Level of measurement
The level of measurement of a variable in mathematics and statistics describes how much information the numbers associated with the variable contain. Different mathematical operations on variables are possible, depending on the level at which a variable is measured. In statistics, the kinds of descriptive statistics and significance tests that are appropriate depend on the level of measurement of the variables concerned. These levels were identified by Stanley Smith Stevens in "Mathematics, Measurement, and Psychophysics" (Handbook of Experimental Psychology, Wiley, 1951).
Four levels of measurement are usually recognised:
- Nominal measurement. The numbers are names or labels. They can and often are replaced by verbal 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. Information entropy is available as a measure of statistical dispersion, but no notion of standard deviation or the like exists. Variables that are measured only nominally are also called categorical variables.
- Ordinal measurement. The numbers have all the features of nominal measures and also 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 and IQ 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 have all the features of ordinal measurement and also are separated by the same interval. In this case, differences between arbitrary pairs of numbers can be meaningfully compared. Operations such as addition and subtraction are therefore meaningful. However, the zero point on the scale is arbitrary, and ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out. On the other hand, negative values on the scale can be used. An example is the year date in many calendars. 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.
- Ratio measurement. The numbers have all the features of interval measurement and also have meaninful ratios between arbitrary pairs of numbers. Operations such as multiplication and division are therefore meaningful. The zero value on a ratio scale is non-arbtrary. 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.
Interval and/or ratio measurement are sometimes referred to as "true measurement", though this usage reflects a lack of understanding of the uses of ordinal measurement. However, it is only quantities measured on ratio scales that can correctly be said to have units of measurement.
There is 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.