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Analysis of Variance (ANOVA). (Click icon for audio). Analysis of Variance Sum of Squares. Analysis of Variance Sum of Squares Between. = individual scores, i.e., the i th observation or test unit in the j th group = grand mean - PowerPoint PPT Presentation
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Dr. Michael R. Hyman, NMSU
Analysis of Variance (ANOVA)
(Click icon for audio)
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Analysis of Variance Sum of Squares
betweenwithintotal SS SS SS
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n
jjjn
1
2between )( SS XX
Analysis of Variance Sum of Squares Between
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Analysis of Variance Sum of Squares Between
= individual scores, i.e., the ith observation or test unit in the jth group = grand meannj = number of all observations or test units in a group
jX
X
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n
i
c
jj
1 1
2within )( SS XX ij
Analysis of Variance Sum of Squares Within
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Analysis of Variance Sum of Squares Within
pi = individual scores, i.e., the ith observation or test unit in the jth grouppi = grand meann = number of all observations or test units in a groupc = number of jth groups (or columns)
ijX
X
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n
i
c
j1 1
2total )( SS XX ij
Analysis of Variance Sum of Squares Total
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Analysis of Variance Sum of Squares
pi = individual scores, i.e., the ith observation or test unit in the jth grouppi = grand meann = number of all observations or test units in a groupc = number of jth groups (or columns)
ijX
X
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1
c
SSMS between
between
Analysis of Variance Mean Squares Between
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ccn
SSMS within
within
Analysis of Variance Mean Square Within
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groupswithinVariance
groupsbetweenVarianceF
Analysis of Variance F-Ratio
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within
between
MS
MSF
Analysis of Variance F-Ratio
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ANOVA Summary Table Source of Variation
• Between groups
• Sum of squares
– SS between
• Degrees of freedom
– c-1 where c=number of groups
• Mean squared-MS between
– SS between / c-1
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ANOVA Summary Table Source of Variation
• Within groups
• Sum of squares – SS within
• Degrees of freedom– cn-c where c=number of groups, and
n = number of observations in a group
• Mean squared – MS within– SS within / cn-c
16WITHIN
BETWEEN
MSMS
F
ANOVA Summary Table Source of Variation
• Total
• Sum of Squares – SStotal
• Degrees of Freedom– cn-1 where c = number of groups, and
n = number of observations in a group
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Examples
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Example #1
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Example #2
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Sales in Units (thousands)
Regular Price$.99
1301188784
X1=104.75X=119.58
Reduced Price$.89
145143120131
X2=134.75
Cents-Off CouponRegular Price
1531299699
X1=119.25
Test Market A, B, or CTest Market D, E, or FTest Market G, H, or ITest Market J, K, or L
MeanGrand Mean
Test Market Pricing Experiment
Example #3
