3.1 Scale of measurement

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.

Nominal	Ordinal	Interval	Ratio
People or objects with the same scale value are the same on some attribute. The values of the scale have no 'numeric' meaning in the way that you usually think about numbers.	People or objects with a higher scale value have more of some attribute. The intervals between adjacent scale values are indeterminate. Scale assignment is by the property of "greater than," "equal to," or "less than."	Intervals between adjacent scale values are equal with respect the attribute being measured. E.g., the difference between 8 and 9 is the same as the difference between 76 and 77. Positive linear affine	There is a rationale zero point for the scale. Ratios are equivalent, e.g., the ratio of 2 to 1 is the same as the ratio of 8 to 4. Positive similarities multiplication

3.1 Scale of measurement

Nominal	Ordinal	Interval	Ratio
One to One (equality (=)) Statistics tools: mode, Chi-squared	Monotonic increasing (order ( < )) Statistics tools: mode, Chi-squared	Statistics tools: mean, standard deviation, correlation, regression, analysis of variance	All statistics permitted for interval scales plus the following: geometric mean, harmonic mean, coefficient of variation, logarithms

Nominal scale

At the nominal scale, i.e., for a nominal category, one uses labels; for example, football players, smoking females, sex, religious and kind of personalities. We can select any number to express these kind without any relation between the qualities of the number and the variable. For examples the 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.

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...

3.1 Scale of measurement

...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".

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. An example of an ordinal scale is the result of a horse race, which says only which horses arrived first, second, or third but...

3.1 Scale of measurement

...include no information about race times.

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, 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. 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.

Interval scale

Quantitative attributes are all measurable on interval scales, as any difference between the levels of an...

3.1 Scale of measurement

...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, inter quartile range, and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as standardized 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.

3.1 Scale of measurement

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.

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 standardized range or coefficient of variation.