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450:{\displaystyle \nu _{\chi '}\approx {\frac {\displaystyle \left(\sum _{i=1}^{n}k_{i}s_{i}^{2}\right)^{2}}{\displaystyle \sum _{i=1}^{n}{\frac {(k_{i}s_{i}^{2})^{2}}{\nu _{i}}}}}}
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Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.",
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Satterthwaite, F. E. (1946), "An
Approximate Distribution of Estimates of Variance Components.",
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cannot be expressed analytically. However, its distribution can be approximated by another
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Michael
Allwood (2008) "The Satterthwaite Formula for Degrees of Freedom in the Two-Sample
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tests. The simplest application of this equation is in performing
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degrees of freedom, often one computes the linear combination.
681:
Neter, John; William
Wasserman; Michael H. Kutner (1990).
524:
Handbook of mathematics and statistics for the environment
31:
is used to calculate an approximation to the effective
270:, whose effective degrees of freedom are given by the
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164:{\displaystyle \chi '=\sum _{i=1}^{n}k_{i}s_{i}^{2}.}
100:
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assumption that the underlying population variances
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707:, Advanced Placement Program, The College Board.
16:Equation to approximate pooled degrees of freedom
480:The result can be used to perform approximate
249:{\displaystyle k_{i}={\frac {1}{\nu _{i}+1}}}
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570:Van Emden, H. F. (Helmut Fritz) (2008).
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201:is a real positive number, typically
7:
573:Statistics for terrified biologists
14:
683:Applied Linear Statistical Models
527:. Whiting, Nancy E. Boca Raton.
473:are equal. This is known as the
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393:
1:
576:. Malden, MA: Blackwell Pub.
272:Welch–Satterthwaite equation
29:Welch–Satterthwaite equation
80:, each respectively having
749:
733:Statistical approximations
685:. Richard D. Irwin, Inc.
45:pooled degrees of freedom
268:chi-squared distribution
258:probability distribution
47:, corresponding to the
723:Theorems in statistics
475:Behrens–Fisher problem
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482:statistical inference
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194:{\displaystyle k_{i}}
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43:, also known as the
25:uncertainty analysis
611:Biometrics Bulletin
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256:. In general, the
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37:linear combination
33:degrees of freedom
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60:sample variances
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603:Further reading
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49:pooled variance
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717:Categories
647:Biometrika
508:References
74:= 1, ...,
21:statistics
728:Equations
592:317778677
551:cite book
543:863225343
460:There is
434:ν
371:∑
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174:where
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627:JSTOR
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671:PMID
635:PMID
588:OCLC
578:ISBN
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539:OCLC
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