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 independent sample variances.
For n sample variances si2 (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
\nu_{\chi'} \approx \frac{\displaystyle\left(\sum_{i=1}^n k_i s_i^2\right)^2}<br />
                          {\displaystyle\sum_{i=1}^n \frac{(k_i s_i^2)^2}<br />
                                               {\nu_i}<br />
                          }
There is no assumption that the underlying population variances σi2 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.

This is an excerpt from the article Welch–Satterthwaite equation from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
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Welch–Satterthwaite equation - Wikipedia, the free 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 ...
en.wikipedia.org/wiki/Welch%E2%80%93Satterthwaite_equation
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Welch's t test - Wikipedia, the free encyclopedia
Welch's t-test is an adaptation of Student's t-test, and is more reliable when the two ... variance estimate is approximated using the Welch–Satterthwaite equation :.
en.wikipedia.org/wiki/Welch's_t_test
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Calculating Effective Degrees of Freedom - isobudgets
Jun 13, 2014 ... Instead, we must use the Welch Satterthwaite approximation equation to calculate the effective degrees of freedom. In this article, we will ...
www.isobudgets.com/calculating-effective-degrees-of-freedom/
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Class of 04/16/09
In practice, it's good enough to use the simpler formula, certainly in this course. 5 ... Calculate df: either Welch-Satterthwaite formula or simpler df = min(n1,n2) ...
www.unc.edu/~rls/s151-09/class26.pdf
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Don't forget the degrees of freedom: evaluating uncertainty from ...
Oct 20, 2011 ... It is called the Welch-Satterthwaite formula and obtains an effective degrees of freedom for the result. That number, νeff , is used to calculate a ...
rf.irl.cri.nz/sites/default/files/files/DoF%20ANAMET35%20(notes).pdf
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2.5.7.1. Degrees of freedom
... components of uncertainty with small degrees of freedom. In this case, the degrees of freedom is approximated by the Welch-Satterthwaite formula ( Brownlee).
www.itl.nist.gov/div898/handbook/mpc/section5/mpc571.htm
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Welch-Satterthwaite T-Test Definition | Statistics Dictionary | MBA ...
... variances, the standard t-test which assumes equality if variances of the two groups, cannot be used. The test then used is called the Welch-Satterthwaite t-test.
www.mbaskool.com/business-concepts/statistics/7377-welch-satterthwaite-t-test.html
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Welch–Satterthwaite equation: proof or derivation? - Talk Stats
Hi, I'm studying now the Welch's t-test in which we use the Welch–Satterthwaite equation to calculate the degrees of freedom of the ...
www.talkstats.com/showthread.php/12753-Welch%E2%80%93Satterthwaite-equation-proof-or-derivation
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hypothesis testing - Should I use Welch's (1947) approximate ...
Sep 24, 2013 ... I'm confused as to the correct formula for approximate degrees of freedom to use for Welch's t-test. Satterthwaite's (1946) formula is the most ...
stats.stackexchange.com/questions/70959/should-i-use-welchs-1947-approximate-degrees-of-freedom-or-satterthwaites-1
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welch.satterthwaite {metRology} | inside-R | A Community Site for R
Details. Implements the Welch-Satterthwaite equation as provided in the ISO Guide to the expression of uncertainty in measurement (1995) (See JCGM ...
www.inside-r.org/node/163623
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Welch–Satterthwaite equation in science
Welch–Satterthwaite equation - Wikipedia, the free 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 ...
Welch's t test - Wikipedia, the free encyclopedia
Welch's t-test is an adaptation of Student's t-test, and is more reliable when the two ... variance estimate is approximated using the Welch–Satterthwaite equation :.
t-distribution and Welch–Satterthwaite formula - Cambridge ...
Cambridge University Press. Utility. University Publishing Online ... to Uncertainty in Measurement; The t-distribution and Welch–Satterthwaite formula pp. 162- ...
[PDF]Michael Lloyd - 2-Sample t-Distribution Approximation - MAA Sections
Languages, and Philosophy at Henderson State University. ... actually an approximation introduced by Welch and Satterthwaite in the late 1940s. We will.
Calculating Effective Degrees of Freedom - isobudgets
Jun 13, 2014 ... Instead, we must use the Welch Satterthwaite approximation equation to ... degree in Engineering from Old Dominion University in Norfolk, VA.
The Welch-Satterthwaite equation gets me... - University of Tulsa ...
The Welch-Satterthwaite equation gets me wet!! ... University of Tulsa Confessions · October 2, 2013 ·. The Welch-Satterthwaite equation gets me wet!
In Stata, how do I conduct a t-test when two samples have unequal ...
By default, the unequal option computes Satterthwaite's approximation ... You also can calculate Welch's approximation using the welch option, as follows: ... Research Analytics is located on the IU Bloomington campus at Woodburn Hall 200, ...
The unequal variance t-test is an underused alternative to Student's t ...
May 17, 2006 ... Authors are unaware that Student's t-test is unreliable when .... variances t-test or Satterwaite's test or the Welch–Satterthwaite test, after Satterwaite (1946). ... Published by Oxford University Press on behalf of the International ...
[PDF]k-sample t-test Redux - UCA
Jan 23, 2013 ... University of Texas Southwestern Medical Center at Dallas ... Welch-Satterthwaite (WS) equation has stood the test of time where it is included ...
[PDF]AP Statistics - AP Central - The College Board
I. Introduction. What's the most complicated formula we encounter in AP Statistics ? ... way of an introduction to the work on Satterthwaite's formula. So this section, too, ..... University, where he gained an undergraduate degree in mathematics.
Books on the term Welch–Satterthwaite equation
Measurement and Data Analysis for Engineering and Science, ...
Patrick F. Dunn, 2010
Note that the covariances in Equation 7.19 should not be ignored simply for convenience when performing an ... for ux. and veff < ∑ Ji=ivi- This equation, known as the Welch-Satterthwaite formula, was presented originally by Welch [9].
Experimentation, Validation, and Uncertainty Analysis for Engineers
Experimentation, Validation, and Uncertainty Analysis for Engineers
2009
This Third Edition helps you assess and manage uncertainty at all stages of experimentation and validation of simulationsIn this greatly expanded Third Edition, the acclaimed Experimentation, Validation, and Uncertainty Analysis for Engineers guides readers through the concepts of experimental uncertainty analysis and the applications in validating...
Experimentation and Uncertainty Analysis for Engineers
Hugh W. Coleman, W. Glenn Steele, 1999
... using the Welch-Satterthwaite formula and thus the need to estimate all of the vt and the VK . ,j BJ Consideration of the Welch-Satterthwaite formula [Eq. (B.l9)] shows that because of the exponent of 4 in each term, v, is most influenced by the  ...
Instrument Engineers' Handbook, Vol. 1: Process Measurement and Analysis
Instrument Engineers' Handbook, Vol. 1: Process Measurement and Analysis
2003
Unsurpassed in its coverage, usability, and authority since its first publication in 1969, the three-volume Instrument Engineers' Handbook continues to be the premier reference for instrument engineers around the world. It helps users select and implement hundreds of measurement and control instruments and analytical devices and design the mos...
Understanding The New Statistics: Effect Sizes, Confidence ...
Geoff Cumming, 2013
BOX 6.2 THE WELCH—SATTERTHWAITE APPROACH You can skip this explanation if you wish. just note the final sentence in the box. For the ... This can be contrasted with 1 1 sP i + i NA NB that appears in Equation (6.2). Recall that sP is ...
Challenges at the Interface of Data Analysis, Computer ...
Wolfgang Gaul, Andreas Geyer-Schulz, Lars Schmidt-Thieme, 2012
The t-statistic as provided by Equation 1 is known to perform reasonably well if the two empirical bootstrap distributions are ... The number of the degrees of freedom for the t-statistic is determined by means of the Welch-Satterthwaite equation ...
Infrared Thermography: Errors and Uncertainties
Infrared Thermography: Errors and Uncertainties
Waldemar Minkina, 2009
In Infrared Thermography , the authors discuss the sources of uncertainty, including how to quantify these sources, associated with the use of thermal imagers. This book explains the common misunderstandings in the interpretation of temperature measurements, and provides a metrological evaluation of commercially available infrared cameras. It sugge...
Encyclopedia of Research Design
Neil J. Salkind, 2010
The test that compares tw against the distribution tðdfwÞ for significance is called Welch's t test. ... It's worth mentioning that the approximation in Example 7 was also obtained by Franklin E. Satterthwaite in 1946 under a different context.
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Welch–Satterthwaite equation
Welch–Satterthwaite equation - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Welch%E2%80%93Satterthwaite_equation
Calculating Effective Degrees of Freedom - isobudgets
www.isobudgets.com/calculating-effective-degrees-of-freedom/
Statistic on aiR: Two sample Student's t-test #2
Comparison of the averages of two independent groups, extracted from two populations at variance unknown; sample variances are not homogeneous.We want to compare the heights in inches of two groups of individuals. Here the measurements:A: 175, 168, 168, 190, 156, 181, 182, 175, 174, 179B: 120, 180, 125, 188, 130, 190, 110, 185, 112, 188As we have seen in a previous exercise, we must first check whether the variances are homogeneous (homoskedasticity) with a F-test of Fisher:a = c(175, 168, 168, 190, 156, 181, 182, 175, 174, 179) b = c(120, 180, 125, 188, 130, 190, 110, 185, 112, 188)var.test(b,a) F test to compare two variancesdata: b and aF = 14.6431, num df = 9, denom df = 9, p-value = 0.0004636alternative hypothesis: true ratio of variances is not equal to 195 percent confidence interval: 3.637133 58.952936sample estimates: ratio of variances 14.64308We obtained p-value less than 0.05, then the two variances are not homogeneous.
statistic-on-air.blogspot.com/2009/07/two-sample-students-t-test-2.html
welch.satterthwaite {metRology} | inside-R | A Community Site for R
www.inside-r.org/node/163623
Welch–Satterthwaite equation: proof or derivation?
Hi, I'm studying now the Welch's t-test in which we use the Welch–Satterthwaite equation to calculate the degrees of freedom of the approxiamate chisquare distribution. However, I can't find a proof or derivation of this equation. Can you please suggest me a paper, site, or book where I can find it? I don't like using formulas without knowing how to prove them :) Thank you in advance Andrea
www.talkstats.com/showthread.php/12753-Welch%E2%80%93Satterthwaite-equation-proof-or-derivation
hypothesis testing - Should I use Welch's (1947) approximate degrees of freedom or Satterthwaite's (1946)? - Cross Validated
stats.stackexchange.com/questions/70959/should-i-use-welchs-1947-approximate-degrees-of-freedom-or-satterthwaites-1
A generalization of the Welch?Satterthwaite formula for use with correlated uncertainty components - Abstract - Metrologia - IOPscience
The Welch?Satterthwaite (W?S) formula described in the Guide to the Expression of Uncertainty in Measurement enables an effective number of degrees of freedom to be associated with the standard uncertainty of a measurement estimate. This facilitates the calculation of an expanded uncertainty interval for the value of the measurand. However, the W?S formula is only applicable when the components of measurement error with finite degrees of freedom are uncorrelated. This paper considers the generalization of the formula to accommodate correlated components with finite degrees of freedom. We show that the number of degrees of freedom to be associated with any estimate derived from repeated observation of several quantities is n ? 1, where n is the number of sets of observations, and we then give two equations by which the number of effective degrees may be calculated.
iopscience.iop.org/0026-1394/44/5/010
Welch-Satterthwaite equation | Project Gutenberg Self-Publishing - eBooks | Read eBooks online
Project Gutenberg Self-Publishing - eBooks
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Stick To Your GUMs Applying Welch-Satterthwaite to Correlated Errors
www.ncsli.org/c/f/p11/165.335.pdf
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