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en>David Eppstein: /* References */ fix chapter title

2014-09-01T23:09:44Z

References: fix chapter title

← Older revision		Revision as of 01:09, 2 September 2014
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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~~.		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.<br><br>Feel free to surf to my site ... [http://Www.Indosfriends.com/profile-253/info/ clairvoyants]

	~~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~~

	~~:<math>~~
	~~\chi' = \sum_{i=1}^n k_i s_i^2~~.
	~~</math>~~

	~~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'''~~

	:~~<math>~~
	~~\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}~~
	}
	~~</math>~~

	~~There is ''no'' assumption that the underlying population variances {{math\|''σ<sub>i<~~/~~sub>''<sup>2<~~/~~sup>}} 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==~~

	* {{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 }}~~
	* {{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 }}
	* {{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~~
	<!-~~- \| pages = 851 -->~~
	~~\| isbn = 0-256-08338-X }}~~

	~~{{DEFAULTSORT:Welch-Satterthwaite equation}}~~
	~~[[Category:Statistical theorems]]~~
	~~[[Category:Equations]]~~
	~~[[Category:Statistical approximations]~~]

en>Billymac00: /* External links */

2014-01-11T18:53:50Z

External links

← Older revision		Revision as of 20:53, 11 January 2014
Line 1:		Line 1:
	~~Hi there. Allow me begin by introducing~~ the ~~author, her title~~ is ~~Sophia~~. I'~~ve usually cherished living in Kentucky but now I~~'~~m considering other options. Since he was 18 he~~'s ~~been working as an info officer but he plans on changing it~~. ~~To play lacross is~~ one of the ~~issues she enjoys most.~~<br><br>~~Feel free~~ to ~~surf to my blog~~ ... ~~psychic readers~~ - [~~http~~:~~//brazil.amor-amore.com/irboothe Read the Full Guide~~],		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.

			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

			:<math>
			\chi' = \sum_{i=1}^n k_i s_i^2.
			</math>

			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'''

			:<math>
			\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}
			}
			</math>

			There is ''no'' assumption that the underlying population variances {{math\|''σ<sub>i</sub>''<sup>2</sup>}} 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==

			* {{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 }}
			* {{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 }}
			* {{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
			<!-- \| pages = 851 -->
			\| isbn = 0-256-08338-X }}

			{{DEFAULTSORT:Welch-Satterthwaite equation}}
			[[Category:Statistical theorems]]
			[[Category:Equations]]
			[[Category:Statistical approximations]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T00:25:52Z

ISBNs (Build KE)

New page

Hi there. Allow me begin by introducing the author, her title is Sophia. I've usually cherished living in Kentucky but now I'm considering other options. Since he was 18 he's been working as an info officer but he plans on changing it. To play lacross is one of the issues she enjoys most.<br><br>Feel free to surf to my blog ... psychic readers - [http://brazil.amor-amore.com/irboothe Read the Full Guide],