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Statistics for Managers
Using Microsoft® Excel
5th Edition
Chapter 11
Analysis of Variance
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-1
Learning Objectives
In this chapter, you learn:
 The basic concepts of experimental design
 How to use the one-way analysis of variance
to test for differences among the means of
several groups
 How to use the two-way analysis of variance
and interpret the interaction
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-2
Chapter Overview
Analysis of Variance (ANOVA)
One-Way
ANOVA
F-test
Two-Way
ANOVA
Interaction
Effects
TukeyKramer
test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-3
General ANOVA Setting
 Investigator controls one or more factors of interest
 Each factor contains two or more levels
 Levels can be numerical or categorical
 Different levels produce different groups
 Think of the groups as populations
 Observe effects on the dependent variable
 Are the groups the same?
 Experimental design: the plan used to collect the
data
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-4
Completely Randomized
Design
 Experimental units (subjects) are assigned
randomly to the different levels (groups)
 Subjects are assumed homogeneous
 Only one factor or independent variable
 With two or more levels (groups)
 Analyzed by one-factor analysis of variance
(one-way ANOVA)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-5
One-Way Analysis of Variance
 Evaluate the difference among the means of three or
more groups
Examples: Accident rates for 1st, 2nd, and 3rd shift
Expected mileage for five brands of tires
 Assumptions
 Populations are normally distributed
 Populations have equal variances
 Samples are randomly and independently drawn
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-6
Hypotheses: One-Way ANOVA

H0 : μ1  μ2  μ3    μc
 All population means are equal
 i.e., no treatment effect (no variation in means among groups)

H1 : Not all of the population means are the same
 At least one population mean is different
 i.e., there is a treatment (groups) effect
 Does not mean that all population means are different (at
least one of the means is different from the others)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-7
Hypotheses: One-Way ANOVA
H0 : μ1  μ2  μ3    μc
H1 : Not all μ j are the same
All Means are the same:
The Null Hypothesis is True
(No Group Effect)
μ1  μ 2  μ 3
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-8
Hypotheses: One-Way ANOVA
At least one mean is different:
The Null Hypothesis is NOT true
(Treatment Effect is present)
H0 : μ1  μ 2  μ 3    μ c
H1 : Not all μj are the same
or
μ1  μ2  μ3
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
μ1  μ2  μ3
Chap 11-9
Partitioning the Variation
 Total variation can be split into two parts:
SST = SSA + SSW
SST = Total Sum of Squares
(Total variation)
SSA = Sum of Squares Among Groups
(Among-group variation)
SSW = Sum of Squares Within Groups
(Within-group variation)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-10
Partitioning the Variation
SST = SSA + SSW
Total Variation = the aggregate dispersion of the individual data
values around the overall (grand) mean of all factor levels (SST)
Among-Group Variation = dispersion between the factor
sample means (SSA)
Within-Group Variation = dispersion that exists among the data
values within the particular factor levels (SSW)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-11
Partitioning the Variation
Total Variation (SST)
=
Among Group
Variation (SSA)
+
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Within Group Variation
(SSW)
Chap 11-12
The Total Sum of Squares
SST = SSA + SSW
c
nj
SST   ( X ij  X )
Where:
2
j 1 i 1
SST = Total sum of squares
c = number of groups
nj = number of values in group j
Xij = ith value from group j
X = grand mean (mean of all data values)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-13
The Total Sum of Squares
c
nj
SST   ( X ij  X )
2
j 1 i 1
SST  ( X 11  X )  ( X 12  X )  ...  ( X nc  X )
2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
2
2
Chap 11-14
Among-Group Variation
SST = SSA + SSW
c
SSA   n j ( X j  X ) 2
Where:
j 1
SSA = Sum of squares among groups
c = number of groups
nj = sample size from group j
Xj = sample mean from group j
X = grand mean (mean of all data values)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-15
Among-Group Variation
SSA  n1 (X1  X) 2  n 2 (X 2  X) 2  ...  n c (X c  X) 2
c
SSA   n j (X j  X) 2
j1
µ1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
µ2
µc
Chap 11-16
Within-Group Variation
SST = SSA + SSW
c
Where:
SSW  
j 1
nj

i 1
( X ij  X j )
2
SSW = Sum of squares within groups
c = number of groups
nj = sample size from group j
Xj = sample mean from group j
Xij = ith value in group j
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-17
Within-Group Variation
c
SSW  
j 1
nj

i 1
( X ij  X j ) 2
SSW  ( X11  X 1 )2  ( X 21  X 1 )2  ...  ( X nc  X c )2
j
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-18
Obtaining the Mean Squares
SSA
MSA 
c 1
Mean Squares Among
SSW
MSW 
nc
Mean Squares Within
SST
MST 
n 1
Mean Squares Within
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-19
One-Way ANOVA Table
Source of
Variation
df
SS
MS
(Variance)
Among
Groups
c-1
SSA
MSA
Within
Groups
n-c
SSW
MSW
Total
n-1
SST =
SSA+SSW
F-Ratio
F
MSA
MSW
c = number of groups
n = sum of the sample sizes from all groups
df = degrees of freedom
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-20
One-Way ANOVA
Test Statistic
H0: μ1= μ2 = … = μc
H1: At least two population means are different
 Test statistic
MSA
F
MSW
 MSA is mean squares among variances
 MSW is mean squares within variances
 Degrees of freedom
 df1 = c – 1
 df2 = n – c
(c = number of groups)
(n = sum of all sample sizes)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-21
One-Way ANOVA
Test Statistic
 The F statistic is the ratio of the among variance to the
within variance
 The ratio must always be positive
 df1 = c -1 will typically be small
 df2 = n - c will typically be large
Decision Rule:
Reject H0 if F > FU,
otherwise do not reject H0
 = .05
0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Do not
reject H0
Reject H0
FU
Chap 11-22
One-Way ANOVA
Example
You want to see if three different
golf clubs yield different
distances. You randomly select
five measurements from trials
on an automated driving
machine for each club. At the
.05 significance level, is there a
difference in mean distance?
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Club 1
254
263
241
237
251
Club 2
234
218
235
227
216
Club 3
200
222
197
206
204
Chap 11-23
One-Way ANOVA
Example
Club 1
254
263
241
237
251
Club 2
234
218
235
227
216
Club 3
200
222
197
206
204
Distance
270
260
250
240
230
220
x1  249.2 x 2  226.0 x 3  205.8
x  227.0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
•
•• X 1
•
•
••
•
••
X2
210
••
••
200
190
•
1
2
Club
X
X3
3
Chap 11-24
One-Way ANOVA
Example
X1 = 249.2
n1 = 5
X2 = 226.0
n2 = 5
X3 = 205.8
n3 = 5
X = 227.0
n = 15
c=3
SSA = 5 (249.2 – 227)2 + 5 (226 – 227)2 + 5 (205.8 – 227)2 = 4716.4
SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = 1119.6
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-25
One-Way ANOVA
Example
2358.2
F
 25.275
93.3
MSA = 4716.4 / (3-1) = 2358.2
MSW = 1119.6 / (15-3) = 93.3
Critical
Value:
FU = 3.89
 = .05
0
Do not
reject H0
Reject H0
FU = 3.89
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
F = 25.275
Chap 11-26
One-Way ANOVA
Example
 H0: μ1 = μ2 = μ3
 H1: μj not all equal
  = .05
 df1= 2
Decision:
Reject H0 at α = 0.05
df2 = 12
Conclusion: There is
evidence that at least one
μj differs from the rest
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-27
One-Way ANOVA in Excel
EXCEL:
Select
Tools
____
Data
Analysis
_____
ANOVA:
single
factor
SUMMARY
Groups
Count
Sum
Average
Variance
Club 1
5
1246
249.2
108.2
Club 2
5
1130
226
77.5
Club 3
5
1029
205.8
94.2
ANOVA
Source of
Variation
SS
df
MS
Between
Groups
4716.4
2
2358.2
Within
Groups
1119.6
12
93.3
Total
5836.0
14
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
F
25.275
P-value
4.99E-05
F crit
3.89
Chap 11-28
The Tukey-Kramer Procedure
 Tells which population means are significantly different
 e.g.: μ1 = μ2 ≠ μ3
 Done after rejection of equal means in ANOVA
 Allows pair-wise comparisons
 Compare absolute mean differences with critical
range
μ1= μ2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
μ3
x
Chap 11-29
Tukey-Kramer Critical Range
Critical Range  QU
MSW  1 1 

2  n j n j' 
where:
QU = Value from Studentized Range Distribution with c
and n - c degrees of freedom for the desired level
of  (see appendix E.9 table)
MSW = Mean Square Within
nj and nj’ = Sample sizes from groups j and j’
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-30
The Tukey-Kramer Procedure
Club 1
254
263
241
237
251
Club 2
234
218
235
227
216
Club 3
200
222
197
206
204
1. Compute absolute mean
differences:
x1  x 2  249.2  226.0  23.2
x1  x 3  249.2  205.8  43.4
x 2  x 3  226.0  205.8  20.2
2. Find the QU value from the table in appendix E.9 with
c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom for
the desired level of  ( = .05 used here):
QU  3.77
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-31
The Tukey-Kramer Procedure
3. Compute Critical Range:
Critical Range  Q U
4. Compare:
MSW  1 1 
93.3  1 1 

 3.77
    16.285


2  n j n j' 
2 5 5
x1  x 2  23.2
x1  x 3  43.4
x 2  x 3  20.2
5. All of the absolute mean differences are greater than
critical range. Therefore there is a significant difference
between each pair of means at the 5% level of significance.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-32
ANOVA Assumptions
 Randomness and Independence
 Select random samples from the c groups (or
randomly assign the levels)
 Normality
 The sample values from each group are from a
normal population
 Homogeneity of Variance
 Can be tested with Levene’s Test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-33
ANOVA Assumptions
Levene’s Test
 Tests the assumption that the variances of each
group are equal.
 First, define the null and alternative hypotheses:
 H0: σ21 = σ22 = …=σ2c
 H1: Not all σ2j are equal
 Second, compute the absolute value of the difference
between each value and the median of each group.
 Third, perform a one-way ANOVA on these absolute
differences.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-34
Two-Way ANOVA
 Examines the effect of
 Two factors of interest on the dependent
variable
 e.g., Percent carbonation and line speed on soft
drink bottling process
 Interaction between the different levels of these
two factors
 e.g., Does the effect of one particular carbonation
level depend on which level the line speed is set?
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-35
Two-Way ANOVA
 Assumptions
 Populations are normally distributed
 Populations have equal variances
 Independent random samples are selected
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-36
Two-Way ANOVA
Sources of Variation
Two Factors of interest: A and B
r = number of levels of factor A
c = number of levels of factor B
n/ = number of replications for each cell
n = total number of observations in all cells
(n = rcn/)
Xijk = value of the kth observation of level i
of factor A and level j of factor B
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-37
Two-Way ANOVA
Sources of Variation
SST = SSA + SSB + SSAB + SSE
SSA
Factor A Variation
SST
Total Variation
SSB
Factor B Variation
Degrees of
Freedom:
r–1
c–1
SSAB
n-1
Variation due to interaction
between A and B
SSE
Random variation (Error)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
(r – 1)(c – 1)
rc(n/ – 1)
Chap 11-38
Two-Way ANOVA
Equations
r
Total Variation:
n
c
SST   ( Xijk  X)
2
i1 j1 k 1
r
Factor A Variation:
2

SSA  cn  ( Xi..  X)
i1
c
Factor B Variation:
2

SSB  rn  ( X. j.  X)
j1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-39
Two-Way ANOVA
Equations
Interaction Variation:
r
c
SSAB  n ( X ij.  X i..  X . j.  X ) 2
i 1 j 1
Sum of Squares Error:
r
c
n
SSE   ( Xijk  Xij. )
2
i1 j1 k 1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-40
Two-Way ANOVA
Equations
r
X
n
c
 X
i1 j1 k 1
rcn
c
Xi.. 
ijk
 Grand Mean
n
 X
j1 k 1
cn
ijk
 Mean of ith level of factor A (i  1, 2, ..., r)
r = number of levels of factor A
c = number of levels of factor B
n/ = number of replications in each cell
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-41
Two-Way ANOVA
Equations
r
X.j. 
n
 X
i 1 k 1
rn
ijk
 Mean of jth level of factor B (j  1, 2, ..., c)
n
Xijk
Xij.  
 Mean of cell ij
k 1 n
r = number of levels of factor A
c = number of levels of factor B
n/ = number of replications in each cell
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-42
Two-Way ANOVA
Equations
SSA
MSA  Mean square factor A 
r 1
SSB
MSB  Mean square factor B 
c 1
SSAB
MSAB  Mean square interactio n 
(r  1)(c  1)
SSE
MSE  Mean square error 
rc(n'1)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-43
Two-Way ANOVA:
The F Test Statistic
F Test for Factor A Effect
H0: μ1.. = μ2.. = • • • = μr..
MSA
F
MSE
H1: Not all μi.. are equal
H0: μ.1. = μ.2. = • • • = μ.c.
F Test for Factor B Effect
MSB
F
MSE
H1: Not all μ.j. are equal
H0: the interaction of A and B is
equal to zero
Reject H0 if
F > FU
Reject H0 if
F > FU
F Test for Interaction Effect
H1: interaction of A and B isn’t zero
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
MSAB
F
MSE
Reject H0 if
F > FU
Chap 11-44
Two-Way ANOVA:
Summary Table
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Factor A
r–1
SSA
Factor B
c–1
SSB
AB
(r – 1)(c – 1)
(Interaction)
SSAB
Error
rc(n’ – 1)
SSE
Total
n–1
SST
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Mean
Squares
F
Statistic
MSA
MSA/
MSE
= SSA /(r – 1)
MSB
= SSB /(c – 1)
MSAB
= SSAB / (r – 1)(c – 1)
MSB/
MSE
MSAB/
MSE
MSE
= SSE/rc(n’ – 1)
Chap 11-45
Two-Way ANOVA:
Features
 Degrees of freedom always add up
 n-1 = rc(n/-1) + (r-1) + (c-1) + (r-1)(c-1)
 Total = error + factor A + factor B + interaction
 The denominator of the F Test is always the same
but the numerator is different
 The sums of squares always add up
 SST = SSE + SSA + SSB + SSAB
 Total = error + factor A + factor B + interaction
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-46
Two-Way ANOVA:
Interaction
 No Significant Interaction:
 Interaction is present:
Factor B Level 3
Factor B Level 2
Mean Response
Mean Response
Factor B Level 1
Factor A Levels
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Factor B Level 1
Factor B Level 2
Factor B Level 3
Factor A Levels
Chap 11-47
Chapter Summary
In this chapter, we have

Described one-way analysis of variance
 The logic of ANOVA
 ANOVA assumptions
 F test for difference in c means
 The Tukey-Kramer procedure for multiple comparisons

Described two-way analysis of variance
 Examined effects of multiple factors
 Examined interaction between factors
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 11-48