Level of measurement

The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order 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. Different mathematical operations on variables are possible, depending on the level at which a variable is measured. 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. Four levels of measurement were proposed by Stevens: nominal, ordinal, interval and ratio.

Nominal measurement
In this classification, numerals are assigned to objects as labels or names. If two entities have the same number associated with them, they belong to the same category, and that is the only significance that they have. The only comparisons that can be made between variable values are equality and inequality. There are no "less than" or "greater than" relations among them, nor operations such as addition or subtraction. Examples include: the international telephone code for a country, the numbers on the shirts of players in a sports team, or the number of a bus. The only kind of measure of central tendency is the mode. Statistical dispersion may be measured with a variation ratio, index of qualitative variation, or via information entropy, but no notion of standard deviation exists. Variables that are measured only nominally are also called categorical variables. In social research, variables measured at a nominal level include gender, race, religious affiliation, political party affiliation, college major, and birthplace.

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 ordinals. Comparisons of greater and less can be made, in addition to equality and inequality. However operations such as conventional addition and subtraction are still without meaning. A physical example is the Mohs scale of mineral hardness. Another example is the results of a horse race; which horses arrived first, second, third, etc. are reported, but the time intervals between the horses are not reported. Most measurement in psychology and other social sciences is at the ordinal level; for example attitudes (like conservatism or prejudice) and social class, are only measured at the ordinal level. If customers surveyed report preferring chocolate- to vanilla-flavored ice cream, the data are of this kind. The central tendency of an ordinally measured variable can be represented by its mode or its median; the latter will give more information. Variables measured at the ordinal level are referred to as ordinal variables or rank variables.

Interval measurement
The numbers assigned to objects have all the features of ordinal measurements. In addition, equal differences between measurements represent equivalent intervals; i.e. differences between arbitrary pairs of measurements can be meaningfully compared. Operations such as addition and subtraction are therefore meaningful. The zero point on the scale is arbitrary; negative values on the scale can be used. An example is the year date in many calendars. Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly on the measurements. Ratios of differences can however 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; the mean will give most information. Variables measured at the interval level are referred to as interval variables, or sometimes as scaled variables, though the latter usage is not obvious and is not recommended. Temperature in Celsius scale or Fahrenheit scale is also measured as interval measure. About the only interval measures commonly used in social scientific research are constructed measures such as standardized intelligence tests (IQ).

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. Most physical quantities, such as mass, length or energy are measured on ratio scales; so is temperature when it is measured in kelvins, i.e. relative to absolute zero. The central tendency of a variable measured at the interval level can be represented by its mode, its median, its arithmetic mean, or its geometric mean; however as with an interval scale, the arithmetic mean will give the most useful information. Variables measured at the interval level are referred to as ratio variables. Social variables of ratio measure would include age, length of residence in a given place, number of organisations belonged to or number of church attendances in a particular times.

Interval and/or ratio measurement are sometimes referred to as "true measurement", though it is often argued 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 classifiction scheme
There has been, and continues to be, debate about the merit of the classifications, particularly in the cases of the nominal and ordinal classifications (Michell, 1986). Thus, while Stevens' classification is widely adopted, it is not universally accepted (for example, Velleman & Wilkinson, 1993).

Among those who accept the classification scheme, there is also some controversy in behavioural sciences over whether the mean is meaningful for ordinal measurement. Mathematically it is not, but some behavioural scientists use it 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. 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.

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.

External link
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 * Hyperstat &mdash; Measurement Scales