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Introduction to statistics
Definitions
Why is statistics important?
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statistics, its past and
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Statistics is the study of the
collection, analysis, interpretation,
presentation, and organization of
data. In applying statistics to, e.g.,
a scientific, industrial, or societal
problem, it is conventional to begin
with a statistical population or a
statistical model process to be
studied.
Populations can be diverse topics
such as "all persons living in a
country" or "every atom
composing a crystal". Statistics
deals with all aspects of data
including the planning of data
collection in terms of the design of
surveys and experiments.
When 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).
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.
A standard statistical procedure involves
the test of the relationship between two
statistical data sets, or a data set and a
synthetic data drawn from idealized
model. An hypothesis is proposed for the
statistical relationship between the two
data sets, and this is compared as an
alternative to an idealized null hypothesis
of no relationship between two data sets.
Rejecting or disproving the
null hypothesis is done
using statistical tests that
quantify the sense in which
the null can be proven false,
given the data that are used
in the test.
Working from a null hypothesis, two basic forms
of error are recognized: Type I errors (null
hypothesis is falsely rejected giving a "false
positive") and Type II errors (null hypothesis fails
to be rejected and an actual difference between
populations is missed giving a "false negative").
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 of missing data and/or censoring
may result in biased estimates and specific
techniques have been developed to address
these problems.
Statistics can be said to have begun in
ancient civilization, going back at least to
the 5th century BC, but it was not until
the 18th century that it started to draw
more heavily from calculus and
probability theory. Statistics continues to
be an area of active research, for example
on the problem of how to analyze Big
data.
Descriptive statistics
Descriptive statistics is the discipline of
quantitatively describing the main features of a
collection of information, or the quantitative
description itself. Descriptive statistics are
distinguished from inferential statistics (or
inductive statistics), in that descriptive statistics
aim to summarize a sample, rather than use the
data to learn about the population that the
sample of data is thought to represent.
Descriptive statistics (continued)
This generally means that descriptive statistics,
unlike inferential statistics, are not developed on
the basis of probability theory. Even when a data
analysis draws its main conclusions using
inferential statistics, descriptive statistics are
generally also presented.
Descriptive statistics (continued)
For example in a paper reporting on a study
involving human subjects, there typically
appears a table giving the overall sample size,
sample sizes in important subgroups (e.g., for
each treatment or exposure group), and
demographic or clinical characteristics such as
the average age, the proportion of subjects of
each sex, and the proportion of subjects with
related comorbidities.
Descriptive statistics (continued)
Some measures that are commonly used to
describe a data set are measures of central
tendency and measures of variability or
dispersion. Measures of central tendency
include the mean, median and mode, while
measures of variability include the standard
deviation (or variance), the minimum and
maximum values of the variables, kurtosis and
skewness.
Inference
Statistical inference is the process of deducing
properties of an underlying distribution by
analysis of data. Inferential statistical analysis
infers properties about a population: this
includes testing hypotheses and deriving
estimates. The population is assumed to be
larger than the observed data set; in other
words, the observed data is assumed to be
sampled from a larger population.
Inference (continued)
Inferential statistics can be contrasted with
descriptive statistics. Descriptive statistics is
solely concerned with properties of the
observed data, and does not assume that the
data came from a larger population.
References and bibliography
• wikipedia.org
• The textbooks in statistics