Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw 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.
Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. The use of modern computers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research for example on the problem of how to analyze big data.
Many statistical methods seek to minimize the residual sum of squares, and these are called \"methods of least squares\" in contrast to Least absolute deviations. The latter gives equal weight to small and big errors, while the former gives more weight to large errors. Residual sum of squares is also differentiable, which provides a handy property for doing regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares, which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve.
Exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.
Applied statistics, sometimes referred to as Statistical science, comprises descriptive statistics and the application of inferential statistics. Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.
A typical statistics course covers descriptive statistics, probability, binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation. Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use of free statistics software.
Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs sampling have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with a new emphasis on \"experimental\" and \"empirical\" statistics. A large number of both general and special purpose statistical software are now available. Examples of available software capable of complex statistical computation include programs such as Mathematica, SAS, SPSS, and R.
In business, \"statistics\" is a widely used management- and decision support tool. It is particularly applied in financial management, marketing management, and production, services and operations management . Statistics is also heavily used in management accounting and auditing. The discipline of Management Science formalizes the use of statistics, and other mathematics, in business. (Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.)
In the last few years, the American Statistical Association has published position statements on the correct use and interpretation of p-values and on avoiding possible traps in applied statistical work. As much of empirical research in demography relies on statistical methods, in order to meet the challenge, Demographic Research has now also issued editorial guidance on our expectations regarding statistical rigour and replicability in empirical work. Please find the full editorial here:demographic-research.org/.../vol41/32
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