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Mathematics is the study of quantity, space, structure, and change. Mathematicians
seek out patterns[4][5] and formulate new conjectures. Mathematicians resolve the
truth or falsity of conjectures by mathematical proofs, which are arguments sufficient
to convince other mathematicians of their validity. The research required to solve
mathematical problems can take years or even centuries of sustained inquiry. However,
mathematical proofs are less formal and painstaking than proofs in mathematical
logic. Since the pioneering work of Giuseppe Peano, David Hilbert, and others on
axiomatic systems in the late 19th century, it has become customary to view mathematical
research as establishing truth by rigorous deduction from appropriately chosen axioms
and definitions. When those mathematical structures are good models of real phenomena,
then mathematical reasoning often provides insight or predictions.
Through the use of abstraction and logical reasoning, mathematics developed from
counting, calculation, measurement, and the systematic study of the shapes and motions
of physical objects. Practical mathematics has been a human activity for as far
back as written records exist. Rigorous arguments first appeared in Greek mathematics,
most notably in Euclid's Elements. Mathematics continued to develop, for example
in China in 300 BC, in India in AD 100[citation needed], and in the Muslim world
in AD 800, until the Renaissance, when mathematical innovations interacting with
new scientific discoveries led to a rapid increase in the rate of mathematical discovery
that continues to the present day.
The mathematician Benjamin Peirce called mathematics "the science that draws necessary
conclusions".[7] David Hilbert said of mathematics: "We are not speaking here of
arbitrariness in any sense. Mathematics is not like a game whose tasks are determined
by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal
necessity that can only be so and by no means otherwise."[8] Albert Einstein stated
that "as far as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality".
Mathematics is used throughout the world as an essential tool in many fields, including
natural science, engineering, medicine, and the social sciences. Applied mathematics,
the branch of mathematics concerned with application of mathematical knowledge to
other fields, inspires and makes use of new mathematical discoveries and sometimes
leads to the development of entirely new mathematical disciplines, such as statistics
and game theory. Mathematicians also engage in pure mathematics, or mathematics
for its own sake, without having any application in mind. There is no clear line
separating pure and applied mathematics, and practical applications for what began
as pure mathematics are often discovered.
History
might be seen as an everincreasing series of abstractions, or alternatively an
expansion of subject matter. The first abstraction, which is shared by many animals,[13]
was probably that of numbers: the realization that a collection of two apples and
a collection of two oranges (for example) have something in common, namely quantity
of their members.
In addition to recognizing how to count physical objects, prehistoric peoples also
recognized how to count abstract quantities, like time – days, seasons, years.[14]
Elementary arithmetic (addition, subtraction, multiplication and division) naturally
followed.
Since numeracy predated writing, further steps were needed for recording numbers
such as tallies or the knotted strings called quipu used by the Inca to store numerical
data.[citation needed] Numeral systems have been many and diverse, with the first
known written numerals created by Egyptians in Middle Kingdom texts such as the
Rhind Mathematical Papyrus.
The earliest uses of mathematics were in trading, land measurement, painting and
weaving patterns and the recording of time. More complex mathematics did not appear
until around 3000 BC, when the Babylonians and Egyptians began using arithmetic,
algebra and geometry for taxation and other financial calculations, for building
and construction, and for astronomy.[15] The systematic study of mathematics in
its own right began with the Ancient Greeks between 600 and 300 BC.
Mathematics has since been greatly extended, and there has been a fruitful interaction
between mathematics and science, to the benefit of both. Mathematical discoveries
continue to be made today. According to Mikhail B. Sevryuk, in the January 2006
issue of the Bulletin of the American Mathematical Society, "The number of papers
and books included in the Mathematical Reviews database since 1940 (the first year
of operation of MR) is now more than 1.9 million, and more than 75 thousand items
are added to the database each year. The overwhelming majority of works in this
ocean contain new mathematical theorems and their proofs."
Fields of mathematics
Mathematics can, broadly speaking, be subdivided into the study of quantity, structure,
space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition
to these main concerns, there are also subdivisions dedicated to exploring links
from the heart of mathematics to other fields: to logic, to set theory (foundations),
to the empirical mathematics of the various sciences (applied mathematics), and
more recently to the rigorous study of uncertainty.
Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic
and set theory were developed. Mathematical logic includes the mathematical study
of logic and the applications of formal logic to other areas of mathematics; set
theory is the branch of mathematics that studies sets or collections of objects.
Category theory, which deals in an abstract way with mathematical structures and
relationships between them, is still in development. The phrase "crisis of foundations"
describes the search for a rigorous foundation for mathematics that took place from
approximately 1900 to 1930.[29] Some disagreement about the foundations of mathematics
continues to the present day. The crisis of foundations was stimulated by a number
of controversies at the time, including the controversy over Cantor's set theory
and the BrouwerHilbert controversy.
Mathematical logic is concerned with setting mathematics within a rigorous axiomatic
framework, and studying the implications of such a framework. As such, it is home
to Gödel's incompleteness theorems which (informally) imply that any formal system
that contains basic arithmetic, if sound (meaning that all theorems that can be
proven are true), is necessarily incomplete (meaning that there are true theorems
which cannot be proved in that system). Whatever finite collection of numbertheoretical
axioms is taken as a foundation, Gödel showed how to construct a formal statement
that is a true numbertheoretical fact, but which does not follow from those axioms.
Therefore no formal system is a complete axiomatization of full number theory. Modern
logic is divided into recursion theory, model theory, and proof theory, and is closely
linked to theoretical computer science[citation needed], as well as to Category
Theory.
Theoretical computer science includes computability theory, computational complexity
theory, and information theory. Computability theory examines the limitations of
various theoretical models of the computer, including the most well known model
– the Turing machine. Complexity theory is the study of tractability by computer;
some problems, although theoretically solvable by computer, are so expensive in
terms of time or space that solving them is likely to remain practically unfeasible,
even with rapid advance of computer hardware. A famous problem is the "P=NP?" problem,
one of the Millennium Prize Problems.[30] Finally, information theory is concerned
with the amount of data that can be stored on a given medium, and hence deals with
concepts such as compression and entropy
Pure mathematics
Quantity
The study of quantity starts with numbers, first the familiar natural numbers and
integers ("whole numbers") and arithmetical operations on them, which are characterized
in arithmetic. The deeper properties of integers are studied in number theory, from
which come such popular results as Fermat's Last Theorem. The twin prime conjecture
and Goldbach's conjecture are two unsolved problems in number theory.
As the number system is further developed, the integers are recognized as a subset
of the rational numbers ("fractions"). These, in turn, are contained within the
real numbers, which are used to represent continuous quantities. Real numbers are
generalized to complex numbers. These are the first steps of a hierarchy of numbers
that goes on to include quarternions and octonions. Consideration of the natural
numbers also leads to the transfinite numbers, which formalize the concept of "infinity".
Another area of study is size, which leads to the cardinal numbers and then to another
conception of infinity: the aleph numbers, which allow meaningful comparison of
the size of infinitely large sets.
Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal
structure as a consequence of operations or relations that are defined on the set.
Mathematics then studies properties of those sets that can be expressed in terms
of that structure; for instance number theory studies properties of the set of integers
that can be expressed in terms of arithmetic operations. Moreover, it frequently
happens that different such structured sets (or structures) exhibit similar properties,
which makes it possible, by a further step of abstraction, to state axioms for a
class of structures, and then study at once the whole class of structures satisfying
these axioms. Thus one can study groups, rings, fields and other abstract systems;
together such studies (for structures defined by algebraic operations) constitute
the domain of abstract algebra. By its great generality, abstract algebra can often
be applied to seemingly unrelated problems; for instance a number of ancient problems
concerning compass and straightedge constructions were finally solved using Galois
theory, which involves field theory and group theory. Another example of an algebraic
theory is linear algebra, which is the general study of vector spaces, whose elements
called vectors have both quantity and direction, and can be used to model (relations
between) points in space. This is one example of the phenomenon that the originally
unrelated areas of geometry and algebra have very strong interactions in modern
mathematics. Combinatorics studies ways of enumerating the number of objects that
fit a given structure.
Space
The study of space originates with geometry – in particular, Euclidean geometry.
Trigonometry is the branch of mathematics that deals with relationships between
the sides and the angles of triangles and with the trigonometric functions; it combines
space and numbers, and encompasses the wellknown Pythagorean theorem. The modern
study of space generalizes these ideas to include higherdimensional geometry, nonEuclidean
geometries (which play a central role in general relativity) and topology. Quantity
and space both play a role in analytic geometry, differential geometry, and algebraic
geometry. Convex and discrete geometry was developed to solve problems in number
theory and functional analysis but now is pursued with an eye on applications in
optimization and computer science. Within differential geometry are the concepts
of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus.
Within algebraic geometry is the description of geometric objects as solution sets
of polynomial equations, combining the concepts of quantity and space, and also
the study of topological groups, which combine structure and space. Lie groups are
used to study space, structure, and change. Topology in all its many ramifications
may have been the greatest growth area in 20th century mathematics; it includes
pointset topology, settheoretic topology, algebraic topology and differential
topology. In particular, instances of modern day topology are metrizability theory,
axiomatic set theory, homotopy theory, and Morse theory. Topology also includes
the now solved Poincaré conjecture. Other results in geometry and topology, including
the four color theorem and Kepler conjecture, have been proved only with the help
of computers.
Change
Understanding and describing change is a common theme in the natural sciences, and
calculus was developed as a powerful tool to investigate it. Functions arise here,
as a central concept describing a changing quantity. The rigorous study of real
numbers and functions of a real variable is known as real analysis, with complex
analysis the equivalent field for the complex numbers. Functional analysis focuses
attention on (typically infinitedimensional) spaces of functions. One of many applications
of functional analysis is quantum mechanics. Many problems lead naturally to relationships
between a quantity and its rate of change, and these are studied as differential
equations. Many phenomena in nature can be described by dynamical systems; chaos
theory makes precise the ways in which many of these systems exhibit unpredictable
yet still deterministic behavior.
Applied mathematics
Applied mathematics concerns itself with mathematical methods that are typically
used in science, engineering, business, and industry. Thus, "applied mathematics"
is a mathematical science with specialized knowledge. The term "applied mathematics"
also describes the professional specialty in which mathematicians work on practical
problems; as a profession focused on practical problems, applied mathematics focuses
on the formulation, study, and use of mathematical models in science, engineering,
and other areas of mathematical practice.
In the past, practical applications have motivated the development of mathematical
theories, which then became the subject of study in pure mathematics, where mathematics
is developed primarily for its own sake. Thus, the activity of applied mathematics
is vitally connected with research in pure mathematics.
Statistics and other decision sciences
Applied mathematics has significant overlap with the discipline of statistics, whose
theory is formulated mathematically, especially with probability theory. Statisticians
(working as part of a research project) "create data that makes sense" with random
sampling and with randomized experiments;[31] the design of a statistical sample
or experiment specifies the analysis of the data (before the data be available).
When reconsidering data from experiments and samples or when analyzing data from
observational studies, statisticians "make sense of the data" using the art of modelling
and the theory of inference – with model selection and estimation; the estimated
models and consequential predictions should be tested on new data.
Statistical theory studies decision problems such as minimizing the risk (expected
loss) of a statistical action, such as using a procedure in, for example, parameter
estimation, hypothesis testing, and selecting the best. In these traditional areas
of mathematical statistics, a statisticaldecision problem is formulated by minimizing
an objective function, like expected loss or cost, under specific constraints: For
example, a designing a survey often involves minimizing the cost of estimating a
population mean with a given level of confidence.[33] Because of its use of optimization,
the mathematical theory of statistics shares concerns with other decision sciences,
such as operations research, control theory, and mathematical economics.
Computational mathematics
Computational mathematics proposes and studies methods for solving mathematical
problems that are typically too large for human numerical capacity. Numerical analysis
studies methods for problems in analysis using functional analysis and approximation
theory; numerical analysis includes the study of approximation and discretization
broadly with special concern for rounding errors. Numerical analysis and, more broadly,
scientific computing also study nonanalytic topics of mathematical science, especially
algorithmic matrix and graph theory. Other areas of computational mathematics include
computer algebra and symbolic computation.
Mathematics as profession
The bestknown award in mathematics is the Fields Medal, established in 1936 and
now awarded every 4 years. It is often considered the equivalent of science's Nobel
Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement,
and another major international award, the Abel Prize, was introduced in 2003. The
Chern Medal was introduced in 2010 to recognize lifetime achievement. These are
awarded for a particular body of work, which may be innovation, or resolution of
an outstanding problem in an established field.
A famous list of 23 open problems, called "Hilbert's problems", was compiled in
1900 by German mathematician David Hilbert. This list achieved great celebrity among
mathematicians, and at least nine of the problems have now been solved. A new list
of seven important problems, titled the "Millennium Prize Problems", was published
in 2000. Solution of each of these problems carries a $1 million reward, and only
one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Mathematics as science
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In
the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften,
the word corresponding to science means a "field of knowledge", and this was the
original meaning of "science" in English, also. Of course, mathematics is in this
sense a field of knowledge. The specialization restricting the meaning of "science"
to natural science follows the rise of Baconian science, which contrasted "natural
science" to scholasticism, the Aristotelean method of inquiring from first principles.
Of course, the role of empirical experimentation and observation is negligible in
mathematics, compared to natural sciences such as psychology, biology, or physics.
Albert Einstein stated that "as far as the laws of mathematics refer to reality,
they are not certain; and as far as they are certain, they do not refer to reality."
Many philosophers believe that mathematics is not experimentally falsifiable, and
thus not a science according to the definition of Karl Popper.[39] However, in the
1930s Gödel's incompleteness theorems convinced many mathematicians[who?] that mathematics
cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical
theories are, like those of physics and biology, hypotheticodeductive: pure mathematics
therefore turns out to be much closer to the natural sciences whose hypotheses are
conjectures, than it seemed even recently."[40] Other thinkers, notably Imre Lakatos,
have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics)
are mathematics with axioms that are intended to correspond to reality. In fact,
the theoretical physicist, J. M. Ziman, proposed that science is public knowledge
and thus includes mathematics.[41] In any case, mathematics shares much in common
with many fields in the physical sciences, notably the exploration of the logical
consequences of assumptions. Intuition and experimentation also play a role in the
formulation of conjectures in both mathematics and the (other) sciences. Experimental
mathematics continues to grow in importance within mathematics, and computation
and simulation are playing an increasing role in both the sciences and mathematics,
weakening the objection that mathematics does not use the scientific method.
The opinions of mathematicians on this matter are varied. Many mathematicians[who?]
feel that to call their area a science is to downplay the importance of its aesthetic
side, and its history in the traditional seven liberal arts; others[who?] feel that
to ignore its connection to the sciences is to turn a blind eye to the fact that
the interface between mathematics and its applications in science and engineering
has driven much development in mathematics. One way this difference of viewpoint
plays out is in the philosophical debate as to whether mathematics is created (as
in art) or discovered (as in science). It is common to see universities divided
into sections that include a division of Science and Mathematics, indicating that
the fields are seen as being allied but that they do not coincide. In practice,
mathematicians are typically grouped with scientists at the gross level but separated
at finer levels. This is one of many issues considered in the philosophy of mathematics.[citation
needed]
Statistics is the study of the collection, organization, analysis, and interpretation
of data.It deals with all aspects of this, including the planning of data collection
in terms of the design of surveys and experiments.
A statistician is someone who is particularly well versed in the ways of thinking
necessary for the successful application of statistical analysis. Such people have
often gained this experience through working in any of a wide number of fields.
There is also a discipline called mathematical statistics that studies statistics
mathematically.
The word statistics, when referring to the scientific discipline, is singular, as
in "Statistics is an art." This should not be confused with the word statistic,
referring to a quantity (such as mean or median) calculated from a set of data,[4]
whose plural is statistics ("this statistic seems wrong" or "these statistics are
misleading").
Scope
Some consider statistics to be a mathematical science pertaining to the collection,
analysis, interpretation or explanation, and presentation of data,[5] while others
consider it a branch of mathematics[6] concerned with collecting and interpreting
data. Because of its empirical roots and its focus on applications, statistics is
usually considered to be a distinct mathematical science rather than a branch of
mathematics.
Statisticians improve the quality of data with the design of experiments and survey
sampling. Statistics also provides tools for prediction and forecasting using data
and statistical models. Statistics is applicable to a wide variety of academic disciplines,
including natural and social sciences, government, and business. Statistical consultants
are available to provide help for organizations and companies without direct access
to expertise relevant to their particular problems.
Statistical methods can be used to summarize or describe a collection of data; this
is called descriptive statistics. This is useful in research, when communicating
the results of experiments. In addition, patterns in the data may be modeled in
a way that accounts for randomness and uncertainty in the observations, and are
then used to draw inferences about the process or population being studied; this
is called inferential statistics. Inference is a vital element of scientific advance,
since it provides a prediction (based in data) for where a theory logically leads.
To prove the guiding theory further, these predictions are tested as well, as part
of the scientific method. If the inference holds true, then the descriptive statistics
of the new data increase the soundness of that hypothesis. Descriptive statistics
and inferential statistics (a.k.a., predictive statistics) together comprise applied
statistics.
Statistics is closely related to probability theory, with which it is often grouped;
the difference is roughly that in probability theory, one starts from the given
parameters of a total population to deduce probabilities pertaining to samples,
but statistical inference moves in the opposite direction, inductive inference from
samples to the parameters of a larger or total population.
History
Some scholars pinpoint the origin of statistics to 1663, with the publication of
Natural and Political Observations upon the Bills of Mortality by John Graunt.[10]
Early applications of statistical thinking revolved around the needs of states to
base policy on demographic and economic data, hence its stat etymology. The scope
of the discipline of statistics broadened in the early 19th century to include the
collection and analysis of data in general. Today, statistics is widely employed
in government, business, and the natural and social sciences.
Its mathematical foundations were laid in the 17th century with the development
of probability theory by Blaise Pascal and Pierre de Fermat. Probability theory
arose from the study of games of chance. The method of least squares was first described
by Carl Friedrich Gauss around 1794. The use of modern computers has expedited largescale
statistical computation, and has also made possible new methods that are impractical
to perform manually.
Statistical methods
Experimental and observational studies
A common goal for a statistical research project is to investigate causality, and
in particular to draw a conclusion on the effect of changes in the values of predictors
or independent variables on dependent variables or response. There are two major
types of causal statistical studies: experimental studies and observational studies.
In both types of studies, the effect of differences of an independent variable (or
variables) on the behavior of the dependent variable are observed. The difference
between the two types lies in how the study is actually conducted. Each can be very
effective. An experimental study involves taking measurements of the system under
study, manipulating the system, and then taking additional measurements using the
same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation.
Instead, data are gathered and correlations between predictors and response are
investigated.
Experiments
The basic steps of a statistical experiment are:
1. Planning the research, including finding the number of replicates of the study,
using the following information: preliminary estimates regarding the size of treatment
effects, alternative hypotheses, and the estimated experimental variability. Consideration
of the selection of experimental subjects and the ethics of research is necessary.
Statisticians recommend that experiments compare (at least) one new treatment with
a standard treatment or control, to allow an unbiased estimate of the difference
in treatment effects.
2. Design of experiments, using blocking to reduce the influence of confounding
variables, and randomized assignment of treatments to subjects to allow unbiased
estimates of treatment effects and experimental error. At this stage, the experimenters
and statisticians write the experimental protocol that shall guide the performance
of the experiment and that specifies the primary analysis of the experimental data.
3. Performing the experiment following the experimental protocol and analyzing the
data following the experimental protocol.
4. Further examining the data set in secondary analyses, to suggest new hypotheses
for future study.
5. Documenting and presenting the results of the study. Experiments on human behavior
have special concerns. The famous Hawthorne study examined changes to the working
environment at the Hawthorne plant of the Western Electric Company. The researchers
were interested in determining whether increased illumination would increase the
productivity of the assembly line workers. The researchers first measured the productivity
in the plant, then modified the illumination in an area of the plant and checked
if the changes in illumination affected productivity. It turned out that productivity
indeed improved (under the experimental conditions). However, the study is heavily
criticized today for errors in experimental procedures, specifically for the lack
of a control group and blindness. The Hawthorne effect refers to finding that an
outcome (in this case, worker productivity) changed due to observation itself. Those
in the Hawthorne study became more productive not because the lighting was changed
but because they were being observed.
Observational study
An example of an observational study is one that explores the correlation between
smoking and lung cancer. This type of study typically uses a survey to collect observations
about the area of interest and then performs statistical analysis. In this case,
the researchers would collect observations of both smokers and nonsmokers, perhaps
through a casecontrol study, and then look for the number of cases of lung cancer
in each group.
Levels of measurement
There are four main levels of measurement used in statistics: nominal, ordinal,
interval, and ratio. Each of these have different degrees of usefulness in statistical
research. Ratio measurements have both a meaningful zero value and the distances
between different measurements defined; they provide the greatest flexibility in
statistical methods that can be used for analyzing the data.[citation needed] Interval
measurements have meaningful distances between measurements defined, but the zero
value is arbitrary (as in the case with longitude and temperature measurements in
Celsius or Fahrenheit). Ordinal measurements have imprecise differences between
consecutive values, but have a meaningful order to those values. Nominal measurements
have no meaningful rank order among values.
Because variables conforming only to nominal or ordinal measurements cannot be reasonably
measured numerically, sometimes they are grouped together as categorical variables,
whereas ratio and interval measurements are grouped together as quantitative variables,
which can be either discrete or continuous, due to their numerical nature.
Key terms used in statistics
Null hypothesis
Interpretation of statistical information can often involve the development of a
null hypothesis in that the assumption is that whatever is proposed as a cause has
no effect on the variable being measured. The best illustration for a novice is
the predicament encountered by a jury trial. The null hypothesis, H0, asserts that
the defendant is innocent, whereas the alternative hypothesis, H1, asserts that
the defendant is guilty. The indictment comes because of suspicion of the guilt.
The H0 (status quo) stands in opposition to H1 and is maintained unless H1 is supported
by evidence"beyond a reasonable doubt". However,"failure to reject H0" in this case
does not imply innocence, but merely that the evidence was insufficient to convict.
So the jury does not necessarily accept H0 but fails to reject H0. While one can
not "prove" a null hypothesis one can test how close it is to being true with a
power test, which tests for type II errors.
Error
Working from a null hypothesis two basic forms of error are recognized:
• Type I errors where the null hypothesis is falsely rejected giving a "false positive".
• Type II errors where the null hypothesis fails to be rejected and an actual difference
between populations is missed giving a "false negative".
Error also refers to the extent to which individual observations in a sample differ
from a central value, such as the sample or population mean. Many statistical methods
seek to minimize the meansquared error, and these are called "methods of least
squares."
Measurement processes that generate statistical data are also subject to error.
Many of these errors are classified as random (noise) or systematic (bias), but
other important types of errors (e.g., blunder, such as when an analyst reports
incorrect units) can also be important.
Interval estimation
Main article: interval estimation
Most studies will only sample part of a population and so the results are not fully
representative of the whole population. Any estimates obtained from the sample only
approximate the population value. Confidence intervals allow statisticians to express
how closely the sample estimate matches the true value in the whole population.
Often they are expressed as 95% confidence intervals. Formally, a 95% confidence
interval for a value is a range where, if the sampling and analysis were repeated
under the same conditions (yielding a different dataset), the interval would include
the true (population) value 95% of the time. This does not imply that the probability
that the true value is in the confidence interval is 95%. From the frequentist perspective,
such a claim does not even make sense, as the true value is not a random variable.
Either the true value is or is not within the given interval. However, it is true
that, before any data are sampled and given a plan for how the confidence interval
will be constructed, the probability is 95% that the yettobecalculated interval
will cover the true value: at this point, the limits of the interval are yettobeobserved
random variables. One approach that does yield an interval that can be interpreted
as having a given probability of containing the true value is to use a credible
interval from Bayesian statistics: this approach depends on a different way of interpreting
what is meant by "probability", that is as a Bayesian probability.
Significance
Main article: Statistical significance
Statistics rarely give a simple Yes/No type answer to the question asked of them.
Interpretation often comes down to the level of statistical significance applied
to the numbers and often refer to the probability of a value accurately rejecting
the null hypothesis (sometimes referred to as the pvalue).
Referring to statistical significance does not necessarily mean that the overall
result is significant in real world terms. For example, in a large study of a drug
it may be shown that the drug has a statistically significant but very small beneficial
effect, such that the drug will be unlikely to help the patient in a noticeable
way.









