
WELCH SATTERTHWAITE
Name:

k  =  the number of components 
s_{i}  =  the standard deviation of the ith component 
\( \nu_{i} \)  =  the standard deviation of the ith component 
a_{i}  =  the sensitivity coefficient of the ith component 
u  = 
the standard uncertainty = \( \sqrt{\sum_{i=1}^{k}{a_{i}^{2} s_{i}^{2}}} \) 
For this command, the s_{i}, ν_{i}, and a_{i} are given as inputs and u will be computed from the a_{i} and s_{i} components.
The sensitivity coefficients are derived from partial derivatives of the measurement equation. For the case of additive, independent uncertainties, these can often be set to 1.
The NIST/SEMATACH eHandbook of Statistical Methods gives some examples of this. In particular, it shows some examples of determining the sensitivity coefficients.
where k_{i} is typically 1/(ν_{i} + 1). The WelchSaitterwaithe approximation for the effective degrees of freedom is given by
A pooled standard deviation is then computed as
T TEST  =  Perform a two sample ttest. 
CONSENSUS MEANS  =  Compute a consensus mean and its associated uncertainty. 
Welch (1947), "The Generalization of Students's Problem when Several Different Population Variances are Involved", Biometrika, 34: 2835.
"Guide to the Expression of Uncertainty in Measurement", ISO, Geneva (1993).
"NIST/SEMATECH Handbook of Statistical Methods", Measurement Process Characterization chapter, " http://www.itl.nist.gov/div898/handbook/mpc/mpc.htm", June, 2003.
SKIP 25 READ AUTO83B.DAT Y1 Y2 LET N1 = SIZE Y1 LET NU1 = N1  1 LET VAR1 = VARIANCE Y1 LET N2 = SIZE Y2 LET NU2 = N2  1 LET VAR2 = VARIANCE Y2 LET YVAR = DATA VAR1 VAR2 LET YDF = DATA NU1 NU2 LET DF POOLSD = VARIANCES WELCH SATTERTHWAITE YVAR YDF LET DF = ROUND(DF,2) LET POOLSD = ROUND(POOLSD,2) PRINT "Degrees of Freedom: ^DF" PRINT "Pooled SD: ^POOLSD"The following output is generated
Degrees of Freedom: 248.09 Pooled SD: 339.52Program 2:
LET YSD = DATA 0.00371 0.00191 0.00191 0.00006 LET YDF = DATA 2 1000 1000 1000 LET YA = DATA 1 1 1 1 LET DF = GUM WELCH SATTERTHWAITE YSD YDF YA LET DF = ROUND(DF,2) PRINT "Degrees of Freedom: ^DF"The following output is generated
Degrees of Freedom: 4.68
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Date created: 07/20/2017
Last updated: 07/20/2017
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