Dr. Michael R. Hyman, NMSU

Analysis of Variance (ANOVA)

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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