# 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