One-Way ANOVA

2. 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

3. 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

4. Results X = 3.00 X = 2.00 X = 1.00

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

6. Hypothesis • Null hypothesis (H0) psych = socio =bio

7. 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!

8. Results X = 3.00 X = 2.00 X = 1.00

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

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

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

12. Results X = 3.00 X = 2.00 X = 1.00

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

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

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

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

17. 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

18. 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

20. Calculating this Variance Ratio

21. Calculating this Variance Ratio

22. Calculating this Variance Ratio

23. Degrees of Freedom • dfbetween • dfwithin • dftotal • dftotal = dfbetween + dfwithin

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

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

26. Sum of Squares • SSBetween • SSWithin • SStotal • SStotal = SSBetween + SSWithin

27. Sum of Squares • SStotal

28. Sum of Squares • SSWithin

29. Sum of Squares • SSBetween

30. Sum of Squares • Ingredients: • X • X2 • Tj2 • N • n

31. To Calculate the SS

32. X Xs = 21 Xp = 14 XB = 7

33. X X = 42 Xs = 21 Xp = 14 XB = 7

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

35. X2 X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11

36. T2 = (X)2 for each group X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11 T2P = 196 T2B = 49 T2s = 441

37. Tj2 X = 42 X2 = 116 Tj2= 686 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11 T2P = 196 T2B = 49 T2s = 441

38. N X = 42 X2 = 116 Tj2= 686 N = 21 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11 T2P = 196 T2B = 49 T2s = 441

39. n X = 42 X2 = 116 Tj2= 686 N = 21 n = 7 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11 T2P = 196 T2B = 49 T2s = 441

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

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

42. X = 42 X2 = 116 Tj2= 686 N = 21 n = 7 Calculate SS • SStotal 42 32 116 21

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

44. X = 42 X2 = 116 Tj2= 686 N = 21 n = 7 Calculate SS • SSWithin 686 18 116 7

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

46. X = 42 X2 = 116 Tj2= 686 N = 21 n = 7 Calculate SS • SSBetween 14 686 42 7 21

47. Sum of Squares • SSBetween • SSWithin • SStotal • SStotal = SSBetween + SSWithin

48. Sum of Squares • SSBetween = 14 • SSWithin = 18 • SStotal = 32 • 32 = 14 + 18

49. Calculating the F value

50. Calculating the F value