University of North Texas Dr. J. Kyle Roberts © 2004

Unit 6: Analysis of Variance (ANOVA)

Lesson 1: Comparing 2 or More Sample Means

EDER 6010: Statistics for Educational Research

Dr. J. Kyle Roberts

University of North Texas

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University of North Texas Dr. J. Kyle Roberts © 2004

ANOVATwo or more different groups measured on the same

construct, typically on the same occasion

Males ↔ Females

Hispanic ↔ Asian ↔ Black ↔ White ↔ Others

School1 ↔ School2 ↔ School3 ↔ School4

“Ways” and “Levels”

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GenderMalesFemales

University of North Texas Dr. J. Kyle Roberts © 2004

Null Hypothesis

43210 : SchoolSchoolSchoolSchool XXXXH

For Ethnicity, the null hypothesis states that there is no difference between the means of the ??? ethnicities on

the dependent variable ???.

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University of North Texas Dr. J. Kyle Roberts © 2004

ANOVA Summary Table

B

B

df

SS

W

B

MS

MS

T

B

SS

SS

W

W

df

SS

n-1

dfT-dfBSST – SSB

K-1SST - SSW

Total

Within

Between

eta2SigFMSdfSSSource

2XX i

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University of North Texas Dr. J. Kyle Roberts © 2004

Source SS df MS F Sig eta2

Between

Within

Total

Practice ANOVA’s

20

80

100

4

10

14

5

8

.625 .20

Page 639 in 4th ed, page 640 in 5th ed

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University of North Texas Dr. J. Kyle Roberts © 2004

Determining F-critical

df Numerator = df Betweendf Denominator = df Within

df Between = 4df Within = 10

F-crit = 3.48at the p = .05 level

If F-calc > F-crit, reject the null hypothesis

Since F-calc was 0.625 in our study, we fail to reject the H0

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University of North Texas Dr. J. Kyle Roberts © 2004

Source SS df MS F Sig eta2

Between

Within

Total

Practice ANOVA’s

300 43

30

5.25

.30

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University of North Texas Dr. J. Kyle Roberts © 2004

Practice ANOVA’s

90

210

300

3

40

5.71

Total

Within

Between

eta2SigFMSdfSSSource

43

30

5.25

.30

F-crit = 2.84

Y

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University of North Texas Dr. J. Kyle Roberts © 2004

Source SS df MS F Sig eta2

Between

Within

Total

Practice ANOVA’s

6.25

125

10

18

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University of North Texas Dr. J. Kyle Roberts © 2004

Source SS df MS F Sig eta2

Between

Within

Total

Practice ANOVA’s

6.25

118.75

125

8

10

18

.78

11.88

.066 .05

F-crit = 3.07

N

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University of North Texas Dr. J. Kyle Roberts © 2004

Assumptions in ANOVA

1. Balanced Design

2. Homogeneity of Variance

- At least 2 people in each “cell”

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XH ...: 3210

University of North Texas Dr. J. Kyle Roberts © 2004

Running an ANOVA in SPSS

Asian Black Hispanic White

9 7 6 8

7 9 5 8

8 3 2 7

7 5 1 7

7 8 3 5

2 Variables:Grouping VariableScore Variable

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University of North Texas Dr. J. Kyle Roberts © 2004

Running an ANOVA in SPSSAnalyzeCompare MeansOne-Way ANOVA

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University of North Texas Dr. J. Kyle Roberts © 2004

SPSS Output for One-Way ANOVADescriptives

SCORE

5 7.6000 .89443 .40000 6.4894 8.7106 7.00 9.00

5 6.4000 2.40832 1.07703 3.4097 9.3903 3.00 9.00

5 3.4000 2.07364 .92736 .8252 5.9748 1.00 6.00

5 7.0000 1.22474 .54772 5.4793 8.5207 5.00 8.00

20 6.1000 2.31471 .51759 5.0167 7.1833 1.00 9.00

Asian

Black

Hispanic

White

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

Test of Homogeneity of Variances

SCORE

2.636 3 16 .085

LeveneStatistic df1 df2 Sig.

ANOVA

SCORE

52.200 3 17.400 5.613 .008

49.600 16 3.100

101.800 19

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

We do not want to reject this H0

We reject the null hypothesis that the mean of the Asian students = the mean of the Black students = the mean of the Hispanic students = the mean of the White students

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University of North Texas Dr. J. Kyle Roberts © 2004

Unit 6: Analysis of Variance (ANOVA)

Lesson 1: Comparing 2 or More Sample Means

EDER 6010: Statistics for Educational Research

Dr. J. Kyle Roberts

University of North Texas