# 11 ANOVA

MBA 6640
Quantitative Analysis for
Managers
Chapter 11
Analysis of Variance

Troy Program – MBA 6640

1

Chapter Goals
After completing this chapter, you should be able to:

Recognize situations in which to use analysis of variance

Understand different analysis of variance designs

Perform a one-way and two-way analysis of variance and
interpret the results

Conduct and interpret a Kruskal-Wallis test

Analyze two-factor analysis of variance tests with more than
one observation per cell

One-Way Analysis of Variance

Evaluate the difference among the means of three
or more groups
Examples: Average production 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

Hypotheses of One-Way ANOVA

H0 : μ1 = μ2 = μ3 =  = μK

All population means are equal

i.e., no variation in means between groups

H1 : μi ≠ μ j

for at least one i, j pair

At least one population mean is different

i.e., there is variation between groups

Does not mean that all population means are different
(some pairs may be the same)

One-Way ANOVA
H0 : μ1 = μ2 = μ3 =  = μK
H1 : Not all μi are the same
All Means are the same:
The Null Hypothesis is True
(No variation between
groups)

μ1 = μ2 = μ3

One-Way ANOVA
(continued)

H0 : μ1 = μ2 = μ3 =  = μK
H1 : Not all μi are the same
At least one mean is different:
The Null Hypothesis is NOT true
(Variation is present between groups)

or

μ1 = μ2 ≠ μ3

μ1 ≠ μ2 ≠ μ3

Variability

The variability of the data is key factor to test the
equality of means

In each case below, the means may look different, but a
large variation within groups in B makes the evidence
that the means are different weak

A

B

A

B
Group

C

Small variation within groups

A

B
Group

C

Large variation within groups

Partitioning the Variation

Total variation can be split into two parts:

SST = SSW + SSG
SST = Total Sum of Squares
Total Variation = the aggregate dispersion of the individual
data values across the various groups

SSW = Sum of Squares Within Groups
Within-Group Variation = dispersion that exists among the
data values within a particular group

SSG = Sum of Squares Between Groups
Between-Group Variation = dispersion between the group
sample means

Partition of Total Variation
Total Sum of Squares
(SST)

=

Variation due to
random sampling
(SSW)

+

Variation due to
differences
between groups
(SSG)

Total Sum of Squares
SST = SSW + SSG
K

ni

SST = ∑∑ (x ij − x)
Where:

2

i=1 j=1

SST = Total sum of squares
K = number of groups (levels or treatments)
ni = number of observations in group i
xij = jth observation from group i
x = overall sample mean

Total Variation
(continued)

SST = (x11 − x )2 + (X12 − x )2 + ... + (x KnK − x )2
Response, X

x

Group 1

Group 2

Group 3

Within-Group Variation
SST = SSW + SSG
K

ni

SSW = ∑∑ (x ij − x i )2
i =1 j=1

Where:

SSW = Sum of squares within groups
K = number of groups
ni = sample size from group i
Xi = sample mean from group i
Xij = jth observation in group i

Within-Group Variation
(continued)
K

ni

SSW = ∑∑ (x ij − x i )
i =1 j=1

Summing the variation
within each group and then

2

SSW
MSW =
n −K
Mean Square Within =
SSW/degrees of freedom

μi

Within-Group Variation
(continued)

SSW = (x11 − x1 )2 + (x12 − x1 )2 + ... + (x KnK − x K )2
Response, X

x1
Group 1

Group 2

x2
Group 3

x3

Between-Group Variation
SST = SSW + SSG
K

SSG = ∑ ni ( x i − x )
Where:

2

i=1

SSG = Sum of squares between groups
K = number of groups
ni = sample size from group i
xi = sample mean from group i
x = grand mean (mean of all data values)

Between-Group Variation
(continued)
K

SSG = ∑ ni ( x i − x )

2

i=1

Variation Due to
Differences
Between
Groups

SSG
MSG =
K −1
Mean Square Between Groups
= SSG/degrees of freedom

μi

μj

Between-Group Variation
(continued)

SSG = n1(x1 − x) + n2 (x 2 − x) + ... + nK (x K − x)
2

2

Response, X

x1
Group 1

Group 2

x2
Group 3

x3

x

2

Obtaining the Mean Squares
SST
MST =
n −1
SSW
MSW =
n −K
SSG
MSG =
K −1

One-Way ANOVA Table
Source of
Variation

SS

df

Between
Groups

SSG

K-1

Within
Groups

SSW

n-K

SST =
SSG+SSW

n-1

Total

MS
(Variance)

F ratio

SSG
MSG
MSG =
K - 1 F = MSW
SSW
MSW =
n-K

K = number of groups
n = sum of the sample sizes from all groups
df = degrees of freedom

One-Factor ANOVA
F Test Statistic
H0: μ1= μ2 = … = μK
H1: At least two population means are different

Test statistic

MSG
F=
MSW

MSG is mean squares between variances
MSW is mean squares within variances

Degrees of freedom

df1 = K – 1

(K = number of groups)

df2 = n – K

(n = sum of sample sizes from all groups)

Interpreting the F Statistic

The F statistic is the ratio of the between estimate of variance and
the within estimate of variance

The ratio must always be positive
df1 = K -1 will typically be small
df2 = n - K will typically be large

Decision Rule:
 Reject H if
0
F > FK-1,n-K,α

α = .05

0

Do not
reject H0

Reject H0

FK-1,n-K,α

One-Factor ANOVA
F Test 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?

Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

One-Factor ANOVA Example:
Scatter Diagram
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

210

x1
••

••

x2

••
••

200
190
1

2
Club

x

3

x3

One-Factor ANOVA Example
Computations
Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

x1 = 249.2

n1 = 5

x2 = 226.0

n2 = 5

x3 = 205.8

n3 = 5

x = 227.0

n = 15

K=3
SSG = 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

MSG = 4716.4 / (3-1) = 2358.2
MSW = 1119.6 / (15-3) = 93.3

2358.2
F=
= 25.275
93.3

One-Factor ANOVA Example
Solution
Test Statistic:

H0: μ1 = μ2 = μ3
H1: μi not all equal
α = .05
df1= 2

df2 = 12
Critical Value:

F2,12,.05= 3.89
α = .05

0

Do not
reject H0

Reject H0

F2,12,.05 = 3.89

MSA 2358.2
F=
=
= 25.275
MSW
93.3

Decision:
Reject H0 at α = 0.05
Conclusion:
There is evidence that
at least one μi differs
F = 25.275
from the rest

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