Welch-Satterthwaite equation

Welch-Satterthwaite equation

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 sample variances.

For "n" sample variances "si"2 ("i" = 1, ..., "n"), each respectively having "νi" degrees of freedom , often one computes the linear combination

: chi' = sum_{i=1}^{n} k_{i} s_{i}^{2}.

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"'

: u_{chi'} approx frac{(sum_{i=1}^{n} k_{i} s_{i}^{2})^{2 {sum_{i=1}^{n} frac{(k_{i} s_{i}^{2})^{2 { u_{i }

There is "no" assumption that the underlying population variances "σi"2 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

*
*
* cite book
last = Neter
first = John
coauthors = John Neter, William Wasserman, Michael H. Kutner
title = Applied Linear Statistical Models
publisher = Richard D. Irwin, Inc.
date = 1990
isbn = 0-256-08338-X


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