---
title: "Level of measurement"
chunk: 2/4
source: "https://en.wikipedia.org/wiki/Level_of_measurement"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:44:10.167435+00:00"
instance: "kb-cron"
---

==== Central tendency and dispersion ====
According to Stevens, for ordinal data, the appropriate measure of central tendency is the median (the mode is also allowed, but not the mean), and the appropriate measure of dispersion is percentile or quartile (the standard deviation is not allowed). Those restrictions would imply that correlations can only be evaluated using rank order methods, and statistical significance can only be evaluated using non-parametric methods (R. M. Kothari, 2004). But the restrictions have not been generally endorsed by statisticians.
In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity according to his rules, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires. Indeed, most psychological data collected by psychometric instruments and tests, measuring cognitive and other abilities, are ordinal (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008). In particular, IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only. There is no zero point that represents an absence of intelligence, and a 10-point difference may carry different meanings at different points of the scale.

=== Interval scale ===
The interval type allows for defining the degree of difference between measurements, but not the ratio between measurements. Examples include temperature scales with the Celsius scale, date when measured from an arbitrary epoch (such as AD), location in Cartesian coordinates, and direction measured in degrees from true or magnetic north. Ratios are not meaningful since 20 °C cannot be said to be "twice as hot" as 10 °C, nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another; for example, the ten-degree difference between 15 °C and 25 °C is twice the five-degree difference between 17 °C and 22 °C.

==== Central tendency and dispersion ====
According to Stevens, the mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range and standard deviation. Since one can only divide by differences, one cannot define measures that require some ratios, such as  the coefficient of variation. More subtly, while one can define moments about the origin, only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments, since ratios of differences are meaningful, but one cannot define the 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 scale ===
See also: Positive real numbers § Ratio scale
The ratio 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 of measurement of the same kind (Michell, 1997, 1999). Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass, length, duration, plane angle, energy and electric charge. In contrast to interval scales, ratios can be compared using division. Ratio scales are often used to express an order of magnitude such as for temperature in Orders of magnitude (temperature).

==== Central tendency and dispersion ====
According to Stevens, the geometric mean and the harmonic mean are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The studentized range and the coefficient of variation are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for the ratio scale.