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title: "Hypothesis"
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A hypothesis (pl.: hypotheses) is a proposed explanation for a phenomenon. A scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educated guess.
If a hypothesis is repeatedly independently demonstrated by experiment to be true, it becomes a scientific theory. In colloquial usage, the words hypothesis and theory are often used interchangeably, but this is incorrect in the context of science.
A working hypothesis is a provisionally-accepted hypothesis used for the purpose of pursuing further progress in research. Working hypotheses are frequently discarded, and often proposed with knowledge (and warning) that they are incomplete and thus false, with the intent of moving research in at least somewhat the right direction, especially when scientists are stuck on an issue and brainstorming ideas.
In formal logic, a hypothesis is the antecedent in a proposition. For example, in the proposition "If P, then Q", statement P denotes the hypothesis (or antecedent) of the consequent Q. Hypothesis P is the assumption in a (possibly counterfactual) "what if" question. The adjective "hypothetical" (having the nature of a hypothesis or being assumed to exist as an immediate consequence of a hypothesis), can refer to any of the above meanings of the term "hypothesis".
== Uses ==
In its ancient usage, hypothesis referred to a summary of the plot of a classical drama. The English word hypothesis comes from the ancient Greek word ὑπόθεσις (hypothesis), whose literal or etymological sense is "putting or placing under" and hence in extended use has many other meanings including "supposition".
In Plato's Meno (86e–87b), Socrates dissects virtue with a method which he says is used by mathematicians, that of "investigating from a hypothesis". In this sense, 'hypothesis' refers to a clever idea or a short cut, or a convenient mathematical approach that simplifies cumbersome calculations. Cardinal Robert Bellarmine gave a famous example of this usage in the warning issued to Galileo in the early 17th century: that he must not treat the motion of the Earth as a reality, but merely as a hypothesis.
In common usage in the 21st century, a hypothesis refers to a provisional idea whose merit requires evaluation. For proper evaluation, the framer of a hypothesis needs to define specifics in operational terms. A hypothesis requires more work by the researcher in order to either confirm or disprove it. In due course, a confirmed hypothesis may become part of a theory or occasionally may grow to become a theory itself. Normally, scientific hypotheses have the form of a mathematical model. Sometimes, but not always, one can also formulate them as existential statements, stating that some particular instance of the phenomenon under examination has some characteristic and causal explanations, which have the general form of universal statements, stating that every instance of the phenomenon has a particular characteristic.
In entrepreneurial setting, a hypothesis is used to formulate provisional ideas about the attributes of products or business models. The formulated hypothesis is then evaluated, where the hypothesis is proven to be either "true" or "false" through a verifiability- or falsifiability-oriented experiment.
Any useful hypothesis will enable predictions by reasoning (including deductive reasoning). It might predict the outcome of an experiment in a laboratory setting or the observation of a phenomenon in nature. The prediction may also invoke statistics and only talk about probabilities. Karl Popper, following others, has argued that a hypothesis must be falsifiable, and that one cannot regard a proposition or theory as scientific if it does not admit the possibility of being shown to be false. Other philosophers of science have rejected the criterion of falsifiability or supplemented it with other criteria, such as verifiability (e.g., verificationism) or coherence (e.g., confirmation holism). The scientific method involves experimentation to test the ability of some hypothesis to adequately answer the question under investigation. In contrast, unfettered observation is not as likely to raise unexplained issues or open questions in science, as would the formulation of a crucial experiment to test the hypothesis. A thought experiment might also be used to test the hypothesis.
In framing a hypothesis, the investigator must not currently know the outcome of a test or that it remains reasonably under continuing investigation. Only in such cases does the experiment, test or study potentially increase the probability of showing the truth of a hypothesis. If the researcher already knows the outcome, it counts as a "consequence" — and the researcher should have already considered this while formulating the hypothesis. If one cannot assess the predictions by observation or by experience, the hypothesis needs to be tested by others providing observations. For example, a new technology or theory might make the necessary experiments feasible.
== Scientific hypothesis ==
A trial solution to a problem is commonly referred to as a hypothesis—or, often, as an "educated guess"—because it provides a suggested outcome based on the evidence. However, some scientists reject the term "educated guess" as incorrect. Experimenters may test and reject several hypotheses before solving the problem.
According to Schick and Vaughn, researchers weighing up alternative hypotheses may take into consideration:
Testability (compare falsifiability as discussed above)
Parsimony (as in the application of "Occam's razor", discouraging the postulation of excessive numbers of entities)
Scope – the apparent applicability of the hypothesis to multiple known phenomena
Fruitfulness – the prospect that the hypothesis may explain further phenomena in the future
Conservatism – the degree of "fit" with existing recognized knowledge-systems.
== Working hypothesis ==

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A working hypothesis is a hypothesis that is provisionally accepted as a basis for further research in the hope that a tenable theory will be produced, even if the hypothesis ultimately fails.
Like all hypotheses, a working hypothesis is constructed as a statement of expectations, which can be linked to the exploratory research purpose in empirical investigation. Working hypotheses are often used as a conceptual framework in qualitative research.
The provisional nature of working hypotheses makes them useful as an organizing device in applied research. Here they act like a useful guide to address problems that are still in a formative phase.
In recent years, philosophers of science have tried to integrate the various approaches to evaluating hypotheses, and the scientific method in general, to form a more complete system that integrates the individual concerns of each approach. Notably, Imre Lakatos and Paul Feyerabend, Karl Popper's colleague and student, respectively, have produced novel attempts at such a synthesis.
== Hypotheses, concepts and measurement ==
Concepts in Hempel's deductive-nomological model play a key role in the development and testing of hypotheses. Most formal hypotheses connect concepts by specifying the expected relationships between propositions. When a set of hypotheses are grouped together, they become a type of conceptual framework. When a conceptual framework is complex and incorporates causality or explanation, it is generally referred to as a theory. According to noted philosopher of science Carl Gustav Hempel,
An adequate empirical interpretation turns a theoretical system into a testable theory: The hypothesis whose constituent terms have been interpreted become capable of test by reference to observable phenomena. Frequently the interpreted hypothesis will be derivative hypotheses of the theory; but their confirmation or disconfirmation by empirical data will then immediately strengthen or weaken also the primitive hypotheses from which they were derived.
Hempel provides a useful metaphor that describes the relationship between a conceptual framework and the framework as it is observed and perhaps tested (interpreted framework). "The whole system floats, as it were, above the plane of observation and is anchored to it by rules of interpretation. These might be viewed as strings which are not part of the network but link certain points of the latter with specific places in the plane of observation. By virtue of those interpretative connections, the network can function as a scientific theory." Hypotheses with concepts anchored in the plane of observation are ready to be tested. In "actual scientific practice the process of framing a theoretical structure and of interpreting it are not always sharply separated, since the intended interpretation usually guides the construction of the theoretician". It is, however, "possible and indeed desirable, for the purposes of logical clarification, to separate the two steps conceptually".
=== Statistical hypothesis testing ===
When a possible correlation or similar relation between phenomena is investigated, such as whether a proposed remedy is effective in treating a disease, the hypothesis that a relation exists cannot be examined the same way one might examine a proposed new law of nature. In such an investigation, if the tested remedy shows no effect in a few cases, these do not necessarily falsify the hypothesis. Instead, statistical tests are used to determine how likely it is that the overall effect would be observed if the hypothesized relation does not exist. If that likelihood is sufficiently small (e.g., less than 1%), the existence of a relation may be assumed. Otherwise, any observed effect may be due to pure chance.
In statistical hypothesis testing, two hypotheses are compared. These are called the null hypothesis and the alternative hypothesis. The null hypothesis is the hypothesis that states that there is no relation between the phenomena whose relation is under investigation, or at least not of the form given by the alternative hypothesis. The alternative hypothesis, as the name suggests, is the alternative to the null hypothesis: it states that there is some kind of relation. The alternative hypothesis may take several forms, depending on the nature of the hypothesized relation; in particular, it can be two-sided (for example: there is some effect, in a yet unknown direction) or one-sided (the direction of the hypothesized relation, positive or negative, is fixed in advance).
Conventional significance levels for testing hypotheses (acceptable probabilities of wrongly rejecting a true null hypothesis) are .10, .05, and .01. The significance level for deciding whether the null hypothesis is rejected and the alternative hypothesis is accepted must be determined in advance, before the observations are collected or inspected. If these criteria are determined later, when the data to be tested are already known, the test is invalid.
The above procedure is actually dependent on the number of the participants (units or sample size) that are included in the study. For instance, to avoid having the sample size be too small to reject a null hypothesis, it is recommended that one specify a sufficient sample size from the beginning. It is advisable to define a small, medium and large effect size for each of a number of important statistical tests which are used to test the hypotheses.
== Honours ==
Mount Hypothesis in Antarctica is named in appreciation of the role of hypotheses in scientific research.
== List ==
Several hypotheses have been put forth, in different subject areas:
Astronomical hypotheses
Authorship debates
Biological hypotheses
Documentary hypothesis
Hypothetical documents
Hypothetical impact events
Hypothetical laws
Linguistic theories and hypotheses
Meteorological hypotheses
Hypothetical objects
Origin hypotheses of ethnic groups
Hypothetical processes
Hypothetical spacecraft
Statistical hypothesis testing
Hypothetical technology
== See also ==
== References ==
== Bibliography ==
Popper, Karl R. (1959), "The Logic of Scientific Discovery", Physics Today, 12 (11): 53, Bibcode:1959PhT....12k..53P, doi:10.1063/1.3060577 1934, 1959.
== External links ==
The dictionary definition of hypothesis at Wiktionary
Learning materials related to Hypothesis at Wikiversity
Media related to Hypotheses at Wikimedia Commons
"How science works", Understanding Science by the University of California Museum of Paleontology.

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The hypothetico-deductive model or method is a proposed description of the scientific method. According to it, scientific inquiry proceeds by formulating a hypothesis in a form that can be falsifiable, using a test on observable data where the outcome is not yet known. A test outcome that could have and does run contrary to predictions of the hypothesis is taken as a falsification of the hypothesis. A test outcome that could have, but does not run contrary to the hypothesis corroborates the theory. It is then proposed to compare the explanatory value of competing hypotheses by testing how stringently they are corroborated by their predictions.
== Example ==
One example of an algorithmic statement of the hypothetico-deductive method is as follows:
One possible sequence in this model would be 1, 2, 3, 4. If the outcome of 4 holds, and 3 is not yet disproven, you may continue with 3, 4, 1, and so forth; but if the outcome of 4 shows 3 to be false, you will have to go back to 2 and try to invent a new 2, deduce a new 3, look for 4, and so forth.
Note that this method can never absolutely verify (prove the truth of) 2. It can only falsify 2. (This is what Einstein meant when he said, "No amount of experimentation can ever prove me right; a single experiment can prove me wrong.")
== Discussion ==
Additionally, as pointed out by Carl Hempel (1905–1997), this simple view of the scientific method is incomplete; a conjecture can also incorporate probabilities, e.g., the drug is effective about 70% of the time. Tests, in this case, must be repeated to substantiate the conjecture (in particular, the probabilities). In this and other cases, we can quantify a probability for our confidence in the conjecture itself and then apply a Bayesian analysis, with each experimental result shifting the probability either up or down. Bayes' theorem shows that the probability will never reach exactly 0 or 100% (no absolute certainty in either direction), but it can still get very close to either extreme. See also confirmation holism.
Qualification of corroborating evidence is sometimes raised as philosophically problematic. The raven paradox is a famous example. The hypothesis that 'all ravens are black' would appear to be corroborated by observations of only black ravens. However, 'all ravens are black' is logically equivalent to 'all non-black things are non-ravens' (this is the contrapositive form of the original implication). 'This is a green tree' is an observation of a non-black thing that is a non-raven and therefore corroborates 'all non-black things are non-ravens'. It appears to follow that the observation 'this is a green tree' is corroborating evidence for the hypothesis 'all ravens are black'.
Attempted resolutions may distinguish:
non-falsifying observations as to strong, moderate, or weak corroborations
investigations that do or do not provide a potentially falsifying test of the hypothesis.
Evidence contrary to a hypothesis is itself philosophically problematic. Such evidence is called a falsification of the hypothesis. However, under the theory of confirmation holism it is always possible to save a given hypothesis from falsification. This is so because any falsifying observation is embedded in a theoretical background, which can be modified in order to save the hypothesis. Karl Popper acknowledged this but maintained that a critical approach respecting methodological rules that avoided such immunizing stratagems is conducive to the progress of science.
Physicist Sean Carroll claims the model ignores underdetermination.
=== Versus other research models ===
The hypothetico-deductive approach contrasts with other research models such as the inductive approach or grounded theory. In the data percolation methodology,
the hypothetico-deductive approach is included in a paradigm of pragmatism by which four types of relations between the variables can exist: descriptive, of influence, longitudinal or causal. The variables are classified in two groups, structural and functional, a classification that drives the formulation of hypotheses and the statistical tests to be performed on the data so as to increase the efficiency of the research.
== See also ==
Confirmation bias
Deductive-nomological
Explanandum and explanans
Inquiry
Models of scientific inquiry
Philosophy of science
Pragmatism
Scientific method
Verifiability theory of meaning
Will to believe doctrine
=== Types of inference ===
Strong inference
Abductive reasoning
Deductive reasoning
Inductive reasoning
Analogy
== Citations ==
== References ==
Brody, Thomas A. (1993), The Philosophy Behind Physics, Springer Verlag, ISBN 0-387-55914-0. (Luis de la Peña and Peter E. Hodgson, eds.)
Bynum, W.F.; Porter, Roy (2005), Oxford Dictionary of Scientific Quotations, Oxford, ISBN 0-19-858409-1.
Godfrey-Smith, Peter (2003), Theory and Reality: An introduction to the philosophy of science, University of Chicago Press, ISBN 0-226-30063-3
Taleb, Nassim Nicholas (2007), The Black Swan, Random House, ISBN 978-1-4000-6351-2

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Interdisciplinary Peer Review (IPR) is a peer review process with an additional focus outside of the area of the author's subject of expertise. Disciplines such as telecommunications, political science, engineering, and medicine require specific subject matter expertise, however, they still cross multiple disciplines and may require review from many alternate functional areas to achieve maximum perspective to prevent duplication or improper publication. Reviews of this nature may also cross cultures, race, and other demographics to gain perspective.
== Reduplication and Interdisciplinarity ==
In Peer Commentary on Peer Review: A case study in scientific quality control, Stevan R. Harnard (p. 15) begins to touch on the concept by addressing how well the review process works and what factors prevent re-duplication. Julie Klein, a professor of interdisciplinary studies at Wayne State University, defined interdisciplinarity as "new divisions of intellectual labor, collaborative research, team teaching, hybrid fields, comparative studies, increased borrowing across disciplines, and a variety of unified, holistic perspectives that have created pressures upon traditional divisions of knowledge". Klein has also described how interdisciplinarity is used to "find answers to complex questions, address broad issues, explore disciplinary and professional relations, to solve problems beyond the scope of any one discipline, and to achieve unity of knowledge whether on a limited or grand scale".
== Differences ==
The difference between an Interdisciplinary Peer Review and Interdisciplinarity is that the peer group in Interdisciplinary Peer Review crosses social, economic, and educational groups. Access to the review process allows greater input from a wider array or potential researchers.
== Open Interdisciplinary Peer Review Via Social Networking ==
Open Interdisciplinary Peer Review via Social Networking –The nature and ease of social networking sites makes Interdisciplinary Peer Review a reality. The issue becomes the informality of the review. This informality makes the review more of an "Open Peer Review" rather than a formalized review. Surprisingly in Encouraging Formative Peer Review Via Social Networking Sites, Bassford (E.67) finds that a high percentage of student's feel social networking is useful for enhancing learning, but only a small portion want to use it for such activities.
== Dilemma ==
Validation of information in an Open Interdisciplinary Peer Review is an ongoing or second Interdisciplinary Peer Review. Interdisciplinary Peer Review is a continual process of review. When publication is instant and prior to a review, the accuracy falls under scrutiny in the Open review nature of social media. The level of accuracy potentially becomes more variable as the non peer group dissemination increases.
== References ==
Harnad, Steven. "Peer Commentary on Peer Review, A Case Study in Scientific Quality Control. Reprinted from the Behavioral and Brain Sciences, an International Journal of Current Research and Theory with Open Peer Commentary." Cambridge University Press, Vol. 18, Issue 2, pages 227–237
Bassford, Marie. "Encouraging formative peer review via social networking sites." British Journal of Educational Technology 41.5 (2010): E67-E69. doi:10.1111/j.1467-8535.2009.00936.x
Kline, Thompson Julie. "Interdisciplinarity: History, Theory, and Practice." (1990): ISBN 0-8143-2088-0 See Google Books
== External links ==
The peer review process; Yahoo Images, Understanding Science
Peer Group Learning Collaboration from Yahoo Images

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In physical science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity (being nothing more than the value of the physical quantity) is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Radar energy expands during both the signal transmission and the reflected return, so the inverse square for both paths means that the radar will receive energy according to the inverse fourth power of the range.
To prevent dilution of energy while propagating a signal, certain methods can be used such as a waveguide, which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one dimension in order to prevent loss of energy transfer to a bullet.
== Formula ==
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see ∝) to the reciprocal of the square of the distance thus:
intensity
∝
1
distance
2
{\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
It can also be mathematically expressed as :
intensity
1
intensity
2
=
distance
2
2
distance
1
2
{\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
or as the formulation of a constant quantity:
intensity
1
×
distance
1
2
=
intensity
2
×
distance
2
2
{\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric, light, sound, and radiation phenomena.
== Justification ==
The inverse-square law generally applies when some force, energy, or other conserved quantity is evenly radiated outward from a point source in three-dimensional space. Since the surface area of a sphere (which is 4πr2) is proportional to the square of the radius, as the emitted radiation gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source. Gauss's law for gravity is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
== Occurrences ==
=== Gravitation ===
Gravitation is the attraction between objects that have mass. Newton's law states:
The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining them.
F
=
G
m
1
m
2
r
2
{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}
If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem. Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's center of mass while calculating the gravitational force.
As the law of gravitation, this law was suggested in 1645 by Ismaël Bullialdus. But Bullialdus did not accept Kepler's second and third laws, nor did he appreciate Christiaan Huygens's solution for circular motion (motion in a straight line pulled aside by the central force). Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion. Robert Hooke and Giovanni Alfonso Borelli both expounded gravitation in 1666 as an attractive force. Hooke's lecture "On gravity" was at the Royal Society, in London, on 21 March. Borelli's "Theory of the Planets" was published later in 1666. Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to Isaac Newton:
my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall.
Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 Principia acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the Solar System, as well as giving some credit to Bullialdus.
=== Electrostatics ===
The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as Coulomb's law. The deviation of the exponent from 2 is less than one part in 1015.
F
=
k
e
q
1
q
2
r
2
{\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}

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=== Light and other electromagnetic radiation ===
The intensity (or illuminance or irradiance) of light or other linear waves radiating from a point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source, so an object (of the same size) twice as far away receives only one-quarter the energy (in the same time period).
More generally, the irradiance, i.e., the intensity (or power per unit area in the direction of propagation), of a spherical wavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by absorption or scattering).
For example, the intensity of radiation from the Sun is 9126 watts per square meter at the distance of Mercury (0.387 AU) but only 1367 watts per square meter at the distance of Earth (1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
For non-isotropic radiators such as parabolic antennas, headlights, and lasers, the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have antenna gain in the direction of the narrow beam relative to a wide beam in all directions of an isotropic antenna.
In photography and stage lighting, the inverse-square law is used to determine the “fall off” or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter; or similarly, to halve the illumination increase the distance by a factor of 1.4 (the square root of 2), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.
The fractional reduction in electromagnetic fluence (Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4πr 2 where r is the radial distance from the center. The law is particularly important in diagnostic radiography and radiotherapy treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. As stated in Fourier theory of heat “as the point source is magnification by distances, its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin”.
==== Example ====
Let P be the total power radiated from a point source (for example, an omnidirectional isotropic radiator). At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases. Since the surface area of a sphere of radius r is A = 4πr 2, the intensity I (power per unit area) of radiation at distance r is
I
=
P
A
=
P
4
π
r
2
.
{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
The energy or intensity decreases (divided by 4) as the distance r is doubled; if measured in dB would decrease by 6.02 dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.
=== Sound in a gas ===
In acoustics, the sound pressure of a spherical wavefront radiating from a point source decreases by 50% as the distance r is doubled; measured in dB, the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):
p
∝
1
r
{\displaystyle p\ \propto \ {\frac {1}{r}}\,}
The same is true for the component of particle velocity
v
{\displaystyle v\,}
that is in-phase with the instantaneous sound pressure
p
{\displaystyle p\,}
:
v
∝
1
r
{\displaystyle v\ \propto {\frac {1}{r}}\ \,}
In the near field is a quadrature component of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound. The sound intensity is the product of the RMS sound pressure and the in-phase component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:
I
=
p
v
∝
1
r
2
.
{\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
== Field theory interpretation ==
For an irrotational vector field in three-dimensional space, the inverse-square law corresponds to the property that the divergence is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in n-dimensional Euclidean space, the intensity "I" of the vector field falls off with the distance "r" following the inverse (n − 1)th power law
I
∝
1
r
n
−
1
,
{\displaystyle I\propto {\frac {1}{r^{n-1}}},}
given that the space outside the source is divergence free.

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== Non-Euclidean implications ==
The inverse-square law, fundamental in Euclidean spaces, also applies to non-Euclidean geometries, including hyperbolic space. The curvature present in these spaces alters physical laws, influencing a variety of fields such as cosmology, general relativity, and string theory.
John D. Barrow, in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He explains that F and Φ obey the relationships F ∝ 1 / R² sinh²(r/R) and Φ ∝ coth(r/R), where R represents the curvature radius and r represents the distance from the focal point.
The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law. Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.
In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts special relativity. General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime.
== History ==
John Dumbleton of the 14th-century Oxford Calculators, was one of the first to express functional relationships in graphical form. He gave a proof of the mean speed theorem stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his Summa logicæ et philosophiæ naturalis (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.
In proposition 9 of Book 1 in his book Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur (1604), the astronomer Johannes Kepler argued that the spreading of light from a point source obeys an inverse square law:
In 1645, in his book Astronomia Philolaica ..., the French astronomer Ismaël Bullialdus (1605–1694) refuted Johannes Kepler's suggestion that "gravity" weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:
In England, the Anglican bishop Seth Ward (1617–1689) publicized the ideas of Bullialdus in his critique In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis (1653) and publicized the planetary astronomy of Kepler in his book Astronomia geometrica (1656).
In 1663–1664, the English scientist Robert Hooke was writing his book Micrographia (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface. Since the atmosphere surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the Earth's surface is a truncated cone (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface). Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the Earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude. Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the Earth's center.
Newton went up to independently develop and derive the inverse-square law for gravity in his Principia (1686). This later led to a priority debate between Newton and Hooke through correspondence.
== See also ==
Antenna (radio)
Distance decay
Fermi paradox
Flux
Gauss's law
Inverse proportionality
Kepler problem
Kepler's laws of planetary motion
Multiplicative inverse
Principle of similitude
Square–cube law
Telecommunications, particularly:
William Thomson, 1st Baron Kelvin
Power-aware routing protocols
== References ==
This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 22 January 2022.
== External links ==
Damping of sound level with distance
Sound pressure p and the inverse distance law 1/r

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Isotope dilution analysis is a method of determining the quantity of chemical substances. In its most simple conception, the method of isotope dilution comprises the addition of known amounts of isotopically enriched substance to the analyzed sample. Mixing of the isotopic standard with the sample effectively "dilutes" the isotopic enrichment of the standard and this forms the basis for the isotope dilution method. Isotope dilution is classified as a method of internal standardisation, because the standard (isotopically enriched form of analyte) is added directly to the sample. In addition, unlike traditional analytical methods which rely on signal intensity, isotope dilution employs signal ratios. Owing to both of these advantages, the method of isotope dilution is regarded among chemistry measurement methods of the highest metrological standing.
Isotopes are variants of a particular chemical element which differ in neutron number. All isotopes of a given element have the same number of protons in each atom. The term isotope is formed from the Greek roots isos (ἴσος "equal") and topos (τόπος "place"), meaning "the same place"; thus, the meaning behind the name is that different isotopes of a single element occupy the same position on the periodic table.
== Early history ==
Analytical application of the radiotracer method is a forerunner of isotope dilution. This method was developed in the early 20th century by George de Hevesy for which he was awarded the Nobel Prize in Chemistry for 1943.
An early application of isotope dilution in the form of radiotracer method was determination of the solubility of lead sulphide and lead chromate in 1913 by George de Hevesy and Friedrich Adolf Paneth. In the 1930s, US biochemist David Rittenberg pioneered the use of isotope dilution in biochemistry enabling detailed studies of cell metabolism.
== Tutorial example ==
Isotope dilution is analogous to the mark and recapture method, commonly used in ecology to estimate population size.
For instance, consider the determination of the number of fish (nA) in a lake. For the purpose of this example, assume all fish native to the lake are blue. On their first visit to the lake, an ecologist adds five yellow fish (nB = 5). On their second visit, the ecologist captures a number of fish according to a sampling plan and observes that the ratio of blue-to-yellow (i.e. native-to-marked) fish is 10:1. The number of fish native to the lake can be calculated using the following equation:
n
A
=
n
B
×
10
1
=
50
{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {10}{1}}=50}
This is a simplified view of isotope dilution but it illustrates the method's salient features. A more complex situation arises when the distinction between marked and unmarked fish becomes fuzzy. This can occur, for example, when the lake already contains a small number of marked fish from previous field experiments; and vice versa, where the amount of marked fish added contains a small number of unmarked fish. In a laboratory setting, an unknown (the "lake") may contain a quantity of a compound that is naturally present in major ("blue") and minor ("yellow") isotopic forms. A standard that is enriched in the minor isotopic form may then be added to the unknown, which can be subsequently analyzed. Keeping to the fish analogy, the following expression can be employed:
n
A
=
n
B
×
R
B
−
R
A
B
R
A
B
−
R
A
×
1
+
R
A
1
+
R
B
{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}
where, as indicated above, nA and nB represent the number of fish in the lake and the number of fish added to the lake, respectively; RA is the ratio of the native-to-marked fish in the lake prior to the addition of marked fish; RB is the ratio of the native-to-marked fish in the amount of marked fish added to the lake; finally, RAB is the ratio of the native-to-marked fish captured during the second visit.
== Applications ==
Isotope dilution is almost exclusively employed with mass spectrometry in applications where high-accuracy is demanded. For example, all National Metrology Institutes rely significantly on isotope dilution when producing certified reference materials. In addition to high-precision analysis, isotope dilution is applied when low recovery of the analyte is encountered. In addition to the use of stable isotopes, radioactive isotopes can be employed in isotope dilution which is often encountered in biomedical applications, for example, in estimating the volume of blood.
== Single dilution method ==
Consider a natural analyte rich in isotope iA (denoted as A), and the same analyte, enriched in isotope jA (denoted as B). Then, the obtained mixture is analyzed for the isotopic composition of the analyte, RAB = n(iA)AB/n(jA)AB. If the amount of the isotopically enriched substance (nB) is known, the amount of substance in the sample (nA) can be obtained:
n
A
=
n
B
R
B
−
R
A
B
R
A
B
−
R
A
×
x
(
j
A
)
B
x
(
j
A
)
A
{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}
Here, RA is the isotope amount ratio of the natural analyte, RA = n(iA)A/n(jA)A, RB is the isotope amount ratio of the isotopically enriched analyte, RB = n(iA)B/n(jA)B, RAB is the isotope amount ratio of the resulting mixture, x(jA)A is the isotopic abundance of the minor isotope in the natural analyte, and x(jA)B is the isotopic abundance of the major isotope in the isotopically enriched analyte.
For elements with only two stable isotopes, such as boron, chlorine, or silver, the above single dilution equation simplifies to the following:

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The Jadad scale, sometimes known as Jadad scoring or the Oxford quality scoring system, is a procedure to assess the methodological quality of a clinical trial by objective criteria. It is named after Canadian-Colombian physician Alex Jadad who in 1996 described a system for allocating such trials a score of between zero (very poor) and five (rigorous). It is the most widely used such assessment in the world, and as of May 2024, its seminal paper has been cited in over 25,000 scientific works.
== Description ==
The Jadad scale independently assesses the methodological quality of a clinical trial judging the effectiveness of blinding. Alejandro "Alex" Jadad Bechara, a Colombian physician who worked as a Research Fellow at the Oxford Pain Relief Unit, Nuffield Department of Anaesthetics at the University of Oxford described the allocating trials a score of between zero (very poor) and five (rigorous) in an appendix to a 1996 paper.
In a 2007 book, Jadad described the randomised controlled trial as "one of the simplest, most powerful and revolutionary forms of research".
== Background ==
Clinical trials are conducted for the purpose of collecting data on the efficacy of medical treatments. The treatment might be, for example, a new drug, a medical device, a surgical procedure, or a preventative regime. Clinical trial protocols vary considerably depending on the nature of the treatment under investigation, but typically, in a controlled trial, researchers gather a group of volunteers and subject some to the test treatment, while giving the others either no treatment (known as a placebo) or an established treatment for comparison. After a defined time period, the patients in the test group are assessed for health improvements in comparison with the control group.
However, trials can vary greatly in quality. Methodological errors such as poor blinding or poor randomisation allow factors such as the placebo effect or selection bias to adversely affect the results of a trial.
=== Randomisation ===
Randomisation is a process to remove potential distortion of statistical results arising from the manner in which the
trial is conducted, in particular in the selection of subjects. Studies have indicated, for example, that nonrandomised trials are more likely to show a positive result for a new treatment than for an established conventional one.
=== Blinding ===
The importance of scientific controls to limit factors under test is well established. However, it is also important that none of those involved in a clinical trial, whether the researcher, the subject patient or any other involved parties, should allow their own prior expectations to affect reporting of results. The placebo effect is known to be a confounding factor in trials; affecting the ability of both patients and doctors to report accurately on the clinical outcome. Experimental blinding is a process to prevent bias, both conscious and subconscious, skewing results.
Blinding frequently takes the form of a placebo, an inactive dummy that is indistinguishable from the real treatment. Blinding can however be difficult to achieve in some trials, for example, surgery or physical therapy. Poor blinding can exaggerate the perceived effects of treatment, particularly if any such effects are small. Blinding should be appropriate to the study, and is ideally double blind, wherein neither the patient nor doctor is aware of whether they are in the control or test group, eliminating any such psychological effects from the study.
=== Withdrawals and dropouts ===
Withdrawals and dropouts are those patients who fail to complete a course of treatment, or fail to report back on its outcome to the researchers. The reasons for doing so might be varied: the individuals may have moved away, abandoned the course of treatment, or died. Whatever the reason, the attrition rate can skew results of a study, particularly for those subjects who ceased treatment due to perceived inefficacy. In smoking cessation studies, for example, it is routine to consider all dropouts as failures.
== Jadad questionnaire ==
A three-point questionnaire forms the basis for a Jadad score. Each question was to be answered with either a yes or a no. Each yes would score a single point, each no zero points; there were to be no fractional points. The Jadad team stated that they expected it should take no longer than ten minutes to score any individual paper. The questions were as follows: Was the study described as randomized?, Was the study described as double blind? and Was there a description of withdrawals and dropouts?
To receive the corresponding point, an article should describe the number of withdrawals and dropouts in each of the study groups, and the underlying reasons. Additional points were given if: The method of randomisation was described in the paper, and that method was appropriate. or The method of blinding was described, and it was appropriate.
Points would be deducted if: The method of randomisation was described, but was inappropriate, or The method of blinding was described, but was inappropriate.
A clinical trial could therefore receive a Jadad score of between zero and five. The Jadad scale is sometimes described as a five-point scale, though there are only three questions.
== Uses ==
The Jadad score may be used in a number of ways:
To evaluate the general quality of medical research in a particular field.
To set a minimum standard for the paper's results to be included in a meta analysis. A researcher conducting a systematic review for example might elect to exclude all papers on the topic with a Jadad score of 3 or less.
For critical analysis of an individual paper.
As of 2008, the Jadad score was the most widely used such assessment in the world, and its seminal paper has been cited in over 3000 scientific works.
== Criticism ==
Critics have charged that the Jadad scale is flawed, being over-simplistic and placing too much emphasis on blinding, and can show low consistency between different raters. Furthermore, it does not take into account allocation concealment, viewed by The Cochrane Collaboration as paramount to avoid bias.
== See also ==
Consolidated Standards of Reporting Trials
Metascience
Unblinding
== References ==

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The Leiden Manifesto for research metrics (LM) is a list of "ten principles to guide research evaluation", published as a comment in Nature on 22 April 2015. It was formulated by public policy professor Diana Hicks, scientometrics professor Paul Wouters, and their colleagues at the 19th International Conference on Science and Technology Indicators, held between 3–5 September 2014 in Leiden, The Netherlands.
The Leiden Manifesto was proposed as a guide to combat misuse of bibliometrics when evaluating scientific research literature. Examples of commonly used bibliometrics for science, or scientometrics, are the h-index, impact factor, other indicators such as Altmetrics. According to the Manifesto's authors, these metrics often pervasively misguide evaluations of scientific material.
== Motivation ==
Motivations for codification of the Leiden Manifesto arose from a growing worry that "impact-factor obsession" was leading to inadequate judgement of scientific material that should be worthy of fair evaluation. Lead author Diana Hicks hoped that publishing the list in Nature would spread its ideas, already commonplace in the scientometrics sphere, to the broader scientific community.
=== DORA and other predecessors ===
The scientific community has long been interested in reforming assessment of the impact of scientific and academic research. The 2013 San Francisco Declaration on Research Assessment (DORA), which has been signed by over 27,000 individuals as of March 2026, was a major influence on the Leiden Manifesto. The Declaration denounced common practices in research assessment, such as using journal impact factor to assess the contributions of individual researchers.
One of the main concerns about overuse of citation-based performance indicators came from the observation that smaller research organizations and institutions may be negatively affected by their metric indices. In one public debate at the Centre for Science and Technology Studies at Leiden University, it was acknowledged that indicators which measure citations may give "more weight to publications from fields with a high expected number of citations than to publications from fields with a low expected number of citations".
Although the main focus of the Leiden Manifesto is the use of scientometrics for research evaluation, the authors also consider how overuse of metrics can adversely affect the wider scholarly community, such as the position of universities in global rankings. According to Hicks et al., scientific metrics such as citation rate are used far too much for ranking the quality of universities (and thus the quality of their research output).
=== Journal impact factor ===
The background of the Leiden Manifesto describes why misusing metrics is becoming a larger problem in the scientific community. The journal impact factor, originally created by Eugene Garfield as a method for librarians to collect data to facilitate selecting journals to purchase, is now mainly used as a method of judging journal quality. This is seen by the authors as an abuse of data in order to examine research too hastily. For example, an impact factor, while a good metric to measure the size and experience of a journal, may or may not be sufficient to accurately describe the quality of its papers, and even less so for a single paper.
== Content ==
The Leiden Manifesto consists of ten principles which aim to reform how research quality is assessed by academic publishers and institutions. It emphasizes detailed and close evaluation of research, rather than relying exclusively on quantitative data. It also aims to remove possible perverse incentives for using scientometrics, such as judgement of academic capability and university quality.
=== Ten principles ===
The ten principles of the Leiden Manifesto are as follows:
Quantitative evaluation should support qualitative, expert assessment.
Measure performance against the research missions of the institution, group, or researcher.
Protect excellence in locally relevant research.
Allow research taking place in a certain area or field to be published in corresponding local research publications, instead of prioritizing high-impact journals. Many high-impact journals are in English, which may decrease needed specificity when publishing a paper meant to study locational characteristics. As an example, in high-impact Spanish-language papers, "topics such as local labor laws" and other features designated for sociologists may be lost.
Keep data collection and analytical processes open, transparent, and simple.
Allow those evaluated to verify data and analysis.
Account for variation by field in publication and citation practices.
Peer-review and citation rate can vary wildly across differing disciplines, for example, "top-ranked journals in mathematics have impact factors of around 3; top-ranked journals in cell biology have impact factors of about 30".
Base assessment of individual researchers on a qualitative judgement of their portfolio.
Avoid misplaced concreteness and false precision.
Use of scientific indicators may precede strong assumptions that are not necessarily correct. For example, when looking at a specific scientist, a low citation rate may lead the investigator to assume low research quality, which is implying causation from correlation. Providing clarification, as well as multiple, robust indicators, may reduce inappropriate concreteness. False precision is possible when indicator producers, such as Clarivate (which publishes the annual Journal Citation Reports) attempt to create an exact journal impact factor (i.e. three decimal places). Conceptual ambiguity and random variability of citation counts make it unnecessary to distinguish indices such as journal impact factors to such a precise extent, because it can foster excessive comparison and competition between publishers.
Recognize the systemic effects of assessment and indicators.
Scrutinize indicators regularly and update them.
== Reception ==
=== 2016 John Ziman Award ===
In 2016, the European Association for the Study of Science and Technology (EASST) gave its John Ziman Award to the Leiden Manifesto for its effort to widen scientometrics knowledge to the scientific community as a whole. EASST president Fred Steward stated that it "emphasizes situatedness, in terms of different cognitive domains and research missions as well as the wider socioeconomic, national and regional context".

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=== Public endorsement ===
LIBER, a collaboration of European research libraries, issued a substantial review of the Leiden Manifesto in 2017, concluding that it was a "solid foundation" on which academic libraries could base their assessment of metrics.
Elsevier, a global leader in research publishing and information analytics, announced on 14 July 2020 that it would use the Leiden Manifesto to guide its development of improved research evaluation. Elsevier stated that the principles of the manifesto were already close in nature to their 2019 CiteScore metrics, which was in summary "improved calculation methodology" for "a more robust, fair and faster indicator of research impact".
Loughborough University's LIS-Bibliometrics committee chose to base their own principles on the Leiden Manifesto, instead of the DORA, because the manifesto takes a "broader approach to the responsible use of all bibliometrics across a range of disciplines and settings", according to their policy manager Elizabeth Gadd. Stephen Curry, chair of the DORA steering committee, commented on this statement by emphasizing that DORA was aiming to extend its "disciplinary and geographical reach".
=== Further applications ===
David Moher and his co-authors referenced the Leiden Manifesto in a perspective for Issues in Science and Technology, writing that academic institutions were not asking the "right questions" (concerning research planning, timeframe, reproducibility, and results) when assessing scientists. They criticize what they see as an obsession with journal impact factors and the "gaming" of scientometrics, advocating that institutions use DORA and the Leiden Manifesto principles instead when assessing individual scientists and research.
In a letter in Science and Engineering Ethics, T. Kanchan and Kewal Krishan called the Leiden Manifesto "one of the best criteria" for assessing scientific research, especially considering the "rat race" for publications in the scholarly community. They also argue that use of the Manifesto will lead to "progress of science and society at large".
== See also ==
San Francisco Declaration on Research Assessment
Nature (journal)
Scientometrics (journal)
Bibliometrics
Scientometrics
H-index
Impact factor
Altmetrics
== References ==

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Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and has since had a complex history, being adopted and extended in some disciplines and by some scholars, and criticized or rejected by others. Other classifications include those by Mosteller and Tukey, and by Chrisman.
Stevens proposed his typology in a 1946 Science article titled "On the theory of scales of measurement". In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval", and "ratio", unifying both "qualitative" (which are described by his "nominal" type) and "quantitative" (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p. 395) wrote:
S. S. Stevens (1946, 1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it is doubtful if he understood it himself ... no measurement theorist I know accepts Stevens's broad definition of measurement... [emphasis added] in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute.
== Stevens's typology ==
=== Nominal scale ===
A nominal scale consists only of a number of distinct classes or categories, for example: [Cat, Dog, Rabbit]. Unlike the other scales, no kind of relationship between the classes can be relied upon. Thus measuring with the nominal scale is equivalent to classifying.
Nominal measurement may differentiate between items or subjects based only on their names or (meta-)categories and other qualitative classifications they belong to. Thus it has been argued that even dichotomous data relies on a constructivist epistemology. In this case, discovery of an exception to a classification can be viewed as progress.
Numbers may be used to represent the variables but the numbers do not have numerical value or relationship: for example, a globally unique identifier.
Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form. In a university one could also use residence hall or department affiliation as examples. Other concrete examples are
in grammar, the parts of speech: noun, verb, preposition, article, pronoun, etc.
in politics, power projection: hard power, soft power, etc.
in biology, the taxonomic ranks below domains: kingdom, phylum, class, etc.
in software engineering, type of fault: specification faults, design faults, and code faults
Nominal scales were often called qualitative scales, and measurements made on qualitative scales were called qualitative data. However, the rise of qualitative research has made this usage confusing. If numbers are assigned as labels in nominal measurement, they have no specific numerical value or meaning. No form of arithmetic computation (+, −, ×, etc.) may be performed on nominal measures.
==== Mathematical operations ====
Equality and other operations that can be defined in terms of equality, such as inequality and set membership, are the only non-trivial operations that generically apply to objects of the nominal type.
==== Central tendency ====
The mode, i.e. the most common item, is allowed as the measure of central tendency for the nominal type.
=== Ordinal scale ===
The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted but still does not allow for a relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as "sick" vs. "healthy" when measuring health, "guilty" vs. "not-guilty" when making judgments in courts, "wrong/false" vs. "right/true" when measuring truth value, and, on the other hand, non-dichotomous data consisting of a spectrum of values, such as "completely agree", "mostly agree", "mostly disagree", "completely disagree" when measuring opinion.
The ordinal scale places events in order, but there is no attempt to make the intervals of the scale equal in terms of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena. A student's rank in his graduation class involves the use of an ordinal scale. One has to be very careful in making a statement about scores based on ordinal scales. For instance, if Devi's position in his class is 10th and Ganga's position is 40th, it cannot be said that Devi's position is four times as good as that of Ganga.
Ordinal scales only permit the ranking of items from highest to lowest. Ordinal measures have no absolute values, and the real differences between adjacent ranks may not be equal. All that can be said is that one person is higher or lower on the scale than another, but more precise comparisons cannot be made. Thus, the use of an ordinal scale implies a statement of "greater than" or "less than" (an equality statement is also acceptable) without our being able to state how much greater or less. The real difference between ranks 1 and 2, for instance, may be more or less than the difference between ranks 5 and 6.

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==== 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.

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== Debate over Stevens's typology ==
While Stevens's typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986). Duncan (1986), for example, objected to the use of the word measurement in relation to the nominal type and Luce (1997) disagreed with Stevens's definition of measurement.
On the other hand, Stevens (1975) said of 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". Hand says, "Basic psychology texts often begin with Stevens's framework and the ideas are ubiquitous. Indeed, the essential soundness of his hierarchy has been established for representational measurement by mathematicians, determining the invariance properties of mappings from empirical systems to real number continua. Certainly the ideas have been revised, extended, and elaborated, but the remarkable thing is his insight given the relatively limited formal apparatus available to him and how many decades have passed since he coined them."
The use of the mean as a measure of the central tendency for the ordinal type is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data anyway. This is often justified on the basis that the ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; 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 the 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 SPSS requires 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 the interval type, based on the law of comparative judgment. A common application of the law is the analytic hierarchy process. Further progress was made by Georg Rasch (1960), who developed the probabilistic Rasch model that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.
=== Other proposed typologies ===
Typologies aside from Stevens's typology have been proposed. For instance, Mosteller and Tukey (1977) and Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991).
==== Mosteller and Tukey's typology (1977) ====
Mosteller and Tukey noted that the four levels are not exhaustive and proposed seven instead:
Names
Grades (ordered labels like beginner, intermediate, advanced)
Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
Counted fractions (bound by 0 and 1)
Counts (non-negative integers)
Amounts (non-negative real numbers)
Balances (any real number)
For example, percentages (a variation on fractions in the Mosteller–Tukey framework) do not fit well into Stevens's framework: No transformation is fully admissible.
==== Chrisman's typology (1998) ====
Nicholas R. Chrisman introduced an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to a range and repeating (like degrees in a circle, clock time, etc.), graded membership categories, and other types of measurement do not fit to Stevens's original work, leading to the introduction of six new levels of measurement, for a total of ten:
Nominal
Gradation of membership
Ordinal
Interval
Log-interval
Extensive ratio
Cyclical ratio
Derived ratio
Counts
Absolute
While some claim that the extended levels of measurement are rarely used outside of academic geography, graded membership is central to fuzzy set theory, while absolute measurements include probabilities and the plausibility and ignorance in Dempster–Shafer theory. Cyclical ratio measurements include angles and times. Counts appear to be ratio measurements, but the scale is not arbitrary and fractional counts are commonly meaningless. Log-interval measurements are commonly displayed in stock market graphics. All these types of measurements are commonly used outside academic geography, and do not fit well to Stevens's original work.
=== Scale types and Stevens's "operational theory of measurement" ===
The theory of scale types is the intellectual handmaiden to Stevens's "operational theory of measurement", which was to become definitive within psychology and the behavioral sciences, despite Michell's characterization as its being quite at odds with measurement in the natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by the British Association for the Advancement of Science to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the Ferguson committee, published a Final Report (Ferguson, et al., 1940, p. 245) in which Stevens's sone scale (Stevens & Davis, 1938) was an object of criticism:

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…any law purporting to express a quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the concept of addition as applied to sensation.
That is, if Stevens's sone scale genuinely measured the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes needed to be produced. The evidence needed was the presence of additive structure—a concept comprehensively treated by the German mathematician Otto Hölder (Hölder, 1901). Given that the physicist and measurement theorist Norman Robert Campbell dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of concatenation operations. This conclusion was later rendered false by the discovery of the theory of conjoint measurement by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens's reaction was not to conduct experiments to test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement:
Paraphrasing N. R. Campbell (Final Report, p. 340), we may say that measurement, in the broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946, p. 677).
Stevens was greatly influenced by the ideas of another Harvard academic, the Nobel laureate physicist Percy Bridgman (1927), whose doctrine of operationalism Stevens used to define measurement. In Stevens's definition, for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationalism object that it confuses the relations between two objects or events for properties of one of those of objects or events (Moyer, 1981a, b; Rogers, 1989).
The Canadian measurement theorist William Rozeboom was an early and trenchant critic of Stevens's theory of scale types.
==== Same variable may be different scale type depending on context ====
Another issue is that the same variable may be a different scale type depending on how it is measured and on the goals of the analysis. For example, hair color is usually thought of as a nominal variable, since it has no apparent ordering. However, it is possible to order colors (including hair colors) in various ways, including by hue; this is known as colorimetry. Hue is an interval level variable.
== Summary table ==
== See also ==
Cohen's kappa
Coherence (units of measurement)
Hume's principle
Inter-rater reliability
Logarithmic scale
Ramsey–Lewis method
Set theory
Statistical data type
Transition (linguistics)
== References ==
== Further reading ==

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Environmental sampling techniques are used in biology, ecology and conservation as part of scientific studies to learn about the flora and fauna of a particular area and establish a habitat's biodiversity, the abundance of species and the conditions in which these species live amongst other information. Where species are caught, researchers often then take the trapped organisms for further study in a lab or are documented by a researcher in the field before the animal is released. This information can then be used to better understand the environment, its ecology, the behaviour of species and how organisms interact with one another and their environment. Here is a list of some sampling techniques and equipment used in environmental sampling:
Quadrats - used for plants and slow moving animals
== Techniques for birds and/or flying invertebrates and/or bats ==
Malaise Trap
Flight Interception Trap
Harp Trap
Robinson Trap
Butterfly Net
Mist Net
== Techniques for Terrestrial Animals ==
Transect
Tullgren Funnel - used for soil-living arthropods
Pitfall Trap - used for small terrestrial animals like insects and amphibians
Netting techniques for terrestrial animals
Beating Net - used for insects dwelling in trees and shrubs
Sweep Netting - used for insects in grasses
Aspirator/Pooter - used for insects
Camera Trap - used for larger animals
Sherman Trap - used for small mammals
== See also ==
Insect Collecting
Wildlife Biology
Sampling
== Sources ==