Welch-Satterthwaite equation
Encyclopedia
In statistics
and uncertainty analysis
, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom
of a linear combination
of independent sample variances.
For sample variances , each respectively having degrees of freedom, often one computes the linear combination
In general, the distribution of 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 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
.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and uncertainty analysis
Propagation of uncertainty
In statistics, propagation of error is the effect of variables' uncertainties on the uncertainty of a function based on them...
, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
of a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of independent sample variances.
For sample variances , each respectively having degrees of freedom, often one computes the linear combination
In general, the distribution of 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 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
Welch's t test
In statistics, Welch's t test is an adaptation of Student's t-test intended for use with two samples having possibly unequal variances. As such, it is an approximate solution to the Behrens–Fisher problem.-Formulas:...
.