Page 215 - Statistics for Environmental Engineers

L1592_frame_C24.fm Page 216 Tuesday, December 18, 2001 2:45 PM

TABLE 24.1
Ten Measurements of Lead Concentration (µg/L)
on Identical Specimens from Five Laboratories
Lab 1 Lab 2 Lab 3 Lab 4 Lab 5
3.4 4.5 5.3 3.2 3.3
3.0 3.7 4.7 3.4 2.4
3.4 3.8 3.6 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
= 4.30 3.97 4.46 3.12 3.34
y i
2
s i = 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:

n t ( y t )
y ti –
s t = ∑ --------------------
2
n t – 1
i=1
where y ti are the n t observations under each treatment.
Assuming that all treatments have the same population variance, we can pool the k sample variances
2
to estimate the within-treatment variance (s w ):
(
k
s w = ∑ t=1 n t – 1)s t 2
2
---------------------------------
(
k
∑ t=1 n t – 1)
()
2
The between-treatment variance s b is calculated using the treatment averages and the grand average, y:
y t
k ( y) 2
∑ t=1 n t y t –
s b = ----------------------------------
2
k – 1
2 2
If there are an equal number of observations in each treatment the equations for s w and s b simplify to:
( n t – k 2
s w = ---------------------------------
2
1)∑ t=1 s t
N – k
and
(
k
n t ∑ t=1 y t – y) 2
s b = ----------------------------------
2
k – 1
2 2
The logic of the comparison between for s w and s b goes like this:
2
1. The pooled variance within treatments (s w ) is based on N − k degrees of freedom. It will be
unaffected by real differences between the means of the k treatments. Assuming no hidden
factors are affecting the results, estimates the pure measurement error variance σ 2 .
2
s w
2. If there are no real differences between the treatment averages other than what would be
2
expected to occur by chance, the variance between treatments (s b ) also reflects only random
2
measurement error. As such, it would be nearly the same magnitude as s w and would give a
second estimate of σ 2 .
© 2002 By CRC Press LLC