must reflect a constant difference in the amount of the underlying inhibition--the thing we would measure directly if we could!

I hope this helps more than it further confuses things. ;-)

Hi, Bruce Weaver .

Yeah, I assume that software implementations of the linear-by-linear test assume equal spacings between categories. At least in one implementation in R you should be able to assign other spacings. Somewhere on my list to do is to confirm that SAS, SPSS, and the most common implementation in R all produce the same results for this and for the Cochran-Armitage test. I wonder if there's any advantage to using this test rather than to use something like Kendall correlation. I also wonder about Jonckheere–Terpstra and Cuzick tests. I'm not really familiar with these, and haven't had a chance to play with them.

On individual Likert items, I think it makes a lot of sense to treat them as ordinal. That being said, I'm becoming more heretical over time. Even with a five-point Likert item, I suspect that respondents tend to view them as essentially interval in nature, especially if there are numbers associated with the levels, or even if there are e.g. equally-spaced bubbles to be filled in.

Hi, Daniel Wright .

I think I'll stick with my choice of the word "can't". If the data are treated as [strictly] ordinal * , then you cannot do arithmetic operations. By the contrapositive, if you are conducting arithmetic operations on the data, then they are not being treated as [strictly] ordinal.

Treating them as interval violates the conditional starting with "if"... You CAN treat the data as interval. (And in most cases my heretical heart isn't bothered by this... which I could explain more). But then you are not treating the data as strictly ordinal.

_____

* I don't think this is an accepted term. But I mean ordinal in nature but _not_ "at least interval".

I think the key point in Bruce Weaver message, and the reason why psychophysicists often point to RT in these examples, is that when coupled with a model it is useful (and appropriate) to claim a specific metric for the variables.

Hi Sal Mangiafico. Regarding the ordinal Chi-square (aka., test of linear-by-linear association), I can confirm that SPSS CROSSTABS is sensitive to the "scores" one assigns to the ordered categories. I've pasted below an example you can try if you can get access to SPSS. (I know you've had issues with PSPP, and would not assume that what it does always matches with SPSS does.) Even if you don't have SPSS, the results are summarized in comment lines at the end of the code. I've also highlighted the key lines that show how the "scores" were modified. I think it should be fairly readable even by folks not familiar with SPSS syntax. HTH.

* Change path on next line as needed.

GET FILE "C:\SPSSdata\survey_sample.sav".

* Variables agecat and degree are both ordinal.

FREQUENCIES agecat degree.

* Make copies without value labels so we can see the actual coding.

COMPUTE age1 = agecat.

COMPUTE degree1 = degree.

FORMATS age1 degree1 (F1).

VALUE LABELS age1 degree1.

FREQUENCIES age1 degree1.

* Make a new version of agecat that uses mean age

* within each category as the "score".

AGGREGATE

/OUTFILE=* MODE=ADDVARIABLES OVERWRITE=YES

/BREAK=agecat

/age2=MEAN(age).

CROSSTABS age1 BY age2.

* For degree, suppose that the step from < HS to HS diploma = 2;

* and likewise, the step from Bachelor to Graduate = 2.

RECODE degree1

(0=0) (1=2) (2=3) (3=4) (4=6) into degree2.

FORMATS degree2 (F1).

CROSSTABS degree1 by degree2.

* Now examine test of linear-by-linear association

* using different combinations of the age and degree variables.

CROSSTABS age1 age2 BY degree1 degree2 /STATISTICS=CHISQ.

* RESULTS.

*

age1 x degree1

Pearson Chi-Square 301.622

Likelihood Ratio 305.280

Linear-by-Linear 16.294

*

age1 x degree2

Pearson Chi-Square 301.622

Likelihood Ratio 305.280

Linear-by-Linear 23.271

*

age2 x degree1

Pearson Chi-Square 301.622

Likelihood Ratio 305.280

Linear-by-Linear 25.655

*

age2 x degree2

Pearson Chi-Square 301.622

Likelihood Ratio 305.280

Linear-by-Linear 34.450

.

* Notice that both Pearson & LR Chi-square tests are

* unaffected by the changes in the "scores" assigned

* to the ordered categories; but as one would hope,

* the linear-by-linear association test is affected

* by changes to those scores. In other words, SPSS

* is using the values of the scores when computing

* the ordinal Chi-square.

Sal Mangiafico , I think we are in agreement that we both think whatever this "intervalness" characteristic of a variable that people talk about is both: 1) in the mind of the researcher and related to the model, not an essence of the variable, and 2) varies. My comment was more aimed if someone thought a variable is something (like interval). And of course we agree without the "if" that for Likert data (1+3)/2 is 2!

Thanks, Bruce Weaver . I'll take a look at it when I can. ... I've had access to SPSS through the university a few times through the years... I don't think I do now...

Daniel Wright , yes: you can have an interval scaled (or an at least ordinal scaled; it may be ratio scaled)* variable and the ranks of this is another interval scaled variable. If you model these (the original and the rank-variable) as random variables, they will typically have different distributions. These are different variables with different meanings and different distributions. The rank transformation is just a monotone transformation, pretty much like the logarithm. While log() can map only numerical values (onto the real line), rank() can map ordinal values as well as numerical values (onto the natural numbers).Since all monotone transformations are rank-invariant and rank() maps to the ranks, this transformation is stable, i.e. repeated transformations do not further change the variable: rank(x) = rank(rank(x)). This is not the case for other monotone transformations.

*I wonder if and how this may apply to continuous variables. The transform must then map an uncountably infinite set ont an only countable infinite set, what is not possible. I found only one paper adressing this: https://www.jstor.org/stable/2578079 - but I have no access and did not read it yet. This may not be a problem in practice assuming that we can in practice only work with rational numbers anyway.

Daniel Wright ,

Hmmm...

I think the distinction between interval and ordinal is probably a lot less useful in the case of Likert-type items than maybe other cases, because, honestly, I think we tend to think of the choices in a Likert-type question as interval anyway. But think of something where it's not so easy to intuit the spacing between categories. Maybe higher education: HS < AA < BA < MS < PhD . If we had a collection of observations, it's sensible to report the median and other quantiles. And it's sensible to use a median test or sign test to compare two groups. But we really don't know the spacing between, say BA, MS, and PhD. * I don't think it makes any sense to say that a BA and PhD can average out to the same as two MS's. Yes, you could convert them to ranks and treat them as interval. But in the end, the interpretation is not the same. With ranks, you can't talk about mean education level or the differences in mean education level. You could talk mean rank or stochastic dominance of one group over the other.

I'm not sure I see the potential complication introduced by your example of converting an interval variable to ranks. The initial data were interval, and the ranked data are interval. This seems perfectly okay to me. It's just that the data were transformed. It's the same as if you have a set of continuous measurements, A. log(A) is also continuous, but the relative spacing between individual observations has changed.

* In my life --- well, I didn't get a Master's degree --- but the work I did at the Master's level was essentially the same as what I did at the PhD level. For other types of programs, a Master's degree may be more like just taking a couple more years of advanced undergraduate courses.

Hi Jochen Wilhelm. I did not have access to O'Brien (1982) via the JSTOR site either. But I can get a PDF via this Google Scholar search:

https://scholar.google.ca/scholar?hl=en&as_sdt=0%2C5&q=Using+Rank-Order+Measures+to+Represent+Continuous+Variables&btnG=

Perhaps it will work for you too. ;-)

Sal Mangiafico , I agree. I think the notion of a qualitative scale where you can say something is greater than something else is easy to see and to explain. The bit that gets tricky is what you do with it. The original questioner asked in the abstract are LIkert data interval or ordinal. WIthout reference what one does with them I don't think this make much sense (but people answered it because that is what RG answers often do). Also people talk about things like linear and distributions as if this relates. Those of course are assumptions for many tests.

Hey folks mathematics goes back a very long way and uses logical arguments. Perhaps we should stick with it and thus ordinal can't be continuous because the definitions don't match. Stevens meant this for psych measurements and not mathematics anyway.

With best wishes to all,, David Booth

The entrie "scales of measurement system" is not relevant for statistics at all. It is important in the field of research to understand that different ways of measurement will provide data with different information content.

When it comes to data analysis and statistics, it's not anymore about the measurement - it's about the variable that is to be analyzed, which typically is some random variable. And there is nothing like an "interval-scaled" or "ratio-scaled" variable. A random variable is defined by its domian and its probability distribution. The domain is the set of possible numerical values it can take, and this can be finite or infinite, and infinite sets can be bounded. Finite domains are neccesarily bounded and discrete, infinite domains may be unbounded and can be continuous. Random variables are only classified as either "discrete" or "continuous". But giving the distribution model of the variable is sufficient to fully characterize the variable (e.g. a Poisson-variable is discrete, with an infinite domain containing all natural numbers including 0 [so is has a lower bound]).

The key question is: how to represent a measured variable by a random variable. What should be the doain and what schould be the probability distribution?

For instance, a binary measured variable can be elegantly represented by a Bernoulli variable. A nominal or ordinal measured variable can be represented by a set of Bernoulli variables that can be analyzed with multlinominal or orderer logistic models. Ranks can be represented by a discrete bounded random variable; It's, however, difficult to give a sensible distribution model, but there exist models for rank sums under particular hypotheses. The distribution of normalized mean ranks may even be resonably approximated by a beta distribution. One can make inference about the mean rank, e.g. if we should expect it to be larger in one group compared to another. This may be a relevant analysis, but it's on the mean ranks, not on the measurement variable. The size of the difference is important to test hypotheses, but it cannot be interpreted as a "measure of importance" on the sclae of the measured variable.

I think this (measurement variable vs. random variable) often is confused and this is a source of many unneccesary discussions.

Jochen Wilhelm Well said!

"I think this (measurement variable vs. random variable) often is confused and this is a source of many unneccesary discussions."

Perhaps 'often' is not strong enough.

@Jochen. Thank you. I couldn't have said it better myself. Best wishes, David

My question to research colleagues in various and diverse disciplines is why anyone should apply arithmetic rules of addition, subtraction and multiplication on a data set that doesn't have equal intervals.

It makes arithmetic and the rest of mathematics meaningless. However, statistics is part and parcel of mathematics.

Likert scale doesn't have equal intervals and should not be modified to interval scale just for the convenience of any disciplinary area.

You are technically correct that one cannot apply "addition, subtraction and multiplication on a data set that doesn't have equal intervals." However, I think this is a case where perfectionism is out weighted by practicality. In a perfect world, we would only perform arithmetic operations on interval data, but in the real world, the fact that these procedures are often "good enough" matters more.

Prof. Morgan, I appreciate your argument and use of the phrase "good enough". My understanding is that the "good enough" issue is for social science disciplines only where even the scientific law of gravity has been diluted and used in the social sciences. However, let distinction between the physical scientific and social scientific use be made amply clear. Let the borrower not corrupt the lender.

Harold Chike I believe this is a case where the perfect is the enemy of the good (or in this case, good enough). The fact that creating interval-level scales from ordinal-level items works is what matters most. Yes, it would be ideal to develop a method that is problem-free, but until that happens, I believe in the olde saying that it is better to light a candle than to curse the darkness.

Dear Dr Alireza Ghasemizad

The Likert scale, which is generally designed with four to seven points, is extensively used in social work research. It is sometimes referred to as an interval scale, however, it is actually an ordinal scale on which no mathematical operations may be performed.

Md. Nayeem Hasan Pramanik There is an important difference between individual LIkert-scored items and scales that consist of summing LIkert-scored items, which are commonly treated as interval.

With regard to whether it is allowable to sum the ordinal-level individual items, please see the posts above.

"A very-good-rated juice is twice as healthy as good-rated juice" - if you are allowed to say this way, you can treat Likert as an interval scale. If not, you are just ranking the options using Likert scale.

But, we have to sacrifice a lot to stay on good terms with our partners; hardness is not a fundamental property of a good relationship. It would be best to decide how you want to treat the Likert scale items based on your analysis plan.

Mosharop Hossian, would you agree that temperature in degrees Celsius and temperature in degrees Fahrenheit example of interval scale measurement? If so, note the following:

• 20 deg C = 68 deg F
• 10 deg C = 50 deg F
• 20/10 = 2
• 68/50 = 1.36

In order for the ratio of two measurements to be meaningful, you need a ratio scale--i.e., a scale with a true 0. For interval scales, the ratio of differences is meaningful, but not the ratio of measurements.

Good point, Bruce Weaver . Or specifically for a 5-point Likert item, if the categories could be considered interval in nature (equally spaced), but could be coded as 1 -5 or -2 to +2, and in either case all addition, subtraction, and averaging would come out effectively the same. (That is, these data would be interval but not ratio.)

Salvatore S. Mangiafico The key issue is whether you can indeed interpret the categories on a Likert-scored item as "equally spaced." I ready don't think you can treat "Strongly Agree" and "Agree" as exactly one unit of agreement apart.

Hi, David L Morgan . There was a missing "if" in my post. But the point was just to give another example for Bruce Weaver 's point about the difference between interval and ratio values.

Jochen Wilhelm

I tend to agree, however I think that you are using the term "statistics" with a particular meaning which is not its only one. I mean, from the point of view of "mathematical statistics" (e.g. inference, estimation, and so on) what you say is absolutely true. But one might also understand the term "statistics" is a more naïve way, closer to its historical origin. Then, "statistics" is concerned abut the organization, representation and visualization of finite sets of data. In this setting, the distinction between norminal, ordinal, interval, ratio, etc. makes complete sense.

My dear friend Antonio, mathematics is often said to be the subject where we say what we mean and mean what we say.. Note Jochen's definition. Is precise so we know what we are talking about as opposed to your proposal which is pretty loose and imprecise doesn't even work. For example where is the normal distribution in your definition? If you mix things together and leave some things out Steven's scales and small washtubs can be included as well. See other comments included on the attached search. Stevens himself said the scales were about measurement and not statistics. Best wishes to all, David Booth

Another search may also add to this discussion. See the attachment below.

David Booth

Antonio M. Oller-Marcén

fair point!

I presume that the background of such questions (what is the scale of measurement of a variable?) is inference rather than summarization/organization/visualization. And inference of such data seems to be stuck in these "ancient times". We could and we should do better today.

David Eugene Booth

I am not your dear friend, and I would appretiate if you did not use such a condescending tone. Of course I do not need you point the wikipedia to me, either.

Mathematics is hardly a "subject where we say what we mean and mean what we say". It may be the case when we present "the final product" but surely it is not during the proccess. Probably an emeritus professor like you already knows the classic by Lakatos "Proofs and refutation"... worth re-reading anyway.

As Jochen Wilhelm points out we should do better, and I agree and I also undertand that if, in a context of inference the question "what is the scale of measurement of a variable?" arises, it means that there is some underlying obstacle and the idea of "scale of measurement" should have not permeated to this point. However, at earlier stages in the learning process of a student, the distinction between nominal, ordinal, etc. is useful and plays a role.

Then, in later stages, it is not that the students should forget about that distinctions. Rather, they should gain enough insight to realize that the distinction is no longer needed or relevant... and it is partly our responsibility as teacher to make them gain that insight.

Dear Professor-Dr Oiler-Marcen, If I was condescending to you I most certainly apologize. I have 2 main concerns whichi raised..1. How can a definition of statistics exclude continuous distributions eg the normal. 2 I don't think a wise teaching strategy is to.tell.my research students to forget what they learned

because they are now research students. I further note that as my screenshot says scales of measurements are widely criticized by scholars in other disciplines. Can you give me an example of ratio measurements in the natural sciences. Stevens was correct when he said essentially said don't use this outside of psych.measuremnt. With best wishes, David Booth NB I owe Bruce Weaver a thank you for his question on temperature measurement which do not IMO fall into the Stevens system because temperatures form a continuous variable as is often the case in the natural sciences.

David, I am surprised to read that you believe temperature measurement does not fall within Stevens' levels of measurement framework. I just double-checked to be sure, and he used temperature in degrees Fahrenheit and Celsius as examples of interval scale measurement. And he used "Absolute Temperature" (i.e., temperature in degrees Kelvin) as an example of ratio scale measurement. See page 679 in his 1946 Science article:

• https://psychology.okstate.edu/faculty/jgrice/psyc3120/Stevens_FourScales_1946.pdf

Perhaps I misunderstood the point you were trying to make. Apologies if I did.

Bruce please recall high school physical science. Where does the temperature of 4/3 degrees fail into the

Stevens system. It's an irrational number but temperature is continuous and a temp of 4/3 must exist by Freshman physics and a Mercury. thermometer. As Feynman a Nobel laureate once said if a theory does not agree with experiment then the theory must be wrong. Nature rules in the sciences. Statistics is no good if if it contracts Nature. David Booth PS Google the history of Zeno paradox from beginning calculus.

@Bruce and others excuse my error please 4/3 is of course rational. Please replace 4/3 by π and the argument is the same. Contracts should have been contradicts. Best wishes to all David Booth

Bruce Weaver , thanks for the Stevens article. I hadn't seen that before. It's pretty useful, and good to have on-hand for a reference.

I guess I'll weigh in on the scales of measurement discussion. In general I find this schema to be very useful and having practical implications.

As an example, if you are considering responses to a single Likert-type item, it's important to know if you are considering the response categories to be merely ordinal in nature or interval in nature. This affects what summary statistics you can use. For example, if it makes sense to calculate a mean, or a median.

It also affects what statistical tests make sense to employ. For example some rank-based tests --- including Wilcoxon signed rank test and aligned ranks transformation anova --- use subtraction in their calculations, so that they shouldn't be used with strictly ordinal data. On the other hand, a test like the Kruskal-Wallis test is fine to use with strictly ordinal data.

Likewise, if you have an interval scale like temperature in °C, and, say, you have one greenhouse at 10 °C and another greenhouse at 20 °C, you should be cognizant that it makes no sense to say that the second greenhouse is twice as warm as the first. But on the other hand, if you are measuring the length of fish --- a ratio measurement --- it makes sense to say a 20 cm fish is twice as long as a 10 cm fish.

David Eugene Booth

You do not need to address me as "professor doctor"... Antonio would do just fine.

So, you do not want your students to forget what they learnt before... bad idea. Let me give you an example. When kids are first taught multiplication, they are told that multiplying is just adding several times... now, try to use this idea to conceptualize, say, complex numbers multiplication. Maybe you do not like the term "forget". Fair enough. But you surely want them to revisit and reelaborate (sometimes deeply) their previous knowledge from a higher standpoint. Maybe not so different.

On the other hand, of course a definition of statistics can "skip" continuous variables... Continuity is just an idealization of reality used to model it. If you are just working with "raw" data and do not want to do inference, everything is discrete and you are just fine. Of course, once you want to use the powerful tools of probability theory you need to model your data (sometimes) using continuous variables.

As I said, the origin of statistics as a discipline is not inference. In the 5th century AD you can already find the concept of mean and I can assure you that the mathematicians using it were not thinking about normal random variables or central limit theorems.

Antonio , you are correct in some assertions. However the power of statistics is the ability to infer not to summarize. Best wishes, David

Dear Dr. Alireza Ghasemizad,

I think the Likert-type scale is ordinal because it is one kind of psychometric scale, and it has the ranking ability.

Ordinal, because you can compare one categories with others

Likert scale should not be used as interval scale in tte pure sciences and engineering to avoid disastrous consequences