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 = 3.00X = 2.00X = 1.00

Hypothesis Alternative hypothesis (H 1 ) H 1: The three population means are not all equal

Hypothesis Null hypothesis (H 0 )  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 = 3.00X = 2.00X = 1.00

Between Variability X = 3.00X = 2.00X = 1.00 S 2 =.66

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 = 3.00X = 2.00X = 1.00

Within Variability X = 3.00X = 2.00X = 1.00 S 2 =.57S 2 =1.43S 2 =.57

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

Degrees of Freedom df between df within df total df total = df between + df within

Degrees of Freedom df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations) df total = N - 1 df total = df between + df within

Degrees of Freedom df between = k = 2 df within = N - k = 18 df total = N = =

Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within

Sum of Squares SS total

Sum of Squares SS Within

Sum of Squares SS Between

Sum of Squares Ingredients:  X  X 2  T j 2 N n

To Calculate the SS

XX  X s = 21  X p = 14  X B = 7

XX  X s = 21  X p = 14  X B = 7  X = 42

X2X2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7  X = 42

X2X2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7  X = 42  X 2 = 116

T 2 = (  X) 2 for each group  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116

Tj2Tj2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686

N  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686 N = 21

n  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

Ingredients  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

Calculate SS  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 SS total

Calculate SS  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 SS total

Calculate SS SS Within  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

Calculate SS SS Within  X = 42  X 2 = 116  T j 2 = 686 N = 21 n =

Calculate SS SS Between  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

Calculate SS SS Between  X = 42  X 2 = 116  T j 2 = 686 N = 21 n =

Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within

Sum of Squares SS Between = 14 SS Within = 18 SS total = =

Calculating the F value

14 2 7

Calculating the F value 7

7 18 1

Calculating the F value 7 1 7

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 Df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations)

Degrees of Freedom Df between = k = 2 df within = N - k = 18 Page 390 Table F Df between are in the numerator Df within 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 H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

Current Example F value = 7.00 F critical = 3.55 Thus, reject H 0, and accept H 1

Alternative hypothesis (H 1 ) H 1: 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