Level of measurement

The level of measurement of a variable in mathematics and statistics is a classification that is used to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. The levels were proposed by Stanley Smith Stevens in his 1946 article On the theory of scales of measurement. According to Stevens' theory of scales, different mathematical operations on variables are possible, depending on the level at which a variable is measured.

Classification levels
According to the classification scheme, in statistics the kinds of descriptive statistics and significance tests that are appropriate depend on the level of measurement of the variables concerned

Stevens proposed four levels of measurement, described below: Interval and ratio variables are also grouped together as continuous variables.
 * nominal (also categorical or discrete)
 * ordinal
 * interval
 * ratio

In the paper in which Stevens introduced the classification Scheme, he also proposed the definition that is widely cited in texts in some version: "Measurement is the assignment of numbers to objects or events according to a rule". This definition has received criticism on a number of grounds (e.g. Duncan, 1984; Michell, 1986, 1999). However, the scheme is widely used.

The levels are in increasing order of mathematical structure—meaning that more operations and relations are defined—and the higher levels are required to define some statistics.

Nominal measurement
In this type of measurement, names are assigned to objects as labels. This assignment is performed by evaluating, by some procedure, the similarity of the to-be-measured instance to each of a set of named exemplars or category definitions. The name of the most similar named exemplar or definition in the set is the "value" assigned by nominal measurement to the given instance. If two instances have the same name associated with them, they belong to the same category, and that is the only significance that nominal measurements have. Variables that are measured only nominally are also called categorical variables. etc.

Nominal numbers
For practical data processing the names may be numerals, but in that case the numerical value of these numerals is irrelevant, and the concept is now sometimes referred to as nominal number. The only comparisons that can be made between variable values are equality and inequality. There are no "less than" or "greater than" relations among the classifying names, nor operations such as addition or subtraction.

Origin and examples
"Nominal measurement" was first identified by psychologist Stanley Smith Stevens in the context of a child learning to categorize colors (red, blue and so on) by comparing the similarity of a perceived color to each of a set of named colors previously learned by ostensive definition. In social research, variables measured at a nominal level include gender, marital status, race, religious affiliation, political party affiliation, college major, and birthplace. Other examples include: geographical location in a country represented by that country's international telephone access code, or the make or model of a car.

Statistical measures
The only kind of measure of central tendency is the mode; median and mean cannot be defined.

Statistical dispersion may be measured with various indices of qualitative variation, but no notion of standard deviation exists.

Ordinal measurement
In this classification, the numbers assigned to objects represent the rank order (1st, 2nd, 3rd etc.) of the entities measured. The numbers are called ordinals. The variables are called ordinal variables or rank variables. Comparisons of greater and less can be made, in addition to equality and inequality. However, operations such as conventional addition and subtraction are still meaningless.

Examples
Examples include the Mohs scale of mineral hardness and also the pH scale); the results of a horse race, which say only which horses arrived first, second, third, etc. but no time intervals; and many measurements in psychology and other social sciences, for example attitudes like preference, conservatism or prejudice, social class, and baseball metaphors for sex.

Statistical measures
The central tendency of an ordinally measured variable can be represented by its mode or its median, but mean cannot be defined.

One can define quantiles, notably quartiles and percentiles, together with maximum and minimum, but no new measures of statistical dispersion beyond the nominal ones can be defined: one cannot define range (statistics) and interquartile range, since one cannot subtract quantities.

Interval measurement
The numbers assigned to objects have all the features of ordinal measurements, and in addition equal differences between measurements represent equivalent intervals. That is, differences between arbitrary pairs of measurements can be meaningfully compared. Operations such as averaging and subtraction are therefore meaningful, but addition is not, and a zero point on the scale is arbitrary; 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 a notion of units of measurement, though the latter usage is not obvious and is not recommended.

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.

Examples
Examples of interval measures are the year date in many calendars, and temperature in Celsius scale or Fahrenheit scale; temperature in the Kelvin scale is a ratio measurement, however.

Statistical measures
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
The numbers assigned to objects have all the features of interval measurement and also have meaningful ratios between arbitrary pairs of numbers. Operations such as multiplication and division are therefore meaningful. The zero value on a ratio scale is non-arbitrary. Variables measured at the ratio level are called ratio variables.

Examples
Most physical quantities, such as mass, length or energy are measured on ratio scales; so is temperature measured in kelvins, that is, relative to absolute zero.

Social variables of ratio measure include age, length of residence in a given place, number of organizations belonged to or number of church attendances in a particular time.

Statistical measures
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.

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.

In a ratio variable, unlike in an interval variable, the moments about the origin are meaningful, since the origin is not arbitrary.

"True measurement"
The interval and ratio measurement levels are sometimes collectively called "true measurement", although it has been argued that this usage reflects a lack of understanding of the uses of ordinal measurement. Only ratio or interval scales can correctly be said to have units of measurement.

Debate on classification scheme
There has been, and continues to be, debate about the merits of the classifications, particularly in the cases of the nominal and ordinal classifications (Michell, 1986). Thus, while Stevens' classification is widely adopted, it is by no means universally accepted (for example, Velleman & Wilkinson, 1993). 

Duncan (1986) observed that Stevens' classification nominal measurement is contrary to his own definition of measurement. Stevens (1975) said on his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to a nonrule". However, so-called nominal measurement involves arbitrary assignment, and the "permissible transformation" is any number for any other. This is one of the points made in Lord's (1953) satirical paper On the Statistical Treatment of Football Numbers.

Among those who accept the classification scheme, there is also some controversy in behavioural sciences over whether the mean is meaningful for ordinal measurement. In terms of measurement theory, it is not, because the arithmetic operations are not made on numbers that are measurements in units, and so the results of computations do not give numbers in units. However, many behavioural scientists use means for ordinal data anyway. This is often justified on the basis that ordinal scales in behavioural science are really somewhere between true ordinal and interval scales; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude. For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across a range of an assessment. Thus, some argue, that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. Statistical analysis software such as PSPP require the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot inadvertently perform meaningless analyses (for example correlation analysis with a variable on a nominal level).

L. L. Thurstone made progress toward developing a justification for obtaining interval-level measurements based on the law of comparative judgment. Further progress was made by Georg Rasch, who developed the probabilistic Rasch model which provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.