Inferential statistics..

In statistics, statistical inference is the process of drawing conclusions from data that is subject to random variation, for example, observational errors or sampling variation. More substantially, the terms statistical inference, statistical induction and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from datasets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations.

The outcome of statistical inference may be an answer to the question "what should be done next?", where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy.

The use of inferential statistics is a cornerstone of research on populations and events, because it is difficult and sometimes impossible to survey every member of a population or to observe every event. Instead, researchers attempt to get a representative sample and use that as a basis for their claims. This differs from descriptive statistics, which describe only the data itself in statistical terms.

More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:

Ø a statistical model of the random process that is supposed to generate the data, which is known when randomization has been used, and

Ø a particular realization of these random process; i.e., a set of data.

The conclusion of a statistical inference is a statistical proposition. Some common forms of statistical proposition are:

Ø an estimate; i.e., a particular value that best approximates some parameter of interest,

Ø a confidence interval or set estimate; i.e., an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the probability at the stated confidence level,

Ø a credible interval; i.e., a set of values containing, for example, 95% of posterior belief,

Ø rejection of a hypothesis

Ø clustering or classification of data points into groups

Statistical inference is generally distinguished from descriptive statistics. In simple terms, descriptive statistics can be thought of as being just a straightforward presentation of facts, in which modeling decisions made by a data analyst have had minimal influence.

Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments.

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Descriptive statistics

Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. It is distinguished from inferential statistics which is also known as 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. This generally means that it is, 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.

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 and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects. Descriptive statistics is also a set of brief descriptive coefficients that summarizes a given data set that represents either the entire population or a sample.

Descriptive statistics provides simple summaries about the sample and about the observations that have been made. Such summaries may be either quantitative, i.e. summary statistics, or visual, i.e. simple-to-understand graphs. These summaries may either form the basis of the initial description of the data as part of a more extensive statistical analysis, or they may be sufficient in and of themselves for a particular investigation.

In the business world, Descriptive statistics provide a useful summary of security returns when performing empirical and analytical analysis, as they provide a historical account of return behavior. Although past information is useful in any analysis, one should always consider the expectations of future events.

The major analysis of Descriptive statistics consists of two analysis which are Univariate and Bivariate analysis.

Univariate analysis

Univariate analysis involves describing the distribution of a single variable, including its central tendency including the mean, median, and mode and dispersion including the range and quantiles of the data-set, and measures of spread such as the variance and standard deviation. 

Bivariate analysis

When a sample consists of more than one variable, descriptive statistics may be used to describe the relationship between pairs of variables. In this case, descriptive statistics include:

Ø Cross-tabulations and contingency tables

Ø Graphical representation via scatter plots

Ø Quantitative measures of dependence

Ø Descriptions of conditional distributions

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Intro to Statistics

Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments. 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.

Much of statistics is non-mathematical: ensuring that data collection is undertaken in a way that allows valid conclusions to be drawn; coding and archiving of data so that information is retained and made useful for international comparisons of official statistics; reporting of results and summarized data tables and graphs in ways that are comprehensible to those who need to make use of them; implementing procedures that ensure the privacy of census information.

Statisticians improve the quality of data by coming up with a specific design of experiments and survey sampling. Statistics itself also provides tools for prediction and forecasting the use of 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 questions.

Statistics is closely related to the 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.

Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This has changed with use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically.

Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.

Ø In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.

Ø Methods of statistics including predictive methods in forecasting are combined with chaos theory and fractal geometry to create video works that are considered to have great beauty.

Ø The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed. With the advent of computers, methods of statistics were applied to formalize such distribution driven natural processes, in order to make and analyze moving video art.

Ø Methods of statistics may be used predicatively in performance art, as in a card trick based on a Markov process that only works some of the time, the occasion of which can be predicted using statistical methodology.

Ø Statistics can be used to predicatively create art, as in the statistical or stochastic music invented by Iannis Xenakis, where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tunable using statistics.

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