# Measurement: Wikis

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# Encyclopedia

In science, measurement is the process of obtaining the magnitude of a quantity, such as length or mass, relative to a unit of measurement, such as a meter or a kilogram. A measurement answers the general question, "how many?", as in how many miles, or millimeters, or gigahertz. As measurement is basically about counting, measurement is conducted in numbers and is quantitative, in comparison to other observations which may be made in words and are qualitative. The term measurement can also be used to refer to a specific result obtained from the measurement process.

## History

The word "measurement" is derived from the Greek word "metron" which means a limited proportion.

The history of measurements is a topic within the history of science and technology.

## Standards

With the exception of a few seemingly fundamental quantum constants, units of measurement are essentially arbitrary; in other words, people make them up and then agree to use them. There is nothing inherent in nature which dictates that an inch has to be a certain length, or that a mile is a better measure of distance than a kilometer. Over the course of human history, however, first for convenience and then for necessity, standards of measurement evolved so that people would have certain common benchmarks. Laws regulating measurement were originally developed to prevent fraud in commerce. Today, units of measurement are generally defined on a scientific basis, overseen by governmental or supra-governmental agencies, and established in international treaties. The meter, for example, was redefined in 1983 as the distance traveled by light in free space in 1⁄299,792,458 of a second. In the United States, the National Institute of Standards and Technology (NIST), a division of the United States Department of Commerce, regulates commercial measurements. In the United Kingdom, the role is performed by the National Physical Laboratory (NPL).

## Units and systems

A baby bottle that measures in all three measurement systems, Imperial (U.K.), U.S. Customary, and metric.

The definition or specification of precise standards of measurement involves two key features, which are evident in the International System of Units (SI). Specifically, in this system the definition of each of the base units refer to specific empirical conditions and, with the exception of the kilogram, also to other quantitative attributes. Each derived SI unit is defined purely in terms of a relationship involving it and other units; for example, the unit of velocity is 1 m/s. Because derived units refer to base units, the specification of empirical conditions is an implied component of the definition of all units.

### Imperial system

Before SI units were widely adopted around the world, the British systems of English units and later Imperial units were used in Britain, the Commonwealth and the United States. The system came to be known as U.S. customary units in the United States and is still in use there and in a few Caribbean countries. These various systems of measurement have at times been called foot-pound-second systems after the Imperial units for distance, weight and time even though the tons, hundredweights, gallons, and nautical miles, for example, are different for the U.S. units. Many Imperial units remain in use in Britain despite the fact that it has officially switched to the SI system. Road signs are still in miles, yards, miles per hour, and so on, people tend to measure their own height in feet and inches and milk is sold in pints, to give just a few examples. Imperial units are used in many other places, for example, in many Commonwealth countries that are considered metricated, land area is measured in acres and floor space in square feet, particularly for commercial transactions (rather than government statistics). Similarly, gasoline is sold by the gallon in many countries that are considered metricated.

### Metric system

The metric system is a decimalized system of measurement based on the metre and the gramme. It exists in several variations, with different choices of base units, though these do not affect its day-to-day use. Since the 1960s, the International System of Units (SI), explained further below, is the internationally recognized standard metric system. Metric units of mass, length, and electricity are widely used around the world for both everyday and scientific purposes. The main advantage of the metric system is that it has a single base unit for each physical quantity. All other units are powers of ten or multiples of ten of this base unit. Unit conversions are always simple because they will be in the ratio of ten, one hundred, one thousand, etc. All lengths and distances, for example, are measured in meters, or thousandths of a meter (millimeters), or thousands of meters (kilometers), and so on. There is no profusion of different units with different conversion factors as in the Imperial system (e.g. inches, feet, yards, fathoms, rods). Multiples and submultiples are related to the fundamental unit by factors of powers of ten, so that one can convert by simply moving the decimal place: 1.234 meters is 1234 millimeters or 0.001234 kilometers. The use of fractions, such as 2/5 of a metre, is not prohibited, but uncommon.

### SI

The International System of Units (abbreviated SI from the French language name Système International d'Unités) is the modern, revised form of the metric system. It is the world's most widely used system of units, both in everyday commerce and in science. The SI was developed in 1960 from the metre-kilogram-second (MKS) system, rather than the centimetre-gram-second (CGS) system, which, in turn, had many variants. At its development the SI also introduced several newly named units that were previously not a part of the metric system. The SI units for the four basic physical quantities: length, time, mass, and temperature are:

1. meter (m)  :SI unit of length
2. second (s)  :SI unit of time
3. kilogram (kg) :SI unit of mass
4. kelvin (K)  :SI unit of temperature

There are two types of SI units, base and derived units. Base units are the simple measurements for time, length, mass, temperature, amount of substance, electric current and light intensity. Derived units are made up of base units, for example, density is kg/m3.

#### Converting prefixes

The SI allows easy multiplication when switching among units having the same base but different prefixes. To convert from meters to centimeters it is only necessary to multiply the number of meters by 100, since there are 100 centimetres in a metre. Inversely, to switch from centimetres to meters one multiplies the number of centimeters by 0.01 or divide centimeters by 100.

### Distance

A 2-metre carpenter's ruler

A ruler or rule is a tool used in, for example, geometry, technical drawing, engineering, and carpentry, to measure distances or to draw straight lines. Strictly speaking, the ruler is the instrument used to rule straight lines and the calibrated instrument used for determining length is called a measure, however common usage calls both instruments rulers and the special name straightedge is used for an unmarked rule. The use of the word measure, in the sense of a measuring instrument, only survives in the phrase tape measure, an instrument that can be used to measure but cannot be used to draw straight lines. As can be seen in the photographs on this page, a two-metre carpenter's rule can be folded down to a length of only 20 centimetres, to easily fit in a pocket, and a five-metre long tape measure easily retracts to fit within a small housing.

## Some special names

We also use some special names for some multiples of some units.

• 100 kilograms = 1 quintal;1000 kilogram = 1 metric tonne;
• 10 years = 1 decade; 100 years = 1 century; 1000 years = 1 millennium

The Australian building trades adopted the metric system in 1966 and the units used for measurement of length are metres (m) and millimetres (mm). Centimetres (cm) are avoided as they cause confusion when reading plans, the length two and a half metres is usually recorded as 2500 mm or 2.5 m.

### Mass

Mass refers to the intrinsic property of all material objects to resist changes in their momentum. Weight, on the other hand, refers to the downward force produced when a mass is in a gravitational field. In free fall, objects lack weight but retain their mass. The Imperial units of mass include the ounce, pound, and ton. The metric units gram and kilogram are units of mass.

A unit for measuring weight or mass is called a weighing scale or, often, simply a scale. A spring scale measures force but not mass, a balance compares masses, but requires a gravitational field to operate. The most accurate instrument for measuring weight or mass is the digital scale, but it also requires a gravitational field, and would not work in free fall.

### Economics

The measures used in economics are physical measures, nominal price value measures and fixed price value measures. These measures differ from one another by the variables they measure and by the variables excluded from measurements. The measurable variables in economics are quantity, quality and distribution. By excluding variables from measurement makes it possible to better focus the measurement on a given variable, yet, this means a narrower approach.

## Difficulties

Since accurate measurement is essential in many fields, and since all measurements are necessarily approximations, a great deal of effort must be taken to make measurements as accurate as possible. For example, consider the problem of measuring the time it takes an object to fall a distance of one metre (39 in). Using physics, it can be shown that, in the gravitational field of the Earth, it should take any object about 0.45 second to fall one metre. However, the following are just some of the sources of error that arise. First, this computation used for the acceleration of gravity 9.8 metres per second per second (32.2 ft/s²). But this measurement is not exact, but only precise to two significant digits. Also, the Earth's gravitational field varies slightly depending on height above sea level and other factors. Next, the computation of .45 seconds involved extracting a square root, a mathematical operation that required rounding off to some number of significant digits, in this case two significant digits.

So far, we have only considered scientific sources of error. In actual practice, dropping an object from a height of a metre stick and using a stopwatch to time its fall, we have other sources of error. First, and most common, is simple carelessness. Then there is the problem of determining the exact time at which the object is released and the exact time it hits the ground. There is also the problem that the measurement of the height and the measurement of the time both involve some error. Finally, there is the problem of air resistance.

Scientific measurements must be carried out with great care to eliminate as much error as possible, and to keep error estimates realistic.

## Definitions and theories

### Classical definition

In the classical definition, which is standard throughout the physical sciences, measurement is the determination or estimation of ratios of quantities. Quantity and measurement are mutually defined: quantitative attributes are those which it is possible to measure, at least in principle. The classical concept of quantity can be traced back to John Wallis and Isaac Newton, and was foreshadowed in Euclid's Elements.[citation needed]

### Representational theory

In the representational theory, measurement is defined as "the correlation of numbers with entities that are not numbers"[1]. The strongest form of representational theory is also known as additive conjoint measurement. In this form of representational theory, numbers are assigned based on correspondences or similarities between the structure of number systems and the structure of qualitative systems. A property is quantitative if such structural similarities can be established. In weaker forms of representational theory, such as that implicit within the work of Stanley Smith Stevens[2], numbers need only be assigned according to a rule.

The concept of measurement is often misunderstood as merely the assignment of a value, but it is possible to assign a value in a way that is not a measurement in terms of the requirements of additive conjoint measurement. One may assign a value to a person's height, but unless it can be established that there is a correlation between measurements of height and empirical relations, it is not a measurement according to additive conjoint measurement theory. Likewise, computing and assigning arbitrary values, like the "book value" of an asset in accounting, is not a measurement because it does not satisfy the necessary criteria.

### Information theory

Information theory recognizes that all data are inexact and statistical in nature. Thus the definition of measurement is: "A set of observations that reduce uncertainty where the result is expressed as a quantity."[3]. This definition is implied in what scientists actually do when they measure something and report both the mean and statistics of the measurements. In practical terms, one begins with an initial guess as to the value of a quantity, and then, using various methods and instruments, reduces the uncertainty in the value. Note that in this view, unlike the positivist representational theory, all measurements are uncertain, so instead of assigning one value, a range of values is assigned to a measurement. This also implies that there is a continuum between estimation and measurement.

### Quantum mechanics

In quantum mechanics, a measurement is the "collapse of the wavefunction". The unambiguous meaning of the measurement problem is an unresolved fundamental problem in quantum mechanics.

## In physics

Measuring the ratios between physical quantities is an important sub-field of physics.

Some important physical quantities include:

## References

1. ^ Ernest Nagel: "Measurement", Erkenntnis, Volume 2, Number 1 / December, 1931, pp. 313-335, published by Springer, the Netherlands
2. ^ Stevens, S.S. On the theory of scales and measurement 1946. Science. 103, 677-680.
3. ^ Douglas Hubbard: "How to Measure Anything", Wiley (2007), p. 21

# Study guide

Up to date as of January 14, 2010

## Meaning and definition of measurement

Measurement is the process of attaching a numeric value to an aspect of a natural phenomenon, such as the volume of the milk produced by a cow, in order to be able to describe that phenomenon accurately and make comparisons to other similar phenomena, like "the crop production of the farm have had a 120% growth over the last 5 years."

To begin the process of measurement, we need to recognize the type of phenomenon, called the physical dimension, that we would like to measure. For example the diameter of the front wheel of a bicycle is of type Length, how fast the bicycle is moving is described by Speed and the amount of air crammed inside the wheel is determined by Pressure. The next thing we are going to need for doing a measurement is a Standard Unit for that type of aspect of natural phenomenon. For example, we can select a special person, like a king, and announce the length of his foot as the standard unit of length. From now on, when we are talking about the height of a sapling, we are talking of how many feet as long as the king's foot do we need to cover the sapling. the number of the feet needed, which might turn out to be 7.75 and there is nothing wrong with that, is now considered the length of that thing. Congratulations! we have managed to correctly attach a numeric value (7.75) to the length of the sapling.

Now, for a more formal definition of measurement, we can refer to wikipedia, which says: "Measurement is the process of estimating the ratio of a magnitude of a quantity to a unit of the same type. A measurement is the result of such a process, expressed as the multiple of a real number and a unit, where the real number is the ratio. An example is 9 metres, which is an estimate of an object's length relative to a unit of length, one metre."

## Importance of measurement

Unless we are able to measure some phenomena, we cannot say we scientifically know anything about that thing.

Measurement gives a base to understand the universe. All around us we are surrounded by various things. We might not note it but unconciously we are actually "measuring" things and understanding them one way or the other. Just imagine how a world would be without being able to measure anything. We are surrounded by Measurement.

## Base Units

In Physics, we have now moved from Imperial units (pounds, yards etc), to the Metric System (metres, grams etc). However, we also use a variety of base units, and from these base units, we are able to derive some of the familiar units we know, i.e. the Newton. This range of units compile into a group known as the Le Système International d'Unités (SI). Most of the units you should recognize from your previous studies; these units are the base units in the SI system measurement:

 Dimension Unit of Measure Description Symbol Base Unit Symbol Length x, y, z, d, l, r, or s meter m Time t second s Mass M or m kilogram kg Current I or i Ampere Amp Temperature T or θ kelvin K Amount of Substance n mole mol Luminous intensity I candela cd

## Derived Units

With these base units, we can combine them to form derived units, such as the Newton, acceleration or speed; as an example, let us look at speed. Speed is described by the following equation:

$Speed=\frac{Distance}{Time}$

As you know, Distance is in Metres, and Time is in Seconds, so m divided by s obviously gives us m/s. Since this is higher physics, it needs to be put into index notation, which means that the derived unit now becomes ms − 1 If we write this as a non-negative power, then we get:

$ms^{-1}=\frac{m}{s^1}$

Now, let us use this derived unit to find another unit commonly met, Acceleration.

Acceleration is the rate of change of Velocity, described by the equation:

$Acceleration=\frac{Final~Velocity-Initial~Velocity}{Time~Taken}$ or in symbols: $a=\frac{v-u}{ \Delta\ t}$

Thus, if we divide ms − 1 by the Time Taken (in seconds) we get ms − 2

Again, if we write this as a non-negative power, we get:

$ms^{-2}=\frac{m}{s^2}$

## Homogeneity

This has shown how base units are used to form the derived units we know, the use of base units can also tell us whether an equation is Homogenous or not. Homogeneity can be used to see whether an equation is correct or not, but be warned, this does not necessarily mean that an equation is correct overall, it only says whether the base units are the same on both sides! The Speed equation is Homogenous because the product of the equation is ms − 1, and we divided the distance (m) by the time (s), so the base units used on one side of the equation are the same as the base units used on the other side of the equation.

## Methods of Measurement

The base quantities are measured in different ways. The Kilogram is measured on scales, length is measure with a ruler, a Micrometre, Vernier Callipers, a Laser, etc. Current is measured with an Ammeter, Temperature with a thermometer, Time with a watch.
The amount of substance is deduced by equations, as is Luminous Intensity.

## Uncertainty

All measurements have a degree of uncertainty. This can be shown as an absolute uncertainty, or percentage uncertainty. (Percentage uncertainty) = 100 $\times$ (absolute uncertainty)/(measured value). The last figure in a number that is measured is called the doubtful figure.

# Wikibooks

Up to date as of January 23, 2010

### From Wikibooks, the open-content textbooks collection

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