Tests
For More Than Two Means
1)
Analysis of Variance (ANOVA) for Completely Randomized Design
Simplification of test statistics
Sample 


Data 

Total 
Mean 
1 
y_{11} 
y_{12} 
y_{13} 
y_{14} 
T_{1} 
Y_{1} 
2 
y_{21} 
y_{22} 
y_{23} 
y_{24} 
T_{2} 
Y_{2} 
3 
y_{31} 
y_{32} 
y_{33} 
y_{34} 
T_{3} 
Y_{3} 
n= The total sample size = Sn_{i}
G= The sum of all sample observations = S T_{i}
Y = The average of all sample observations = G / n
Total sum of squares TSS = S y_{ij}^{2} – G^{2}/n
Withinsample of squares SSW = S (y_{ij} – Y_{i })^{2}
Betweensample sum of squares SSB = S (T_{i}^{2}/n_{i}) – G^{2}/n
TSS = SSW + SSB
The analysis of variance (ANOVA) table is as follows
Source 
Sum of Squares 
Degrees of Freedom 
Mean Sum of Squares 
F Test 
B samples 
SSB 
T – 1 
SSB / (t1) 
MSB/MSW 
W samples 
SSW 
n – t 
SSW / (nt) 

Totals 
TSS 
n – 1 


Ho: m_{1} = m_{2}= m_{3}= ……
Ha: At least one of the means is different from the
rest of means
Reject Ho if calculated F test > F _{(t1),
(nt)} a
2) ANOVA for Randomized Block Design



Block 



Treatment 
1 
2 
…. 
b 
Total 
Mean 
1 
y_{11} 
y_{12} 

y_{1b} 
T1 
Y1 
2 
y_{21} 
y_{22} 

y_{2b} 
T2 
Y2 
: 
: 
: 

: 
: 
: 
t 
y_{t1} 
y_{t2} 

y_{tb} 
Tt 
Yt 







Total 
B1 
B2 

Bb 
G 


b1 
b2 

bb 

Ÿ 
y_{ij} = The observation for treatment i and
block j
t = The number of treatments
b = The number of blocks
n = The total number of sample measurements = bt
T_{i} = The total of all observations
receiving treatment i
B_{j} =
The total of all observations in block j
G = The total of all sample observations
Y_{i} = The sample mean for treatment i = T_{i}/b
b_{j} = The sample mean fro block j
= B_{j}/t
Ÿ = The overall sample mean =
G/n
TSS = S (y_{ij} – Ÿ)^{2} Total sum of
squares
SST = b S (Y_{i} – Ÿ)^{2} Between
treatment sum of squares
SSB = t S (b_{j} – Ÿ)^{2} between block sum of squares
SSE = TSS – SST – SSB Sum of squares of error
Shortcut formulas
TSS = S y_{ij}^{2} –
G^{2} / n
SST = S T_{i}^{2}
/ b – G^{2} / n
SSB = S B_{j}^{2}
/ t – G^{2} / n
SSE = TSS – SST – SSB
Source 
SS 
df 
Ms 
F 
Treatment 
SST 
t1 
SST/t1 
MST/MSE 
Block 
SSB 
b1 
SSB/b1 
MSB/MSE 
Error 
SSE 
(b1)(t1) 
SSE/(b1)(t1) 

Totals 
TSS 
bt1 


Reject Ho if Treatment F > Fa, t1, (b1)(t1)
Reject Ho if Block F > Fa, b1, (b1)(t1)
3) ANOVA for t x t


Column 


Row 
1 
2 
3 
4 
1 
I 
II 
III 
IV 
2 
II 
III 
IV 
I 
3 
III 
IV 
I 
II 
4 
IV 
I 
II 
III 
y_{i}_{j}_{k} = the response on treatment i in row j and column k
t = the number of treatments, also the number of rows
and columns
n = the total number of samples
T_{i} = the total for all observations
receiving treatment i
Y_{i} = the sample mean for treatment i (T_{i}/t)
R_{j} =
the total for all observations in row j
Â_{j} = the sample mean for row j
(R_{j}/t)
C_{k} = the total for all observations in
column k
Ç_{k} = the sample mean for column
k (C_{k}/t)
G = the total for all sample measurements
Ÿ = the overall sample mean
(G/n)
TSS = SST + SSR + SSC + SSE
Shortcut formulas
TSS = S y_{i}_{j}_{k}^{2} – G^{2}/n
SST = S T_{i}^{2}/t –
G^{2}/n
SSR = S R_{j}^{2}/t – G^{2}/n
SSC = S C_{k}^{2}/t –
G^{2}/n
SSE = TSS – SST – SSR – SSC
Source 
SS 
df 
MS 
F 
Treatment 
SST 
t1 
SST/t1 
MST/MSE 
Rows 
SSR 
t1 
SSR/t1 
MSR/MSE 
Columns 
SSC 
t1 
SSC/t1 
MSC/MSE 
Error 
SSE 
t^{2}3t+2 
SSE/t^{2}3t+2 

Total 
TSS 
t^{2}1 


Reject Ho for treatment if F > Fa, t1, t^{2}3t+2
Reject Ho for rows if F > Fa, t1, t^{2}3t+2
Reject Ho for columns if F > Fa, t1, t^{2}3t+2