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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.

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:

- nominal (also categorical or discrete)
- ordinal
- interval
- ratio

Interval and ratio variables are also grouped together as continuous variables.

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.

Level | Can define… | Relation or Operation | Mathematical structure |
---|---|---|---|

nominal | mode | equality (=) | [[set]No meaning of Arithmetic Operation] |

ordinal | median | order (<) | totally ordered set |

interval | mean, standard deviation | subtraction (−) and weighted average | affine line |

ratio | geometric mean, coefficient of variation | addition (+) and multiplication (×) | field |

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

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. The corresponding variable can be called an ordered categorical variable.

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 or interquartile range, since one cannot subtract quantities.

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.

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.

A ratio measurement scale is one in which the ratio between any two measurements is meaningful. To achieve this a ratio scale has to have a non-arbitrary zero value. Then operations such as multiplication and division become meaningful as well. For a ratio scale one can thus say "This value is double this other value".

"If it's twice as cold today as it was yesterday," runs a popular joke, "and it was zero degrees yesterday, how cold is it today?" This illustrates the limitation of interval measurements such as Celsius and Fahrenheit temperature: by setting zero at an arbitrary point, they make it impossible to multiply and divide meaningfully.

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.

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.

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.

- Babbie, E. (2004). The Practice of Social Research, 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0-534-62029-9
- Duncan, O. D. (1984). Notes on social measurement: historical and critical. New York: Russell Sage Foundation.
- Lord, F.M. (1953). On the Statistical Treatment of Football Numbers. Reprint in Readings in Statistics, Ch. 3, (Haber, A., Runyon, R.P., and Badia, P.) Reading, Mass: Addison-Wesley, 1970.
- Michell, J. (1986). Measurement scales and statistics: a clash of paradigms. Psychological Bulletin, 3, 398-407.
- Stevens, S.S. (1946). On the theory of scales of measurement. Science, 103, 677-680.
- Stevens, S.S. (1951). Mathematics, measurement and psychophysics. In S.S. Stevens (Ed.), Handbook of experimental psychology (pp. 1-49). New York: Wiley.
- Stevens, S.S. (1975). Psychophysics. New York: Wiley.
- Velleman, P. F. & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), 65-72. [On line] http://www.spss.com/research/wilkinson/Publications/Stevens.pdf
- Briand, L. & El Emam, K. & Morasca, S. (1995). On the Application of Measurement Theory in Software Engineering. Empirical Software Engineering, 1, 61-88. [On line] http://www2.umassd.edu/swpi/ISERN/isern-95-04.pdf

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