Step 1 denote the number of replication by r the

Step 1. Denote the number of replication by r , the number of levels of factor A ( i.e ., spacing) by
a, and that of factor B ( i.e ., age) by b. Construct the outline of the analysis of variance as
follows:
Table 4.16. Schematic representation of ANOVA of a factorial experiment with two levels of
factor A, three levels of factor B and with three replications in RCBD.
Source of
variation
Degrees of
freedom
( df )
Sum of
squares
( SS )
Mean square Computed
Replication r -1 SSR
Treatment ab - 1 SST
f
MSR
MST

MSA
MSB
MSE
Step 2.Compute treatment totals (Tij ), replication totals ( Rk ), and the grand total ( G ), as shown in
Table 4.15 and
compute the SSTO , SSR , SST and SSE following the procedure described
Section 4.3.3.
Let yijk refer to the observation corresponding to the i th level of factor A and
level factor B in the k th replication.
in
j th
22

23
SST (4.25)
= 14251.87
SSE = SSTO - SSR - SST (4.26)
= 17479.10 - 2040.37 - 14251.87
= 1186.86
The preliminary analysis of variance is shown in Table 4.17.
Table 4.17. Preliminary analysis of variance for data in Table 4.15.
Source of
variation
Degree
of
freedom
Sum of
squares
Mean
square
Computed F Tabular
F 5%
Replication 2 2040.37 1020.187 8.59567* 4.10
Treatment 5 14251.87 2850.373 24.01609* 3.33
Error 10 1186.86 118.686
Total 17 17479.10
*Significant at 5% level.
1.8 Degrees of Freedom
To determine the degrees of freedom (the number of variables whose values may be
independently specified) in our model we could simply count the number of independent
variables (the number of

24
variables which remain on the right- hand side) in our modified equations. This suggests a
possible definition:
degrees of freedom = # variables - # equations
Definition:
The degrees of freedom for a given problem are the number of independent problem variables
which must be specified to uniquely determine a solution. In our distillation example, there are:
16 equations 16 variables (recall that F and XF are fixed by upstream processes). This seems to
indicate that there are no degrees of freedom.
Consider the three equations relating QC, QR, and qvapour:
QR - QC = 0
QR - DHvap qvapour
= 0 QC - DHvap
qvapour = 0
Notice that if we subtract the last from the second equation:
QR - DHvap qvapour = 0
- QC - DHvap qvapour = 0
QR - QC = 0 the result is the first equation.
It seems that we have three different equations, which contain no more information than two of
the equations. In fact any of the equations is a linear combination of the other two equations. We
require a clearer, more precise definition for degrees of freedom.
Measures that ignore time value of money
net profit
payout
time
return on investment, ROI
Measures that recognize the time value of money
Internal rate of
return, IRR Net
present value,
NPV Discounted return on investment,
DROI A measure of the total
profitability
• Alternative, Benefits-to-cost ratio
• Strengths