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Level of Measurement Level of measurement From Wikipedia, the free encyclopedia The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement".[1] In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval" and "ratio". Contents 1 The theory of scale types o 1.1 Nominal scale o 1.2 Ordinal scale o 1.3 Interval scale o 1.4 Ratio measurement 2 Debate on classification scheme 3 Scale types and Stevens' "operational theory of measurement" 4 Notes 5 See also 6 References 7 External links The theory of scale types Stevens (1946, 1951) proposed that measurements can be classified into four different types of scales. These are shown in the table below as: nominal, ordinal, interval, and ratio. Admissible Scale Mathematical Scale Type Permissible Statistics Transformation structure nominal (also standard set One to One denoted as mode, Chi-square structure (equality (=)) categorical) (unordered) Monotonic ordinal median, percentile totally ordered set increasing (order (<)) mean, standard deviation, correlation, Positive linear interval affine line regression, analysis of variance (affine) All statistics permitted for interval scales plus the following: geometric mean, Positive similarities ratio field harmonic mean, coefficient of variation, (multiplication) logarithms Nominal scale 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. 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. 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. A Likert Scale is a type of ordinal scale and may also use names with an order such as: "bad", "medium", and "good"; or "very satisfied", "satisfied", "neutral", "unsatisfied", "very unsatisfied." 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 conclusion from such statistical data. Psychometricians like to theorise 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 score Score minus 8 Tripled score Cubed score 3 x x-8 3x x 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 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. 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). The Likert scale, which is one of the most common scales used in survey research, would be a popular example and practical application of the 'interval scale'. 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. 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 non- arbitrary 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 non arbitrary as the particles that compose matter at this temperature have zero kinetic energy.
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