Talk:Level of measurement/Archive 1

IQ
Article states: Most measurement in psychology and other social sciences is at the ordinal level; for example attitudes and IQ are only measured at the ordinal level.. But Babbie's specifically states that 'about the only interval measures commonly used in social scientifc research are constructed measures such as standarized intelligence tests' (IQ). --Piotr Konieczny aka Prokonsul Piotrus Talk 19:24, 16 October 2005 (UTC)

Raw scores, and standardized scores derived from raw scores, for IQ tests do not have an underlying unit of a scale and so, in Stevens' schema, don't constitute interval level measurements. Having said that, when analysed with Rasch models many such tests meet the criteria for interval level measurement to a reasonable extent. The transformation of raw scores is non-linear when these models are applied, but all the same, the transformation is close to linear in a substantial range of the raw scores, for most tests. So it depends how strict your criteria are. There's an argument both ways, depending on this. Certainly, though, person estimates produced by IQ test are no more or less 'interval-level' than those produced by many attainment tests, or various other tests for that matter. Stephenhumphry 10:57, 17 October 2005 (UTC)

On measurement and numbers
I have reverted the most recent edits. First, placing "ordinal numbers are ..." directly after the term Ordinal measurement seems to me to imply ordinal measurement is definable simply in terms of a number system. This runs the risk of going down a troubling path, and there is enough confusion about the meaning of measurement and scale as it is. Stevens (1946, p. 677) said: "The isomorphism between ... properties of the numeral series and certain empirical operations which we perform with objects permits the use of the series as a model to represent aspects of the empirical world" - which clearly implies that he had more than just a formal system in mind when proposing his measurement scales. Sophisticated works such as the Foundations of Measurement by Luce, Krantz, Suppes, & Tversky have been devoted to establishing formal foundations for measurement in which the assignment of numbers to objects in terms is justified in terms of structural correspondences between numbers and empirical qualitative relations. Secondly, I find the usefulness of the term nominal number dubious, and quickly find that I'm not alone. Lastly, to refer to alphabetical sorting as a type of measurement reminds me of Lord's (1953) article On the statistical treatment of football numbers (i.e. it just starts getting silly). smhhms 07:02, 3 December 2005 (UTC)

Source
I seem to recall that some physicist classified these "levels of measurement" in some paper published in the early 20th century, and perhaps proved mathematically that under specified assumptions, no other levels than these four are possible. Are there scholarly references that can be added to the article, that are not just textbook explanations of the same information that's already here? Michael Hardy 22:58, 28 Aug 2004 (UTC)

Perhaps someone tried to prove that these were the only four levels, but Stevens in the 1959 article cited in the article presents a fifth "level" that he called "logarithmic-interval". This handled scales such as Richter's scale of earthquake intensity that had a fixed zero but non-linear units. In later work, a number of researchers presented a higher scale of measurement called "absolute" that could not be rescaled by a multiplier. This includes probability. A more complete axoimatization of measurement is in the three volumes: Krantz, David H., Luce, R. Duncan, Suppes, Patrick and Tversky, Amos 1971. Foundations of Measurement: Volume 1: Additive and Polynomial Representations. New York: Academic Press. Suppes, Patrick, Krantz, David H., Luce, R. Duncan and Tversky, Amos 1989. Foundations of Measurement: Geometrical, Threshold, and Probabilistic Representations. San Diego: Academic Press. Luce, R. Duncan, Krantz, David H., Suppes, Patrick and Tversky, Amos 1990. Foundations of Measurement: Representation, Axiomatization, and Invariance. San Diego: Academic Press. This is very dense material, and not cited in normal textbook versions. Still, I think it would be important to say that the four common levels are not the end of the story.
 * NChrisman 18:44, 4 September 2006 (UTC)

I added a reference to Stevens's article which is often quoted as the origin of this subject. I have also seen references to Guttman's "A Basis for Scaling Qualitative Data" which was published 7 years earlier (American Sociological Review, 1944, 9:139-150) but I didn't include that since I haven't read or even seen the article myself. Perhaps someone else has read this source and can add or comment? :Jlefeaux 14:29, 19 Nov 2004 (UTC)

Guttman scales are was to generate an interval measurement from ordinal scales. It is not a work on this level of measurement debate. Although Stevens had limited citations in Science, his work derives more from the operationalism debate (Campbell and others).
 * NChrisman 18:44, 4 September 2006 (UTC)

Those levels are described in Earl Babbies 'The practice of social research', a popular textbook for students in social sciences. --Piotr Konieczny aka Prokonsul Piotrus Talk 19:24, 16 October 2005 (UTC)

I published an article about limitations in Stevens' taxonomy for the field of cartography: Chrisman, N. R. (1998) Rethinking levels of measurement in cartography. Cartography and GIS, 25, 231-242.
 * NChrisman 18:44, 4 September 2006 (UTC)

Levels and their implications for formula; & intensive, extensive
In the past for my own amusement I did work out the permissible mathematical relationships or transformations of different levels. I was interested in this as it could help specifiy an unknown formula in the same way that dimensional analysis does. The piece of paper I wrote my results on has become lost long ago. I would like to read more about this.

I've also heard something about intensive and extensive measurements. I would like to read more about these. --62.253.52.46 10:15, 16 July 2006 (UTC)


 * See intensive quantity and extensive quantity and also the quantity article itself. Holon 02:37, 17 July 2006 (UTC)

Stevens thought that the distinctions between intensive and extensive were not needed. He argued that the only issue of importance was invariance under transformation. I, like many others, disagree since "ratio" scales behave differently when aggregated or disaggregated. The extensive measure "population" must be partitioned among sub units, while density (a derived intensive measure) behaves differently. NChrisman 18:45, 4 September 2006 (UTC)

Most information
I removed 'the mean gives the most information'; I think that this will just confuse people. Alternative measures of central tendency are under-utilized, and saying that the mean gives the most information may just exacerbate that. Plf515 02:15, 24 November 2006 (UTC)plf515

The mean gives the most information - why?
I notice that a lot of time has elapsed since Plf515 deleted the above comment, and the delete appears to have been reverted with no explanation. I also believe it should be removed, as it is wrong as it stands. E.g. if you actually want to know the mode, knowing the mean is unhelpful and gives no information! OK, one could argue that the arithmetic mean is based on information from all the points whereas the mode is not, but the geometric mean depends on all the points as well. Surely it depends what kind of information you are trying to distil from the data. In summary, I don't think the phrase adds anything to the article. A point which is perhaps worth making is that the the classification levels are successively more informative and can undergo successively more different meaningful operations (equality, comparison, arithmetic mean, geometric mean). Comments anyone? Kiwi137 (talk) 17:06, 15 January 2008 (UTC)
 * On my first quick reading of the page, this phrase also struck me as odd. But perhaps it depends on whether you think that more information is necessarily a good thing. Using the mode rather than the median may be most appropriate in context, as with choosing an appropriate level of approximation for numbers. Itsmejudith (talk) —Preceding comment was added at 09:57, 21 January 2008 (UTC)
 * I agree with the above that saying "the mean gives more information" is incorrect: the mean gives different information than the median or mode (likewise for the standard deviation compared to the MAD or maximum deviation), but it does not give strictly more information, nor is it in general preferable. I've thus removed the statements saying that "the mean gives more information."
 * Nbarth (email) (talk) 18:47, 2 March 2008 (UTC)

Related article?
Can someone look at the oprhaned article Nominal category and see if it is correct, if it should be linked to from here or merged, or whatever. Melcombe (talk) 09:11, 22 May 2008 (UTC)
 * I worked it into the text. Thanks. --Ancheta Wis (talk) 12:26, 25 April 2009 (UTC)
 * Some of the critiques of Stevens are pointing to dead links.


 * To all editors: a 'dead link' condition can be flagged with if one follows the bad link and get a 404 error. It will help other readers, if we then return to the article and tag the bad link.  --Ancheta Wis (talk) 14:00, 25 April 2009 (UTC)

Operationalism
I have never heard of 'Operationism', a term used in this article, but there is an article on Bridgman's Operationalism. It appears that this article ought to be reviewed for accuracy; if this is a typo, then it should be possible to backtrack to the edits which introduced this, using the article history. --Ancheta Wis (talk) 15:15, 25 April 2009 (UTC)

Interval data example actually ratio?
Isn't height data ratio data? A height is a length and the zero point is not arbitrary. It means "no height". Either I'm correct or something else is wrong with this article. I'm not sure which, so I didn't make an edit.

There's already a good example in the interval data section: temperature in Celsius. Just delete the height example? will (talk) 02:57, 5 January 2010 (UTC)


 * I agree - the height example is not appropriate. I've removed it. -- Avenue (talk) 08:28, 5 January 2010 (UTC)

Vicson?
I reverted a change that somebody made. They deleted the reference to the Stevens material, which is important to the history of this topic. They added a claim about Vicson, but didn't provide a reference or any description of his/her fifth type of scale. --Lou Sander (talk) 13:00, 7 June 2010 (UTC)

Intelligence Citations Bibliography for Articles Related to IQ Testing
This is a great article from which I have already learned a lot. I'd like to help it improve more and to help other articles improve more by sharing information. I have posted a bibliography of  Intelligence Citations for the use of all Wikipedians who have occasion to edit articles on human intelligence and related issues. I happen to have circulating access to a huge academic research library at a university with an active research program in those issues (and to another library that is one of the ten largest public library systems in the United States) and have been researching these issues since 1989. You are welcome to use these citations for your own research and to suggest new sources to me by comments on that page. It would be particularly helpful to Wikipedia articles on IQ testing to bring in more levels of measurement perspective with citations to reliable sources. -- WeijiBaikeBianji (talk) 23:17, 2 July 2010 (UTC)

Likert scales as interval?
I've been analyzing Likert scale data with Rasch models for years and have yet to see an instance in which those data were anything but ordinal. The spacing between the rating categories is interval in name only. Anyone can demonstrate this for themselves. Obtain ratings on a group of items that vary in the challenges, importance, frequency, or agreeability they pose from a group of people who vary across all the possible score groups. Divide the items into two groups by their total sum scores. Now sum the ratings for each respondent twice, once for the low-scoring group of items (the hard test) and again for the high scoring group (the easy test). Plot the pairs of summed ratings. This plot will always be a curve. If the scores were linear, equal-interval measures, the plot would be a line.

Scores summed from ratings or counts must be ordinal and plot in a curve. This is because the maximum and minimum possible scores are bounded by the concrete limits of the variation observed in the particular questions asked and the sample responding. Say we start with a twenty-item survey and 5-point Likert scale. The person who obtains a maximum score of 50 on the low scoring group of ten items (the hard test) is highly likely to obtain the maximum of 50 on the high scoring items (the easy test), too. Conversely, the person who obtains a minimum score of 10 on the high scoring group of ten items (the easy test) is highly likely to also obtain the minimum score on the low scoring group of items (the hard test).

Thus, the maximums and minimums from both sets of items will plot on the identity line. But what happens in between? By definition, with one set of items easier, or more important, frequent, or agreeable, than the other, the paired values must move away from the identity line. Someone who obtains a score of 45 on the low scoring items will have a higher score on the other group of items, but there are only five available higher score groups, so the range of possible variation is restricted by the arbitrary ceiling of 50. But someone who scores 25 on the low scoring items might score 35 or 40 on the other, high scoring, group (I tried to create a figure to illustrate the point, but I couldn't get it to display properly so I deleted it).

So, though Likert scales are widely referred to as providing interval measures, it is mathematically impossible for them to do so. The ordinal unit of measurement provided by Likert scales must change size as the score moves away from the middle of the scale to either extreme. One way to obtain interval measures that vary from negative infinity to positive infinity, and so do not suffer from the range restriction, is to employ the two-stretch transformation effected by a log-odds unit, as in Rasch model applications. WPFisherJr (talk) 20:56, 26 December 2010 (UTC)WPFisherJr


 * I am working with the Likert as well, and I wonder if Likert should not be the type of scale.--John Bessa (talk) 16:24, 31 May 2011 (UTC)

Table in theory section
This needs a column for examples.--John Bessa (talk) 16:26, 31 May 2011 (UTC)

Ordinal scales
I have made several carefully explained edits to the section. They were reverted without explanation. Please explain.Miradre (talk) 00:38, 8 July 2011 (UTC)
 * Carefully explained vandalism is still vandalism. See WP:Vandalism. aprock (talk) 00:43, 8 July 2011 (UTC)
 * I corrected and removed factual errors and incorrect arguments. That is not vandalism.Miradre (talk) 00:46, 8 July 2011 (UTC)
 * Here is the edits I made, each one explained in edit comments:
 * Note that year of the study. He is talking about old ratio IQs, not the modern development deviation IQs
 * OR synthesis and unsourced conclusion which is incorrect. Yes, psychologists do construct the tests so that one SD is constant number of points but that does not mean one SD is linearly related to IQ.
 * Minor edit.
 * Please explain what is incorrect with the above.Miradre (talk) 05:03, 9 July 2011 (UTC)
 * You can dress up your vandalism however you like, that does not excuse it. aprock (talk) 06:41, 9 July 2011 (UTC)
 * Your continued incivility is noted. Vandalism does not include content disputes.
 * What is wrong with my comments and reason for removing this incorrect material? Miradre (talk) 06:47, 9 July 2011 (UTC)

Level or Scale?
The original citation for this idea [Stevens, 1946] referred to them as scales of measurement. The Handbook of Parametric and Nonparametric Statistical Procedures [Sheshkin, 2007] refers to it as scales of measurement. Almost all of the citations referenced use "scale" rather than "level": of the 35 references+notes, 6 contain "scale" in its title, while only 1 contains "level". All the references that are from Psychology and Statistics, including review papers and textbooks cited used the term "scale."

Searching for the phrase "scales of measurement" on Google returns about 3x more results than "levels of measurement," providing additional, albeit weak evidence regarding the preference by frequency.

I believe both are correct terms, and we should recognize that. However, "scales of measurement" is a more widely accepted scholarly term, as well, as a more accessible term for the student learning about this, and we should rename the article to reflect this.

[1] Stevens, S.S (June 7, 1946). "On the Theory of Scales of Measurement". Science 103 (2684): 677–680. doi:10.1126/science.103.2684.677.

[2] Sheskin, David J. (2007). Handbook of Parametric and Nonparametric Statistical Procedures (Fourth ed.)

-- Embryonix (talk) 09:42, 8 December 2011 (UTC)

Nominal Data is Categorical, Ordinal Data not?
According to Hastie, Tibshirani, and Friedman: The Elements of Statistical Learning the concept Categorical Data is used in two ways. Either unordered (a.k.a. Nominal or Qualitative Data) and ordered (a.k.a. Ordinal and a more special concept Rank Data). While this article appears to be a bit more restrictive on this at the moment, I would opt for adjusting to this. Any other opinions? 134.102.219.52 (talk) 09:42, 10 January 2012 (UTC)

Cooking Abilities
Why are there more females listed than males for the cooking abilities? Isn't that a little sexist? — Preceding unsigned comment added by 98.218.229.41 (talk) 21:27, 30 May 2012 (UTC)

Ordinal scales and monotonic increasing functions.
I know the treatment here is pretty standard, but I for one find the claim that ordinal scales are preserved by monotonic increasing functions a little misleading. It's true if you take the relevant ordering to be the non-strict ordering, but that is not the intuitive ordering to worry about; the intuitive ordering is the strict ordering, and that is preserved only by strictly increasing functions.

I note that all the functions used in the example are strictly increasing functions; I suspect it is for this reason.

Is this something worth bothering about? — Preceding unsigned comment added by 209.6.37.235 (talk) 16:12, 30 September 2012 (UTC)

Order Theory
I'm parking this here for now, where it can be fixed by someone with more knowledge of Stanley Smith Stevens coupled with what I'm about to tell you. First note that a layperson would understand 'order' or 'sorting' without qualification, to mean a total order. Total orders in general allow for 'tied' relations that the removed text suggested was special to preorders (a total order that does not allow ties is known as a 'strict total order'). A preorder is a lot more general than this and includes many relations that are contrary to our intuitive notion of 'ordering' (such as "A has a path to B") and even if it is a total preorder it can still have cycles (if A has a path to B then B also has a path to A so A>B and B>A even though A is not B).

The point about monotonically increasing function is true for strictly monotonic functions on total orders and needs no reference to abstract mathematical structures such as order isomorphism unless Stanley Smith Stevens himself referred to such things and then only with proper explanation.

The following is the offending text: 86.5.179.220 (talk) 00:22, 7 March 2013 (UTC)


 * In mathematical order theory, an ordinal scale defines a total preorder of objects (in essence, a way of sorting all the objects, in which some may be tied). The scale values themselves (such as labels like "great", "good", and "bad"; 1st, 2nd, and 3rd) have a total order, where they may be sorted into a single line with no ambiguities. If numbers are used to define the scale, they remain correct even if they are transformed by any monotonically increasing function. This property is known as the order isomorphism. A simple example follows:


 * {| class="sortable wikitable" style="font-size: 95%"

!class="unsortable"| !Judge's score x !Score minus 8 x-8 !Tripled score 3x !Cubed score x3
 * Alice's cooking ability
 * 10
 * 2
 * 30
 * 1000
 * Bob's cooking ability
 * 9
 * 1
 * 27
 * 729
 * Claire's cooking ability
 * 8.5
 * 0.5
 * 25.5
 * 614.125
 * Dana's cooking ability
 * 8
 * 0
 * 24
 * 512
 * Edgar's cooking ability
 * 5
 * -3
 * 15
 * 125
 * }
 * 5
 * -3
 * 15
 * 125
 * }
 * }


 * Since x-8, 3x, and x3 are all monotonically increasing functions, replacing the ordinal judge's score by any of these alternate scores does not affect the relative ranking of the five people's cooking abilities. Each column of numbers is an equally legitimate ordinal scale for describing their abilities. However, the numerical (additive) difference between the various ordinal scores has no particular meaning.


 * See also Strict weak ordering.

Temperature not an interval scale?
In the same way that height is not an interval scale because there is a fixed zero point, Isn't temperature also not an interval scale? the zero point is not arbitrary, it is 0K or -273.15 degrees celcius. It is impossible to have a -274. Also, it is wrong to say that 2Degrees C is twice as warm as 1 degree C. Temperature is the average kinetic energy of the molecules. 64.211.58.60 (talk) 01:20, 30 April 2012 (UTC)


 * By the terminology proposed by Stevens, temperature is an interval scale. The issue of zero points refers to ratio scales. -- WeijiBaikeBianji (talk, how I edit) 15:11, 22 March 2013 (UTC)

I think I'll pass through for a coherency copy-edit, and then update with sources.
I've been following the article edits and the talk page discussion here for a while, and I think I'll try to do a top-to-bottom close reading of the article for proofreading soon, and then add in references (with article text updates as needed) from the books I've had out of the local academic library for a while. I look forward to discussing further edits with all of you. -- WeijiBaikeBianji (talk, how I edit) 02:31, 30 April 2013 (UTC)

Thanks for the table improvement
I was going to reformat the table that is so visually dominant in the article, but I see another editor did that first. Thanks. I've been checking all the references for this article, and have most of the references in my office at the moment, so I will be checking the article text soon. -- WeijiBaikeBianji (talk, how I edit) 16:00, 24 July 2013 (UTC)

Dichotomous data
Dichotomous data (aka Binary data) is incorrectly placed under Ordinal. It should be under Nominal. Both by definition (see reference below), and common sense (their needs to be minimum of 3 categories in order to have an inherent order and fall under Ordinal). http://en.wikipedia.org/wiki/Binary_data

Furthermore, I would argue we should subdivide Ordinal into Short Ordinal Scales and Long Ordinal Scales. See reference from Cochrane below. This is helpful since different statistical tests/methods are often used for long ordinal scales.

http://www.cochrane-net.org/openlearning/html/mod11-2.htm — Preceding unsigned comment added by 70.171.0.180 (talk) 15:01, 4 October 2013 (UTC)

negative numbers for ratio measurements
Could someone talk about ratio measurements being negative? This has always confused me. If I take "5 kg" and subtract from it "7 kg", is the result "-2 kg" also a ratio measurement? — Preceding unsigned comment added by 2620:0:E50:1004:2E0:81FF:FEB1:B0A2 (talk) 15:53, 23 April 2014 (UTC)


 * Ratio measurement is generally related to measurements that have an ABSOLUTE zero, a quantity below which one cannot go. So it works for temperature, but not so well for weight. (One could have negative weight, that is buoyancy, with a balloon.) In my opinion, the whole rigid four-type framework by Stevens has some grave problems. We editors should use the sources mentioned in the article references to update this article to reflect that scholarly consensus. -- WeijiBaikeBianji (talk, how I edit) 16:42, 23 April 2014 (UTC)


 * There should be a 'zero allowed' column, with a checkmark for ratio scales, added to the synoptic chart. Given that a zero can be defined for that type of measurement (which implies that a zero calibration is necessary for the measuring instrument), you can subtract a larger from a smaller to net a negative. I find the comment about the zero Kelvin scale to conflict with the comment about the Celsius scale, however. One of those sentences ought to be deleted for consistency, Perhaps we ought to delete the Celsius scale from the ratio section.✅ 18:02, 7 June 2014 (UTC)
 * What would you think about money measurements as an example. Even children understand "I have no money.". --Ancheta Wis   (talk  &#124; contribs) 16:55, 23 April 2014 (UTC)

Let's rethink the table
For a while this article has included a table of different measurements, relating them to the Stevens scheme of classifying levels of measurement. I've just deleted the entire table because of its nonstandard formatting (see the Manual of Style on accessibility of tables for the user-friendliness considerations that motivated this decision). Perhaps restoring a table with much more simplified markup would be worthwhile, but note that the following section of this article has long pointed out that the Stevens scheme of classification is controversial and possibly incorrect. Let's discuss this based on current sources. You can find some of those sources in a source list I keep in user space for all Wikipedians to use, and I would be delighted to hear from you about other authoritative sources on the topic of this article. -- WeijiBaikeBianji (talk, how I edit) 13:49, 7 June 2014 (UTC)


 * The table entry for nominal scale had some important mathematical operations which are now deleted. I will re-enter them in the text section. --Ancheta Wis   (talk  &#124; contribs) 16:13, 7 June 2014 (UTC)


 * Thank you for your reply. Do you have sources at hand that discuss the issue of mathematical operations on different kinds of measurements? It will be good to cite those sources carefully in article text, whether that article text is a paragraph of written prose or a data table with accessible markup. -- WeijiBaikeBianji (talk, how I edit) 16:17, 7 June 2014 (UTC)
 * This is the subject of category theory. Marshall Stone, the son of the supreme court justice, started the subject. Justice Stone is said to have remarked, 'my son has written a book which I cannot understand'. The table had this kind of understanding of relationships (which are symbolized by arrows) embedded in it. I did not start or work on the table, but perhaps it might be reinstituted in some form somehow? --Ancheta Wis   (talk  &#124; contribs) 16:38, 7 June 2014 (UTC)
 * This source is meant for computer people and students; it supplements MacLane's Categories for the Working Mathematician, which is meant for mathematicians. But the subject is not meant to be abstract; certainly Stevens' work is concrete; computer people certainly think at the nominal level when programming, which is an application of the mathematics of categories.
 * I took care a while ago (last year, I think) to gather most of the sources then cited by this article. I'll check my mathematics reference books for what I have at hand about category theory, as I usually look at the psychology reference books in my office more when I think about this article and its topic. Thanks for pointing out that connection. -- WeijiBaikeBianji (talk, how I edit) 21:28, 7 June 2014 (UTC)

To all editors: it may be useful, to perform the following: ( under your Preferences, at the top of your screen, right side | Appearance Gadgets| check the Navigation Popups checkbox | Scrolldown |Save, wait for "Your preferences have been saved" in green ). I find that checking the Reference tooltips checkbox is useful as well, if the checkbox is not already checked for you.

The result of this selection is to display a summary of a wikilink (or citation) when you hover your mouse cursor over the wikilink. This feature may not be available for devices such as tablets, that do not display a cursor.

I have found this feature to be useful when studying an article. --Ancheta Wis   (talk  &#124; contribs) 08:19, 8 June 2014 (UTC)


 * Thanks for the tip. I found the navigation popup tooltip setting under the Gadgets tab in my preferences. -- WeijiBaikeBianji (talk, how I edit) 12:14, 8 June 2014 (UTC)
 * Thank you for your correction, --Ancheta Wis   (talk  &#124; contribs) 14:02, 8 June 2014 (UTC)

Would any editor object if we were to add elements of this list to the See Also? I will wait a week or so before adding some of the proposed elements, if no one objects. Of course, if we see support before the time period ends, we could add some sooner. --Ancheta Wis   (talk  &#124; contribs) 15:45, 22 June 2014 (UTC)
 * Peirce's categories
 * Categorization
 * Hypostatic abstraction
 * Nominalism
 * The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Nominal scale examples
The article could use a concrete example of a nominal scale. It mentions gender, for example, but it isn't clear if we are talking about something like 1-male, 2-female, 3-prefer not to say, or just the terms themselves. I suspect that "nominal scale" refers to the former, but I don't know. The article doesn't help. Lou Sander (talk) 15:48, 18 August 2014 (UTC)
 * It's pretty hard to list items which aren't also quantifiable, at least on the order of a mass noun. The best example I can come up with is types (and the names we have assigned to each type). If we talk about sets, categories, and types at least we have the domain right. Would that suffice? It's possible to talk about things without assigning a type, as in the  Ramsey-Lewis method. My difficulty is the example in that article is electrons, which are well quantified. Ramsey-Lewis can be applied to things that are not as well understood as electrons. Might that be what you are asking about?
 * The example you list could just as well be 'M', 'F', 'U' just to lift the example up beyond numbers. --Ancheta Wis    (talk  &#124; contribs) 16:12, 18 August 2014 (UTC)
 * I am no expert on levels of measurement. As I understand a nominal scale, we could be asking a number of people about their preference for different American football teams. We could tabulate the answers on a numbered sheet as follows:
 * Steelers - 8 people
 * Cowboys - 4 people
 * Packers - 6 people
 * Ravens - 1 person
 * Notre Dame - 3 people
 * Alabama - 3 people
 * Green Bay - 1 person


 * By my understanding:
 * The numbers 1-7 would have no meaning as to measurement; they are just convenient ways of identifying the different teams.
 * The fact that the Steelers have 8 fans could be used as a measurement in analysis, since the 8 is the modal/most frequent number of fans
 * Some of the operations listed in the article (and I only vaguely understand them) might be used to account for the facts that Notre Dame and Alabama are college teams, while the rest are professional teams.
 * Similarly, those operations could account for the fact that Packers and Green Bay are both the same team (there being no non-professional team in Green Bay, which is the home of the Packers). Lou Sander (talk) 19:10, 25 August 2014 (UTC)
 * I agree with your statements. In database language, identifiers 1-7 serve as primary keys to a record for that team, and 8 serves as an alternate key to the Packers. And to be more precise about the Nominal scale, the identifiers are candidate keys to putative records about some category of objects, such as draft picks. --Ancheta Wis   (talk  &#124; contribs) 20:00, 25 August 2014 (UTC)

Citation style is a mixture of inline and the standard refs
Anon 2602:304:cfb2:7240:41bf:e3b2:4c7c:a42c has kindly pointed out that the citation style is a mixture of inline (e.g. Ferguson et.al., 1940) and the standard refs. --Ancheta Wis   (talk  &#124; contribs) 17:24, 17 September 2014 (UTC)


 * I really like the citation style suggested in User:RexxS/Cite_multiple_pages and would be happy to move this article over to that style, as I have several of the references at hand in my office. -- WeijiBaikeBianji (talk, how I edit) 18:31, 17 September 2014 (UTC)
 * Support. I like your proposal! --Ancheta Wis   (talk  &#124; contribs) 19:37, 17 September 2014 (UTC)

Citations for lede
@Danielkueh: The citations needed for the lede need to be taken from a variety of domains of discourse, ranging from mathematics, logic, and computer science, to measure theory, to probability theory and statistics, to engineering, natural science, and social science.

S.S. Stevens (1946), in the original citation, identifies just what what is controversial: "the meaning of measurement". But the topic is understood in a variety of domains of discourse. For example, Max Born Natural philosophy of cause and chance Oxford 1949, states that a statement of function y = f(x), or law in the domain of discourse, is the whole point: "This logical dependence needs no further analysis (I even think it cannot be further analysed)."

In pictures, the ovals are colored, red for X in this case, denoted 'the domain', blue for Y, denoted 'co-domain, or range': The dots from the red oval, or domain of definition, are the inputs to the experiment, in this case, a measurement. Paul R. Halmos Naive set theory p.2: "the principal concept of set theory ... is that of belonging." Thus the dots in the red oval belong to set X.

The dots in the blue oval, which is highlighted in yellow, are the result of the measurement. For the mathematicians, measure theory became interesting with the real numbers, which for Stevens is the 'interval'. But for computer scientists and logicians, even categories or types are interesting, which for Stevens is the 'nominal' type, and the arrows from X to Y, for f: X -> Y are their livelihood.

--Ancheta Wis   (talk  &#124; contribs) 10:01, 25 July 2015 (UTC)
 * , Three things: This article is about level of measurement, not about the multiple disciplines that use these levels of measurement. Squeezing in six disciplines in the lead contributes to clutter and takes the focus away from the main topic of this article. So I have pushed them back to the end of the lead and added a citation tag. Second, if you want to include citations from multiple disciplines, then by all means do so. Previously, there was none. Now there is at least one. Finally, interesting example about set theory. But Stevens is not mentioned in that book. At least, not through a search on Google books. Examples such as those need to be verified. danielkueh (talk) 13:25, 25 July 2015 (UTC)

Suggested reorganization
Perhaps the organization should be into 1. Introduction 2. Stevens' method and 3. Other methods with subcategories under the third?

Stevens was not the only one to propose a scheme. PeterLFlomPhD (talk) 11:01, 25 July 2015 (UTC)
 * If you would like to add a third, you should add it to the main body of this article first before adding it to the lead as the lead is supposed to provide a summary of the article. danielkueh (talk) 13:31, 25 July 2015 (UTC)
 * danielkueh What I'm proposing would be a pretty big reorganization. As a new editor (10 days) I'm a bit hesitant to do it. PeterLFlomPhD (talk) 13:56, 25 July 2015 (UTC)
 * No worries. Just write up a draft using the sandbox WP:sandbox. Once you have something ready, post or link it here for further discussion. danielkueh (talk) 14:07, 25 July 2015 (UTC)
 * danielkueh Thanks! I wrote something up in my sandbox.  How do I link it here? PeterLFlomPhD (talk) 14:35, 25 July 2015 (UTC)
 * Like this: User:PeterLFlomPhD/sandbox. Also, take a gander at:
 * WP:RS and WP:V for recommended sources. These two pages will really serve you well, especially if you intend to edit on Wikipedia in the long-run.
 * WP:OR and WP:SYNTH for WP's policies on original research.
 * WP:MOS for writing and organizational style guidelines. Also, check WP:lead for writing the lead of a WP article.
 * WP:CT for citation formats and templates.
 * A lot to digest but take your time. Good luck. Have fun! danielkueh (talk) 14:51, 25 July 2015 (UTC)

A draft of a suggested reorganization
User:PeterLFlomPhD/sandbox PeterLFlomPhD (talk) 16:33, 25 July 2015 (UTC)
 * I have two comments. First, I recommend changing "Introduction" to "Overview" under Stevens's typology. Second, I strongly recommend that you cite secondary sources (e.g., mainstream and reputable textbooks, review journal articles, monograms, etc) along with the primary sources by Chrisman and Mosteller (see WP:SECONDARY). The general rule in WP is that all the interpretation, summary, and documentation of information should done by the authors of secondary sources and not by WP editors. This is to guard against WP:OR and WP:fringe. All in all, the proposed changes look good. Thanks you. danielkueh (talk) 20:50, 25 July 2015 (UTC)
 * danielkueh Thanks! I will post them. PeterLFlomPhD (talk) 20:59, 25 July 2015 (UTC)

Parametric versus non-parametric
Shouldn't this article clarify how nominal data and ordinal data are commonly clarified as non-parametric data, whereas interval and ratio data are classified as parametric data? Vorbee (talk) 14:24, 30 August 2017 (UTC)

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Dichotomous data as example of ordinal level?
Similar to the comment above, I was surprised to see dichotomous data given as an example of ordinal level of measurement. Dichotomous data doesn't have a natural order and should be listed as nominal. Its mathematical operation is only equal or not equal. It doesn't become ordinal until a metric is applied (i.e. positive is better than negative when the metric is blood type match, while negative is better than positive when the metric is being tested for an incurable disease). This is also true for data that is 'trichotomous' or 'n-chotomous' (if you'll excuse my abuse of English). Additionally, a median value for 'n-chotomous' data doesn't naturally occur (i.e. if a survey asks a participant for their gender, and the answers are 'male', 'female', and 'other').

I'm reluctant to make an edit on the page since I'm not an active WP editor, but I propose changing the second sentence under ordinal data to:

"Examples include nominal data whose values are naturally transitive, such as 'completely agree', 'mostly agree', 'mostly disagree', 'completely disagree' when measuring opinion, or 'No Education', 'High School Diploma', 'Bachelor's Degree', 'Master's Degree', 'PhD' when assessing level of education."

BAusmus, May 2019 — Preceding unsigned comment added by BAusmus (talk • contribs) 17:55, 31 May 2019 (UTC)

What is a yardstick?
I am confused as what "yardstick" advanced operation for intervals stand for? The linked page does not help, and some clarification would be much appreciated.

Thanks, User:boramalper — Preceding undated comment added 08:29, 26 February 2021 (UTC)


 * I think that having the "Comparison" table at the start of the article is useful, but it has some problems:
 * I also have no idea what a "yardstick" is in this context.
 * The "Interval" row has plus and minus as "Mathematical operators". Minus is correct because the definition of "interval" is that it is meaningful to talk about the difference between items.  But, in general, it is not meaningful to add values.  For example, dates are interval variables but it is not sensible to add the year numbers 1999 and 2021.  An exception to this is when calculating an arithmetic mean (the values can be added, and then divided by the count).  This exception can be explained as taking the differences (the deviations) from an arbitrary origin, averaging them, and then adding back the origin.  I suggest removing the plus sign from the table, and putting "Arithmetic mean" as the "Advanced operation".
 * The column heading "Incremental progress" is not very helpful. It just needs to say "Level".
 * I do not understand the note which says "This is inverted for the 'Measure property'."
 * JonH (talk) 16:00, 15 May 2021 (UTC)