*(A typical tape measure with both metric and imperial units and two US pennies for comparison)*

Physical *measurement* is the action of determining the value or values of a quantity (length, capacity, etc.) by comparison with a constant quantity of the same species taken as a reference term (standard or unit). According to the canonical definition:

To measure a magnitude is to compare it to another magnitude of the same kind taken as unit.

The comparison is quantitative; the quantity is expressed by a rational number multiplying the unit. Physical measurement is to give an accurate communication about an object. Units common to all measures are needed; otherwise, conversion rules must be established. The International Bureau of Weights and Measures has established an International System of Units based on a small number of standards and inter-unit relationships.

Unlike a count, giving an integer defined by a logical response, a physical measure is an estimate and therefore has some uncertainty.

In physics, physical measurement is defined as the determination of the numerical value of a quantity by the interpretation of the result of an experiment or an observation. The conditions and methods of this experiment or observation determine the uncertainty. Uncertainty is an integral part of the measure, and may even be its main result.

#### Overview

“In physical science the first essential step in the direction of learning any subject is to find principles of numerical reckoning and practicable methods for measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of

Science, whatever the matter may be.”

– William Thomas Thomson, Lord Kelvin, PLA, vol. 1, “Electrical Units of Measurement”, 1883-05-03

In the natural sciences, the act of measuring an object involves comparing a characteristic of the object with a standard unit using a dedicated instrument under controlled conditions. Examples of measuring instruments are the thermometer, the voltmeter, the odometer, the tachometer, the dynamometer, etc. In order to perform a precise physical measurement, the measuring instruments must be constructed carefully and correctly calibrated. Each measurement has an associated uncertainty, which estimates the quality of the measurement. Although a measurement is given in the usual way by a number followed by a unit, it actually has three components: the estimate, the uncertainty and the probability that the measurement is within a given interval. For example, measuring the length of a board can be 9 meters plus or minus 0.01 meters, with a probability of 0.95. In other words, if one proceeds 100 times to the measurement of this length, about 95 times one will find a value included between 8.99 meter and 9.01 meter (one speaks about the 95% confidence interval).

Physical measurement is fundamental in most scientific fields (chemistry, physics, biology, etc.). Physical measurement is also essential for a large number of industrial and commercial applications ranging from engineering to manufacturing, pharmaceutical production and electronics.

#### Metrology

In physics, *metrology* is considered as the study of the measurement of physical quantities, the definition of their units, and the practical realization of manipulations to arrive at a numerical result.

In general, a *metric* is a measurement system, direct or indirect, defined with respect to a clearly defined standard and a scale. The quantification of a quantity by the measurement process is based on the implicit or explicit existence of a metric.

##### Measurement and counting

The measure is usually different from counting. A *counting* is a natural whole number and can be exact. For example, there are twelve eggs in a box when counting them. Formally, counting depends on logic. It depends on answers to a question whose answer has only two possible values, such as “Is this object an egg?”. As a result, an expression like “this carton contains 12 eggs” can be broken down into a combination of similar questions, and can be “true” or “false”.

It is not the same for a measurement. A *measure* is the result of an experiment or manipulation of an object, making it possible to calculate a rational number. One can always find a small difference in size of the object for which two consecutive experiments give the same result, and one can always find a small difference of result between two consecutive experiments.

However, some groups can not be easily counted, and their numbers are estimated by statistical inference from a sample whose size determines the uncertainty, as for a physical measure.

*Example – Estimation of the number of patients*: To study the evolution of a pandemic, one counts the statements made by doctors, and one extrapolates a probable number of affected people, with a margin of error.

The methods of physical measurement are thus not limited to physical quantities. Metrological results and theories can be extended to the quantification of intensities of all kinds, provided that a metric has been defined. In the social sciences, as well as in other fields of research such as health, biology and market research, probabilistic models such as the Rasch model for measurements are applied with, for measuring instruments, questionnaires and surveys, that allow comparisons between people.

The results of these assessments are rational numbers, which are often proportions relative to a total population, not counts.

#### Choice of units

* (The seven base units in the SI system. Arrows point from units to those that depend on them. )*

It is sometimes better to choose a non-standard unit to measure certain physical quantities. This choice may depend in particular on the order of magnitude (it is easier to count the interstellar distances in light-years, for example) or the use that we need.

However, giving two measurement results of the same size in the same unit makes it possible to compare and facilitate the interpretation of the data, even if it means using a unit that does not correspond to its size.

#### International system

*(The International Prototype Kilogram (IPK) is an artefact standard or prototype that is defined to be exactly one kilogram mass. )*

The international system of units (abbreviated as SI) is the modern and revised form of the metric system. It is the most widespread system of units in the world, both in everyday life and in scientific fields. The SI was developed in the 1960s from the MKS system (meter-kilogram-second) preferentially to the CGS system (centimeter-gram-second), which has multiple variants. The SI introduces from the beginning many new units that are not part of the metric system.

There are two types of SI units, base units and secondary units. The basic units are the measurements corresponding to time, length, mass, temperature, quantity (of objects), electric current, and light intensity. Secondary units are built on base units; for example the density which is expressed in kg/m^{3}.

##### Conversion prefixes

To “eliminate” the prefixes, the use of multiplication is the simplest. To convert the meters to centimeters is to multiply the quantities in meters by 100, since there is 100 centimeters in one meter. And conversely.

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