**STATISTICS**

**Statistics**is the study of the collection, analysis, interpretation, presentation, and organization ofdata.

^{[1]}In applying statistics to, e.g., a scientific, industrial, or societal problem, it is necessary to begin with a population or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal". It deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.

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In case census data cannot be collected, statisticians collect data
by developing specific experiment designs and survey samples.
Representative sampling assures that inferences and conclusions can safely
extend from the sample to the population as a whole. 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.

Two main statistical methodologies are used in data
analysis: descriptive statistics, which summarizes data from a sample using indexes such
as the mean or standard deviation,
and inferential statistics, which draws conclusions from data that are subject
to random variation (e.g., observational errors, sampling variation).

^{[2]}Descriptive statistics are most often concerned with two sets of properties of a*distribution*(sample or population):*central tendency*(or*location*) seeks to characterize the distribution's central or typical value, while*dispersion*(or*variability*) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. To make an inference upon unknown quantities, one or more estimators are evaluated using the sample.
Standard statistical procedure involve the development
of a null hypothesis, a general statement or default position that there
is no relationship between two quantities. Rejecting or disproving the null hypothesis
is a central task in the modern practice of science, and gives a precise sense
in which a claim is capable of being proven false. What statisticians call an alternative hypothesis is simply an hypothesis that contradicts the null
hypothesis. Working from a null hypothesis two basic forms of error are
recognized: Type I errors (null hypothesis is falsely rejected giving a
"false positive") andType II errors (null hypothesis fails to be rejected and an actual
difference between populations is missed giving a "false negative").
A critical region is the set of values of the estimator that leads to
refuting the null hypothesis. The probability of type I error is therefore the
probability that the estimator belongs to the critical region given that null
hypothesis is true (statistical significance) and the probability of type II
error is the probability that the estimator doesn't belong to the critical
region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects
the null hypothesis when the null hypothesis is false. Multiple problems have
come to be associated with this framework: ranging from obtaining a sufficient
sample size to specifying an adequate null hypothesis.

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. The presence ofmissing data and/or censoring may result in biased estimates and specific techniques
have been developed to address these problems. Confidence intervals allow
statisticians to express how closely the sample estimate matches the true value
in the whole population. 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 in 95% of all possible cases. Ways to avoid misuse of
statistics include using proper diagrams and avoiding bias. In statistics,
dependence is any statistical relationship between two random variables or two
sets of data. Correlation refers to any of a broad class of statistical
relationships involving dependence. If two variables are correlated, they may
or may not be the cause of one another. The correlation phenomena could be
caused by a third, previously unconsidered phenomenon, called a lurking
variable or confounding variable.

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