Statistics for Environmental Engineers

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3.1


2.7


5.0


3.9


5.0


3.0


3.2


5.1


4.3


3.6


3.9


3.3


5.5


3.9


4.5


2.0


2.9


5.4


4.1


4.6


1.9


4.4


4.2


4.0


5.3


2.7


3.4


3.8


3.0


3.9


3.8


4.8


4.2


4.5


4.1


4.2


3.0


y, = 4.30


3.97


4.46


3.12


3.34


s2 = 0.82


0.19


0.41


0.58


0.54

The within-treatment sum of squares is calculated from the residuals of the observations within a treatment and the average for that treatment. The variance within each treatment is:


2


St



nt


I


i= 1



(yti-y<)


nt — 1


where y, are the nt observations under each treatment.


Assuming that all treatments have the same population variance, we can pool the k sample variances to estimate the within-treatment variance (s2W):


2 = Iil(nt— 1)s)


Sw    X’k {    — —


It=1 (nt — 1 )


4*11


The between-treatment variance (sj!) is calculated using the treatment averages yt and the grand average,


2 = Ik=1nt(yt-y)


Sb    k-1


If there are an equal number of observations in each treatment the equations for sW and sj2 simplify to:


s



2



w



( n t — 1) I k= 1 s] N — к

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