In statistics Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. 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 and uncertainty analysis In statistics, propagation of uncertainty is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g. instrument precision) which propagate to the combination of variables in the, the Welch-Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article of sample variances.
For n sample variances si2 (i = 1, ..., n), each respectively having νi degrees of freedom , often one computes the linear combination
In general, the distribution of χ' cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution, whose effective degrees of freedom are given by the Welch-Satterthwaite equation
There is no assumption that the underlying population variances σi2 are equal.
The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t test.
References
- Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110–114, doi The Digital Object Identifier System is a managed system for persistent identification of content-related entities on digital networks. These entities may be content items (digital files, physical objects, abstract works), or any related entities in a content transaction (e.g. licenses, parties, etc.). "DOI" is sometimes used to mean the:10.2307/3002019
- Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika 34: 28–35
- Neter, John; John Neter, William Wasserman, Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc.. ISBN 0-256-08338-X.
Categories: Statistical theorems | Equations | Statistical approximations
