Ordinal Scale

When items are classified according to whether they have more or less of a characteristic, the scale used is referred to as an ordinal scale (definition of ordinal scale). The main characteristic of the ordinal scale is that the categories have a logical or ordered relationship to each other.

These types of scale permit the measurement of degrees of difference, but not the specific amount of difference. This scale is very common in marketing, satisfaction and attitudinal research. Any questions that ask the respondent to rate something are using ordinal scales. For example,

How would you rate the service of our wait-staff?

Excellent 0 Very good 0 Good 0 Fair 0 Poor 0

Although we would know that respondent X ("very good") thought the service to be better than respondent Y ("good"), we have no idea how much better nor can we even be sure that both respondents have the same understanding of what constitutes "good service" and therefore, whether they really differ in their opinion about its quality.

Likert scales are commonly used in attitudinal measurements. This type of scale uses a five-point scale ranging from strongly agree, agree, neither agree nor disagree, disagree, strongly disagree to rate people’s attitudes. Variants of the Likert-scale exist that use any number of points between three and ten, however it is best to give at least four or five choices. Be sure to include all possible responses: sometimes respondents may not have an opinion or may not know the answer, and therefore you should include a "neutral" category or the possibility to check off "undecided/uncertain", "no opinion" or "don’t know".

Although some researchers treat them as an interval scale, we do not really know that the distances between answer alternatives are equal. Hence only the mode and median can be calculated, but not the mean. The range and percentile ranking can also be calculated.

Interval Scale

Interval scales (definition of interval scale) take the notion of ranking items in order one step further, since the distance between adjacent points on the scale are equal. For instance, the Fahrenheit scale is an interval scale, since each degree is equal but there is no absolute zero point. This means that although we can add and subtract degrees (100° is 10° warmer than 90°), we cannot multiply values or create ratios (100° is not twice as warm as 50°). What is important in determining whether a scale is considered interval or not is the underlying intent regarding the equal intervals: although in an IQ scale, the intervals are not necessarily equal (e.g. the difference between 105 and 110 is not really the same as between 80 and 85), behavioural scientists are willing to assume that most of their measures are interval scales as this allows the calculation of of averages – mode, median and mean – , the range and standard deviation.

Although Likert scales are really ordinal scales (definition of ordinal scale), they are often treated as interval scales. By treating this type of agreement scale or attitudinal measurement as interval, researchers can calculate mean scores which can then be compared. For instance, the level of agreement for men was 3.5 compared to 4.1 for women, or it was 3.3 for first time visitors compared to 2.8 for repeat visitors.

Ratio Scale

When a scale consists not only of equidistant points but also has a meaningful zero point, then we refer to it as a ratio scale. If we ask respondents their ages, the difference between any two years would always be the same, and ‘zero’ signifies the absence of age or birth. Hence, a 100-year old person is indeed twice as old as a 50-year old one. Sales figures, quantities purchased and market share are all expressed on a ratio scale.

Ratio scales should be used to gather quantitative information, and we see them perhaps most commonly when respondents are asked for their age, income, years of participation, etc. In order to respect the notion of equal distance between

adjacent points on the scale, you must make each category the same size. Therefore, if your first category is $0-$19,999, your second category must be $20,000-$39,999. Obviously, categories should never overlap and categories should follow a logical order, most often increasing in size.

Ratio scales are the most sophisticated of scales, since it incorporates all the characteristics of nominal (definition of nominal scale), ordinal (definition of ordinal scale) and interval scales (definition of interval scale). As a result, a large number of descriptive calculations are applicable.

At the nominal scale, i.e., for a nominal category, one uses labels; for example, rocks can be generally categorized as igneous, sedimentary and metamorphic. For this scale, some valid operations are equivalence and set membership. Nominal measures offer names or labels for certain characteristics.

Variables assessed on a nominal scale are called categorical variables; see also categorical data. Categorically-typed random variables that have only two possible outcomes (often termed "yes" vs. "no" or "success" vs. "failure") are known as binary variables (or Bernoulli variables), and characterized using a Bernoulli distribution. A categorical variable with 3 or more outcomes is sometimes termed multi-way (or K-way for some specific value of K), and characterized by a categorical distribution.

Stevens (1946, p. 679) must have known that claiming nominal scales to measure obviously non-quantitative things would have attracted criticism, so he invoked his theory of measurement to justify nominal scales as measurement:

“ …the use of numerals as names for classes is an example of the assignment of numerals according to rule. The rule is: Do not assign the same numeral to different classes or different numerals to the same class. Beyond that, anything goes with the nominal scale. ”

The central tendency of a nominal attribute is given by its mode; neither the mean nor the median can be defined.

We can use a simple example of a nominal category: first names. Looking at nearby people, we might find one or more of them named Aamir. Aamir is their label; and the set of all first names is a nominal scale. We can only check whether two people have the same name (equivalence) or whether a given name is in on a certain list of names (set membership), but it is impossible to say which name is greater or less than another (comparison) or to measure the difference between two names. Given a set of people, we can describe the set by its most common name (the mode), but cannot provide an "average name" or even the "middle name" among all the names. However, if we decide to sort our names alphabetically (or to sort them by length; or by how many times they appear in the US Census), we will begin to turn this nominal scale into an ordinal scale.

[edit] Ordinal scale

Rank-ordering data simply puts the data on an ordinal scale. Ordinal measurements describe order, but not relative size or degree of difference between the items measured. In this scale type, the numbers assigned to objects or events represent the rank order (1st, 2nd, 3rd, etc.) of the entities assessed. An example of an ordinal scale is the result of a horse race, which says only which horses arrived first, second, or third but include no information about race times. Another is the Mohs scale of mineral hardness, which characterizes the hardness of various minerals through the ability of a harder material to scratch a softer one, saying nothing about the actual hardness of any of them. Yet another example is military ranks; they have an order, but no well-defined numerical difference between ranks.

When using an ordinal scale, the central tendency of a group of items can be described by using the group's mode (or most common item) or its median (the middle-ranked item), but the mean (or average) cannot be defined.

In 1946, Stevens observed that psychological measurement usually operates on ordinal scales, and that ordinary statistics like means and standard deviations do not have valid interpretations. Nevertheless, such statistics can often be used to generate fruitful information, with the caveat that caution should be taken in drawing conclusions from such statistical data.

Psychometricians like to theorize that psychometric tests produce interval scale measures of cognitive abilities (e.g. Lord & Novick, 1968; von Eye, 2005) but there is little prima facie evidence to suggest that such attributes are anything more than ordinal for most psychological data (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008). In particular,[2] IQ scores reflect an ordinal scale, in which all scores are only meaningful for comparison, rather than an interval scale, in which a given number of IQ "points" corresponds to a unit of intelligence.[3][4][5] Thus it is an error to write that an IQ of 160 is just as different from an IQ of 130 as an IQ of 100 is different from an IQ of 70.[6][7]

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:

Judge's scorex

Score minus 8x-8

Tripled score3x

Cubed scorex3

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 8 0 24 512

Judge's scorex

Score minus 8x-8

Tripled score3x

Cubed scorex3

ability

Edgar's cooking ability

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.

[edit] Interval scale

Quantitative attributes are all measurable on interval scales, as any difference between the levels of an attribute can be multiplied by any real number to exceed or equal another difference. A highly familiar example of interval scale measurement is temperature with the Celsius scale. In this particular scale, the unit of measurement is 1/100 of the temperature difference between the freezing and boiling points of water under a pressure of 1 atmosphere. The "zero point" on an interval scale is arbitrary; and negative values can be used. The formal mathematical term is an affine space (in this case an affine line). Variables measured at the interval level are called "interval variables" or sometimes "scaled variables" as they have units of measurement.

Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly. But ratios of differences can be expressed; for example, one difference can be twice another.

The central tendency of a variable measured at the interval level can be represented by its mode, its median, or its arithmetic mean. Statistical dispersion can be measured in most of the usual ways, which just involved differences or

averaging, such as range, interquartile range, and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as studentized range or coefficient of variation. More subtly, while one can define moments about the origin, only central moments are useful, since the choice of origin is arbitrary and not meaningful. One can define standardized moments, since ratios of differences are meaningful, but one cannot define coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.

[edit] Ratio measurement

Most measurement in the physical sciences and engineering is done on ratio scales. Mass, length, time, plane angle, energy and electric charge are examples of physical measures that are ratio scales. The scale type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). Informally, the distinguishing feature of a ratio scale is the possession of a zero value. For example, the Kelvin temperature scale has a non-arbitrary zero point of absolute zero, which is denoted 0K and is equal to -273.15 degrees Celsius. This zero point is accuracy representing the particles that compose matter at this temperature having zero kinetic energy.

Examples of ratio scale measurement in the behavioral sciences are all but non-existent. Luce (2000) argues that an example of ratio scale measurement in psychology can be found in rank and sign dependent expected utility theory.

All statistical measures can be used for a variable measured at the ratio level, as all necessary mathematical operations are defined. The central tendency of a variable measured at the ratio level can be represented by, in addition to its mode, its median, or its arithmetic mean, also its geometric mean or harmonic mean. In addition to the measures of statistical dispersion defined for interval variables, such as range and standard deviation, for ratio variables one can also define measures that require a ratio, such as studentized range or coefficient of variation.

Attitude Scale Construction (Likert’s Method)

A Likert scale is a psychometric scale commonly involved in research that employs questionnaires. It is the most widely used approach to scaling responses in survey research, such that the term is often used interchangeably with rating scale, or more accurately the Likert-type scale, even though the two are not synonymous. The scale is named after its inventor, psychologist Rensis Likert.[2]

Likert distinguished between a scale proper, which emerges from collective responses to a set of items (usually eight or more), and the format in which responses are scored along a range. Technically speaking, a Likert scale refers only to the former. The difference between these two concepts has to do with the distinction Likert made between the underlying phenomenon being investigated and the means of capturing variation that points to the underlying phenomenon. [3]

When responding to a Likert questionnaire item, respondents specify their level of agreement or disagreement on a symmetric agree-disagree scale for a series of statements. Thus, the range captures the intensity of their feelings for a given item,[4] while the results of analysis of multiple items (if the items are developed appropriately) reveals a pattern that has scaled properties of the kind Likert identified.

Sample question presented using a five-point Likert item

An important distinction must be made between a Likert scale and a Likert item. The Likert scale is the sum of responses on several Likert items. Because Likert items are often accompanied by a visual analog scale (e.g., a horizontal line, on which a subject indicates his or her response by circling or checking tick-marks), the items are sometimes called scales themselves. This is the source of much confusion; it is better, therefore, to reserve the term Likert scale to apply to the summed scale, and Likert item to refer to an individual item.

A Likert item is simply a statement which the respondent is asked to evaluate according to any kind of subjective or objective criteria; generally the level of agreement or disagreement is measured. It is considered symmetric or "balanced" because there are equal amounts of positive and negative positions.[5]

Often five ordered response levels are used, although many psychometricians advocate using seven or nine levels; a recent empirical study [6] found that a 5- or 7- point scale may produce slightly higher mean scores relative to the highest

possible attainable score, compared to those produced from a 10-point scale, and this difference was statistically significant. In terms of the other data characteristics, there was very little difference among the scale formats in terms of variation about the mean, skewness or kurtosis.

The format of a typical five-level Likert item, for example, could be:

1. Strongly disagree2. Disagree3. Neither agree nor disagree4. Agree5. Strongly agree

Likert scaling is a bipolar scaling method, measuring either positive or negative response to a statement. Sometimes an even-point scale is used, where the middle option of "Neither agree nor disagree" is not available. This is sometimes called a "forced choice" method, since the neutral option is removed [7]. The neutral option can be seen as an easy option to take when a respondent is unsure, and so whether it is a true neutral option is questionable. It has been shown that when comparing between a 4-point and a 5-point Likert scale, where the former has the neutral option unavailable, the overall difference in the response is negligible. Likert scales may be subject to distortion from several causes. Respondents may avoid using extreme response categories (central tendency bias); agree with statements as presented (acquiescence bias); or try to portray themselves or their organization in a more favorable light (social desirability bias). Designing a scale with balanced keying (an equal number of positive and negative statements) can obviate the problem of acquiescence bias, since acquiescence on positively keyed items will balance acquiescence on negatively keyed items, but central tendency and social desirability are somewhat more problematic.

Scoring and analysis

After the questionnaire is completed, each item may be analyzed separately or in some cases item responses may be summed to create a score for a group of items. Hence, Likert scales are often called summative scales.

Whether individual Likert items can be considered as interval-level data, or whether they should be treated as ordered-categorical data is the subject of considerable disagreement in literature[9][10], with strong convictions on what are the most applicable methods. This disagreement can be traced back, in many respects, to the extent in which Likert items are interpreted as being ordinal data.

There are two primary considerations in this discussion. Firstly, a key factor to accept is that Likert scales are arbitrary. The value assigned to a Likert item has no unique mathematical property, either in terms of measure theory or scale (from which a distance metric can be determined). The value assigned for each Likert item is simply determined by the researcher as providing the necessary detail for their research. However, for convention, Likert items tend to take progressive positive integer values. Likert scales typically range from 2 to 10 – with 5 or 7 being the most common. In this, the typical structure of the Likert scale is such that each progressive Likert item is treated as having a ‘better’ response than the preceding value. (This may differ in cases where reverse ordering of the Likert Scale is needed).

The second, and possibly more important point, is whether the ‘distance’ between each successive Likert item is equidistant – which is traditionally inferred. For example, in the above 5-point Likert Scale, the inference is that the ‘distance’ between items ‘1’ and ‘2’ is the same as between items ‘3’ and ‘4’. In terms of good research ethics, an equidistant presentation by the researcher is important; otherwise it will introduce a research bias into the analysis. For example, a 4-point Likert Scale – Poor, Average, Good, Very Good – is unlikely to be equidistant as there is only one item that can receive a below average rating. This would clearly bias any result in favor of a better outcome. However, even if a researcher presents an equidistant scale, this may not be interpreted as such by the respondent.

A good Likert scale, as above, will present a symmetry of Likert items about a middle category that have clearly defined linguistic qualifiers for each item. In such symmetric scaling, equidistant attributes will typically be more clearly observed or, at least, inferred. It is when a Likert scale is symmetric and equidistant that it will behave more like an interval-level measurement. So while a Likert scale is ordinal (which cannot be denied) – if it is well presented, then it may be possible the Likert Scale can approximate an interval-level measurement. This is beneficial as, if it was treated just as an ordinal scale, then some valuable information could be lost if the ‘distance’ between Likert items were not available for consideration. The important idea here is that the appropriate type of analysis is dependent on how the Likert scale has been presented.

Given its ordinal basis, it remains more correct to summarize the central tendency of responses from a Likert scale by using either the median or the mode, with ‘spread’ measured by quartiles or percentiles[11]. Non-parametric tests should be preferred for statistical inferences, such as chi-squared test, Mann–Whitney test, Wilcoxon signed-rank test, or Kruskal–Wallis test.[12] . While some commentators[13] consider that parametric analysis is justified for a Likert scale using the Central Limit Theorem, this should be reserved for when the Likert scale has suitable symmetry and equidistance so an interval-level measurement can be approximated and reasonably inferred.

Responses to several Likert questions may be summed, providing that all questions use the same Likert scale and that the scale is a defensible approximation to an interval scale, in which case they may be treated as interval data measuring a latent variable. If the summed responses fulfill these assumptions, parametric statistical tests such as the analysis of variance can be applied. These can be applied only when more than 5 Likert questions are summed.]

Data from Likert scales are sometimes reduced to the binomial level by combining all agree and disagree responses into two categories of "accept" and "reject". The chi-squared, Cochran Q, or McNemar test are common statistical procedures used after this transformation.

Consensus based assessment (CBA) can be used to create an objective standard for Likert scales in domains where no generally accepted standard or objective standard exists. Consensus based assessment (CBA) can be used to refine or even validate generally accepted standards.

Level of measurement

The five response categories are often believed to represent an Interval level of measurement. But this can only be the case if the intervals between the scale points correspond to empirical observations in a metric sense. Reips and Funke (2008)[14] show that this criterion is much better met by a visual analogue scale. In fact, there may also appear phenomena which even question the ordinal scale level in Likert scales. For example, in a set of items A,B,C rated with a Likert scale circular relations like A>B, B>C and C>A can appear. This violates the axiom of transitivity for the ordinal scale.

THE LIKERT-TYPE SCALE In a recent article in Today's Speech, the writers described an instrument frequently used in persuasion studies, the semantic differential. Prior to the development of the semantic differential, several other attitude measuring instruments were used, some of which are still in common use. One of the most popular methods of measuring attitudes is the method of summated ratings, commonly referred to as the Likert-type scale.

The Likert-type scale has been used by persuasion researchers for over three decades. The original scale of this type was developed by Rensis Likert and is explained in his article, "A Technique for the Measurement of Attitudes," in Achieves of Psychology (1932). He reported very satisfactory reliability data for the scales developed with his procedure. In addition, Likert reported that results obtained from his scales compared favorably with those obtained by the "granddaddy" of the attitude scales--the Thurstone scale. Subsequent research has generally confirmed the fact that the Likert-type attitude scales are quite reliable and valid instruments for the measurement of attitude.

A Likert-type scale consists of a series of declarative statements. The subject is asked to indicate whether he agrees or disagrees with each statement. Commonly, five options are provided: "strongly agree," "agree," "undecided," "disagree," and "strongly disagree." Other Likert-type scales include four or six steps rather than five, excluding the undecided position.

Scales developed by the Likert method will ordinarily include from six to thirty declarative statements. Some of these statements will be worded in a positive manner and other will be worded in a negative manner. For example, two statements which might be used in a scale to measure attitude toward capital punishment would be, "Capital punishment is nothing but legalized murder," and "Capital punishment gives the murderer just what he deserves." A person who is in favor of capital punishment would be expected to disagree with the first statement but agree with the second statement. Of course, the person opposed to capital punishment would be expected to give opposite responses. The individual responses "strongly agree" through "strongly disagree" are assigned numbers, usually 1-5. In this manner the responses to the various items are quantified and may be summed across statements to give a total score for the individual on the scale. It is necessary, of course, that the assigned numbers are consistent with the meaning of the response. For example, the first statement above could be scored 1-5 and the second statement scored 5-1. In this way a person with a strongly favorable attitude toward capital punishment would receive a score of 10 for these two items while a person strongly opposed to capital punishment would receive a score of 2.

Attitude scores produced by Likert-type scales have been found by Likert and others to yield data which may be analyzed by "normal curve" statistics. Because this type of data is desired by researchers in most cases, the Likert-type scale provides a very useful and relatively uncomplicated method of obtaining data on people's attitudes for persuasion research.

SOME APPLICATIONS

Likert attitude scales can be used in much the same way as semantic differential scales. In an earlier Today's Speech article, the writers suggested four areas of research in which the semantic differential could be used: (1) to measure the credibility of speakers, (2) to measure listener attitudes, (3) for classroom evaluation of speakers and speeches, and (4) to assess the worth of speech courses to the students. The same purpose could be served by a Likert-type scale.

Rather than repeat the discussion of these four uses in detail, two special applications of Likert scaling are presented. First, Likert scales can be used by speech teachers to assist a student in audience analysis. Second, Likert scales can be used profitably as a means of course evaluation.

One of the primary functions of an introductory course in speech is to help a student understand the concept of audience analysis. To analyze an audience is to achieve some understanding of audience attitudes or opinions on numerous issues. Once a speaker picks a topic for a speech, he needs to know what the audience believes about that topic if he is going to give an effective speech. Likert attitude scales can be administered to the audience prior to the speech to ascertain those beliefs. All too often, the speaker is told after his speech by some listener, "I agreed with you before you started the speech." It is not recommended that these attitude scales be administered before every speech, but an occasional assessment of listener attitudes should improve the student's understanding of audience analysis.

Like the semantic differential, Likert scales can be used as a means of course evaluation. Unlike the semantic differential, these scales can accomplish much in little time. For example, it would take twelve separate responses to discover a person's attitude toward a textbook and an instructor with the semantic differential. With Likert scaling, two responses would yield the same data. This efficiency can best be appreciated by the overloaded instructor. Students can evaluate numerous facets of instruction, course content, and the instructor in a relatively short period of time.

Today's Speech -- WORTHWHILE?

A discussion of this research procedure would be incomplete without examples of Likert scales. Let us again try to get you to react to a number of items related to Today's Speech. Again, you are invited to fill out these scales.

Today's Speech 1. The covers on Today's Speech add to the caliber of the journal.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ strongly disagree (1)

2. The articles appearing in Today's Speech are valuable.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

3. The Editor of Today's Speech should be relieved of his position.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

4. The format of this journal is not attractive.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

Now that you have completed these scales, add up the number value beside each of the verbs that you checked. This number represents your overall attitude toward Today's Speech. How does this score compare with your score on the semantic differential scale for Today's Speech? Have you changed your attitude?

The second example is taken from a speech course evaluation questionnaire developed by the first author. Try example two. Remember these questions pertain to a person's evaluation of a particular speech course.

1. Written assignments in speech classes are necessary.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

2. Speech is a very worthwhile course.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

3. The speech instructor does not synthesize, integrate or summarize the material.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

4. The course material is poorly organized.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)

5. The textbook in this course is very good.

_____ Strongly agree (5)

_____ Agree (4)

_____ Undecided (3)

_____ Disagree (2)

_____ Strongly disagree (1)