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 = 18
dftotal = N = 20
20 =

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

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

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

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

39. n X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Xs = 21 Xp = 14 XB = 7
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. Calculate SS 42 32 116 21 SStotal X = 42 X2 = 116 Tj2 = 686 N = 21

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

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

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

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

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

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

49. Calculating the F value

50. Calculating the F value

51. Calculating the F value14 7 2

52. Calculating the F value7

53. Calculating the F value7 18 1 18

54. Calculating the F value7 7 1

55. How to write it out

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

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

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

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

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

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

62. 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!!

63. How to write it out