One-Way ANOVA ANOVA = Analysis of Variance This is a technique used to analyze the results of an experiment when you have more than two groups

Example You measure the number of days 7 psychology majors, 7 sociology majors, and 7 biology majors are absent from class You wonder if the average number of days each of these three groups was absent is significantly different from one another

Results X = 3.00 X = 2.00 X = 1.00

Hypothesis Alternative hypothesis (H1) H1: The three population means are not all equal

Hypothesis Null hypothesis (H0) psych = socio = bio

Between and Within Group Variability Two types of variability Between the differences between the mean scores of the three groups The more different these means are, the more variability!

Results X = 3.00 X = 2.00 X = 1.00

Between Variability S2 = .66 X = 3.00 X = 2.00 X = 1.00

Between Group Variability What causes this variability to increase? 1) Effect of the variable (college major) 2) Sampling error

Between and Within Group Variability Two types of variability Within the variability of the scores within each group

Results X = 3.00 X = 2.00 X = 1.00

Within Variability S2 =.57 S2 =1.43 S2 =.57 X = 3.00 X = 2.00 X = 1.00

Within Group Variability What causes this variability to increase? 1) Sampling error

Between and Within Group Variability Between-group variability Within-group variability

Between and Within Group Variability sampling error + effect of variable sampling error

Between and Within Group Variability sampling error + effect of variable sampling error Thus, if null hypothesis was true this would result in a value of 1.00

Between and Within Group Variability sampling error + effect of variable sampling error Thus, if null hypothesis was not true this value would be greater than 1.00

Calculating this Variance Ratio

Calculating this Variance Ratio

Calculating this Variance Ratio

Degrees of Freedom dfbetween dfwithin dftotal dftotal = dfbetween + dfwithin

Degrees of Freedom dfbetween = k - 1 (k = number of groups) dfwithin = N - k (N = total number of observations) dftotal = N - 1 dftotal = dfbetween + dfwithin

Degrees of Freedom dfbetween = k - 1 3 - 1 = 2 dfwithin = N - k 21 - 3 = 18 dftotal = N - 1 21 - 1 = 20 20 = 2 + 18

Sum of Squares SSBetween SSWithin SStotal SStotal = SSBetween + SSWithin

Sum of Squares SStotal

Sum of Squares SSWithin

Sum of Squares SSBetween

Sum of Squares Ingredients: X X2 Tj2 N n

To Calculate the SS

X Xs = 21 Xp = 14 XB = 7

X X = 42 Xs = 21 Xp = 14 XB = 7

X2 X = 42 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11

X2 X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38

T2 = (X)2 for each group X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 T2P = 196 T2B = 49 T2s = 441

Tj2 X = 42 X2 = 116 Tj2 = 686 Xs = 21 Xp = 14 XB = 7 X2s = 67 T2P = 196 T2B = 49 T2s = 441

N X = 42 X2 = 116 Tj2 = 686 N = 21 Xs = 21 Xp = 14 XB = 7 T2P = 196 T2B = 49 T2s = 441

n X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Xs = 21 Xp = 14 XB = 7 T2P = 196 T2B = 49 T2s = 441

X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Ingredients

X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Calculate SS SStotal

Calculate SS 42 32 116 21 SStotal X = 42 X2 = 116 Tj2 = 686 N = 21

X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Calculate SS SSWithin

Calculate SS 686 18 116 7 SSWithin X = 42 X2 = 116 Tj2 = 686 N = 21

X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Calculate SS SSBetween

Calculate SS 14 686 42 7 21 SSBetween X = 42 X2 = 116 Tj2 = 686

Sum of Squares SSBetween SSWithin SStotal SStotal = SSBetween + SSWithin

Sum of Squares SSBetween = 14 SSWithin = 18 SStotal = 32 32 = 14 + 18

Calculating the F value

Calculating the F value

Calculating the F value 14 7 2

Calculating the F value 7

Calculating the F value 7 18 1 18

Calculating the F value 7 7 1

How to write it out

Significance Is an F value of 7.0 significant at the .05 level? To find out you need to know both df

Degrees of Freedom Dfbetween = k - 1 (k = number of groups) dfwithin = N - k (N = total number of observations)

Degrees of Freedom dfbetween = k - 1 3 - 1 = 2 dfwithin = N - k 21 - 3 = 18 Page 394 Table F dfbetween are in the numerator dfwithin are in the denominator Write this in the table

Critical F Value F(2,18) = 3.55 The nice thing about the F distribution is that everything is a one-tailed test

Decision Thus, if F value > than F critical Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Current Example F value = 7.00 F critical = 3.55 Thus, reject H0, and accept H1

Alternative hypothesis (H1) H1: The three population means are not all equal In other words, psychology, sociology, and biology majors do not have equal class attendence Notice: It does not say where this difference is at!!

How to write it out