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| In [[statistics]] and [[propagation of uncertainty|uncertainty analysis]], the '''Welch–Satterthwaite equation''' is used to calculate an approximation to the effective [[degrees of freedom (statistics)|degrees of freedom]] of a [[linear combination]] of independent [[sample variance]]s.
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| For {{math|''n''}} sample variances {{math|''s''<sub>''i''</sub><sup>2</sup> (''i'' {{=}} 1, ..., ''n'')}}, each respectively having {{math|''ν''<sub>''i''</sub>}} degrees of freedom, often one computes the linear combination
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| :<math>
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| \chi' = \sum_{i=1}^n k_i s_i^2.
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| </math>
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| In general, the [[probability distribution]] of {{math|''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'''
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| :<math> | |
| \nu_{\chi'} \approx \frac{\displaystyle\left(\sum_{i=1}^n k_i s_i^2\right)^2}
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| {\displaystyle\sum_{i=1}^n \frac{(k_i s_i^2)^2}
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| {\nu_i}
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| }
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| </math>
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| There is ''no'' assumption that the underlying population variances {{math|''σ<sub>i</sub>''<sup>2</sup>}} are equal.
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| The result can be used to perform approximate [[statistical inference]] tests. The simplest application of this equation is in performing [[Welch's t test]].
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| ==References==
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| * {{Citation | last = Satterthwaite | first = F. E. | title = An Approximate Distribution of Estimates of Variance Components.| journal = Biometrics Bulletin | volume = 2 | pages = 110–114 | year = 1946 | doi = 10.2307/3002019 }}
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| * {{Citation | last = Welch | first = B. L. | title = The generalization of "student's" problem when several different population variances are involved. | journal = Biometrika | volume = 34 | pages = 28–35 | year = 1947 }}
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| * {{cite book
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| | last = Neter
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| | first = John
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| | coauthors = John Neter, William Wasserman, Michael H. Kutner
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| | title = Applied Linear Statistical Models
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| | publisher = Richard D. Irwin, Inc.
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| | date = 1990
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| <!-- | pages = 851 -->
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| | isbn = 0-256-08338-X }}
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| {{DEFAULTSORT:Welch-Satterthwaite equation}}
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| [[Category:Statistical theorems]]
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| [[Category:Equations]]
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| [[Category:Statistical approximations]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. The favorite hobby for him and his kids is to play lacross and he would never give it up. Alaska is exactly where I've always been living. My day job is an info officer but I've already applied for an additional 1.
Feel free to surf to my site ... clairvoyants