What is a one way anova?

What is a one way anova?

1. One-Way Analysis of Variance

2. We will now venture into the world of One-way Analysis of Variance.

3. How did we get here?

4. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA.

5. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA . We want to determine if there is a statistically significant difference between English comprehension test scores

6. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA . We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor

7. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA . We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German ,

8. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA . We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French ,

9. How did we get here? We will begin with a real-world problem to consider how we got to a One-Way ANOVA . We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French, or Dutch Speaker.

10. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question.

11. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem

12. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem Inferential Descriptive

13. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem Inferential Descriptive We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

14. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem Inferential Descriptive We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

15. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem Inferential Descriptive We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker. Statistical Significance means we are dealing with the probability that something has or has not occurred which means we are dealing with INFERENTIAL STATISTICS

16. How did we get here? First, we have to determine if the problem we are solving poses an inferential or descriptive question. Problem Inferential Descriptive We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

17. How did we get here? Second, we determine if the inferential problem poses a difference , relationship, independence or goodness of fit question.

18. How did we get here? Second, we determine if the inferential problem poses a difference , relationship, independence or goodness of fit question. Problem Inferential Descriptive Difference Relationship Goodness of Fit Independence

19. How did we get here? Second, we determine if the inferential problem poses a difference , relationship, independence or goodness of fit question. Problem Inferential Descriptive Difference Relationship Goodness of Fit Independence We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

20. How did we get here? Second, we determine if the inferential problem poses a difference , relationship, independence or goodness of fit question. Problem Inferential Descriptive Difference Relationship Goodness of Fit Independence We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker. Notice the word DIFFERENCE here.

21. How did we get here? Second, we determine if the inferential problem poses a difference , relationship, independence or goodness of fit question. Problem Inferential Descriptive Difference Relationship Goodness of Fit Independence We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

22. How did we get here? Third, we determine if we will use parametric or nonparametric analytical methods by determining if the distributions are normal , skewed or kurtotic .

23. How did we get here? Third, we determine if we will use parametric or nonparametric analytical methods by determining if the distributions are normal , skewed or kurtotic . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence

24. How did we get here? Third, we determine if we will use parametric or nonparametric analytical methods by determining if the distributions are normal , skewed or kurtotic . Let’s say we find that the distribution is NORMAL Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence

25. How did we get here? Third, we determine if we will use parametric or nonparametric analytical methods by determining if the distributions are normal , skewed or kurtotic . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Let’s say we find that the distribution is NORMAL

26. How did we get here? Third, we determine if we will use parametric or nonparametric analytical methods by determining if the distributions are normal , skewed or kurtotic . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Let’s say we find that the distribution is NORMAL

27. How did we get here? Fourth, we determine if we will use parametric or nonparametric analytical methods by determining if the data are ratio/interval , ordinal , or nominal .

28. How did we get here? Fourth, we determine if we will use parametric or nonparametric analytical methods by determining if the data are ratio/interval , ordinal , or nominal . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal

29. How did we get here? Fourth, we determine if we will use parametric or nonparametric analytical methods by determining if the data are ratio/interval , ordinal , or nominal . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal Let’s say we find that the data are Ratio/Interval

30. How did we get here? Fourth, we determine if we will use parametric or nonparametric analytical methods by determining if the data are ratio/interval , ordinal , or nominal . Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal Let’s say we find that the data are Ratio/Interval

31. How did we get here? Fifth, we determine the number of dependent variables.

32. How did we get here? Fifth, we determine the number of dependent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables

33. How did we get here? Fifth, we determine the number of dependent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

34. How did we get here? Fifth, we determine the number of dependent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

35. How did we get here? Fifth, we determine the number of dependent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

36. How did we get here? Sixth, we determine the number of independent variables.

37. How did we get here? Sixth, we determine the number of independent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables

38. How did we get here? Sixth, we determine the number of independent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables

39. How did we get here? Sixth, we determine the number of independent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

40. How did we get here? Sixth, we determine the number of independent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

41. How did we get here? Sixth, we determine the number of independent variables. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

42. How did we get here? Seventh, we determine the number of levels of the independent variable.

43. How did we get here? Seventh, we determine the number of levels of the independent variable. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels

44. How did we get here? Seventh, we determine the number of levels of the independent variable. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German, French or Dutch Speaker.

45. How did we get here? Seventh, we determine the number of levels of the independent variable. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German , French or Dutch Speaker.

46. How did we get here? Seventh, we determine the number of levels of the independent variable. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German , French or Dutch Speaker.

47. How did we get here? Seventh, we determine the number of levels of the independent variable. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German , French or Dutch Speaker. 3

48. Problem Inferential Descriptive Difference Relationship Distributions Normal Distributions Skewed or Kurtotic Goodness of Fit Independence Data: Ratio/Interval Data: Ordinal Data: Nominal 1 Dependent Variable 2 or more Dependent Variables 1 Independent Variable 2 or more Independent Variables 2 levels 3 or more levels How did we get here? Seventh, we determine the number of levels of the independent variable. These elements point to a one-way ANOVA as an appropriate method to answer this question. We want to determine if there is a statistically significant difference between English comprehension test scores of students who have an English speaking instructor who is a native German , French or Dutch Speaker.

49. Now let’s consider the concepts that support the use of a One-Way Analysis of Variance

50. Once we have made the leap from simple means-differences embedded in t-tests to the logic of sums of squares, a wide range of procedures opens up to us.

51. Once we have made the leap from simple means-differences embedded in t-tests to the logic of sums of squares, a wide range of procedures opens up to us. mean 1 mean 2 Simple Mean Difference

52. Once we have made the leap from simple means-differences embedded in t-tests to the logic of sums of squares, a wide range of procedures opens up to us. mean 2 mean 3 mean 1 Sums of Squares Logic mean 1 mean 2 Simple Mean Difference

53. Once we have made the leap from simple means-differences embedded in t-tests to the logic of sums of squares, a wide range of procedures opens up to us. mean 2 mean 3 mean 1 Sums of Squares Logic mean 1 mean 2 Simple Mean Difference The possibilities are endless !

54. Once we have made the leap from simple means-differences embedded in t-tests to the logic of sums of squares, a wide range of procedures opens up to us. mean 2 mean 3 mean 1 Sums of Squares Logic mean 1 mean 2 Simple Mean Difference The possibilities are endless ! (well, almost … )

55. As reminder, t -tests can compare means only between two levels of one independent variable.

56. As reminder, t -tests can compare means only between two levels of one independent variable . First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player

57. As reminder, t -tests can compare means only between two levels of one independent variable . First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant

58. As reminder, t -tests can compare means only between two levels of one independent variable . First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant Third example : One Independent variable: Age Two levels : 19-45 yrs old and 31-64 yrs old

59. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest

60. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest Fifth example : One Independent variable: People from Texas Two levels : Population of Texans and a sample of Texans who ate raw fish

61. … across one dependent variable.

62. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player

63. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player One Dependent variable : degree of flexibility

64. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player One Dependent variable : degree of flexibility Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant

65. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player One Dependent variable : degree of flexibility Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant One Dependent variable : level of education

66. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player One Dependent variable : degree of flexibility Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant One Dependent variable : level of education Third example : One Independent variable: Age Two levels : 19-45 yrs old and 31-64 yrs old

67. … across one dependent variable. First example : One Independent variable: Athlete Two levels : Soccer player and Basketball player One Dependent variable : degree of flexibility Second example : One Independent variable: Religious Affiliation Two levels : Catholic and Protestant One Dependent variable : level of education Third example : One Independent variable: Age Two levels : 19-45 yrs old and 31-64 yrs old One Dependent variable : level of comfort with technology

68. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest

69. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest One Dependent variable : scores on a basic statistics test

70. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest One Dependent variable : scores on a basic statistics test Fifth example : One Independent variable: People from Texas Two levels : Population of Texans and a sample of Texans who ate raw fish

71. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest One Dependent variable : scores on a basic statistics test Fifth example : One Independent variable: People from Texas Two levels : Population of Texans and a sample of Texans who ate raw fish One Dependent variable : IQ scores

72. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable.

73. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Two levels : Soccer players and Basketball players One Dependent variable: degree of flexibility

74. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Three levels : Soccer players and Basketball players and Football players One Dependent variable: degree of flexibility

75. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Three levels : Soccer players and Basketball players and Football Players One Dependent variable: degree of flexibility Second example : One Independent variable : Religious Affiliation Two levels: Catholic and Protestant One Dependent variable: level of education

76. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Three levels : Soccer players and Basketball players and Football players One Dependent variable: degree of flexibility Second example : One Independent variable : Religious Affiliation Four levels: Catholic and Protestant and Mormon and Muslim One Dependent variable : level of education

77. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Three levels : Soccer players and Basketball players and Football players One Dependent variable: degree of flexibility Second example : One Independent variable : Religious Affiliation Four levels: Catholic and Protestant and Mormon and Muslim One Dependent variable : level of education Third example : One Independent variable: Age Two levels : 19-30 yrs old and 31-64 yrs old One Dependent variable: level of comfort with technology

78. With a one way analysis of variance, on the other hand, we are generally comparing more than two levels of an independent variable. First example : One Independent variable: Athlete Three levels : Soccer players and Basketball players and Football players One Dependent variable: degree of flexibility Second example : One Independent variable : Religious Affiliation Four levels: Catholic and Protestant and Mormon and Muslim One Dependent variable : level of education Third example : One Independent variable: Age Three levels: 19-30 yrs old and 31-64 yrs old and 65-79 yrs old One Dependent variable: level of comfort with technology

79. Fourth example : One Independent variable: Test takers in a stats class Two levels : Pre-test and posttest One Dependent variable: scores on a basic statistics test

80. Fourth example : One Independent variable: Test takers in a stats class Three levels : Pre-test and midterm test and posttest One Dependent variable: scores on a basic statistics test

81. Fourth example : One Independent variable: Test takers in a stats class Three levels : Pre-test and midterm test and posttest One Dependent variable: scores on a basic statistics test Fifth example : One Independent variable: People from Texas Two levels: Population of Texans and a sample of Texans who ate raw fish One Dependent variable: IQ scores

82. Fourth example : One Independent variable: Test takers in a stats class Three levels : Pre-test and midterm test and posttest One Dependent variable: scores on a basic statistics test Fifth example : One Independent variable: People from Texas Three levels: Population of Texans and a sample of Texans who ate raw fish and a sample of Texans who ate cooked fish One Dependent variable: IQ scores

83. While there is a whole family of procedures under the umbrella of Analysis of Variance we will begin with the most basic: One-Way Analysis of Variance .

84. While there is a whole family of procedures under the umbrella of Analysis of Variance we will begin with the most basic: One-Way Analysis of Variance . First, let’s remind ourselves what variance is before we do an “analysis” of it.

85. While there is a whole family of procedures under the umbrella of Analysis of Variance we will begin with the most basic: One-Way Analysis of Variance . First, let’s remind ourselves what variance is before we do an “analysis” of it. Technically, the variance is the average squared distance of the scores around the mean.

86. Variance Conceptually, the variance represents the average squared deviations of scores from the mean .

87. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . So, what does that mean?

88. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . So, what does that mean? Here is a visual depiction of the variance:

89. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . So, what does that mean? Here is a visual depiction of the variance: 1 2 3 4 5 6 7 8 9 1 2 3 4 5

90. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . So, what does that mean? Here is a visual depiction of the variance: 1 2 3 4 5 6 7 8 9 1 2 3 4 5 So, the mean of this distribution is 5

91. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . So, what does that mean? Here is a visual depiction of the variance: 1 2 3 4 5 6 7 8 9 1 2 3 4 5 And this observation is 9 So we subtract this observation “9” from the mean “5” and we get +4 This means that this observation is 4 units from the mean. And this observation is 3 units from the mean. And this observation is 2 units from the mean. And this observation is 1 unit from the mean. And this observation is 1 unit from the mean. And this observation is units from the mean. And this observation is -1 units from the mean. And this observation is 2 units from the mean. And this observation is 1 unit from the mean. And this observation is units from the mean. And this observation is units from the mean. And this observation is units from the mean. And this observation is -1 units from the mean. And this observation is -1 units from the mean. And this observation is -2 units from the mean. And this observation is -2 units from the mean. And this observation is -3 units from the mean. And this observation is -4 units from the mean.

92. Variance Conceptually, the variance represents the average squared deviations of scores from the mean . You might think that the variance is just the average of all of the deviations (the average distance between each score from the mean), right? 1 2 3 4 5 6 7 8 9 1 2 3 4 5 But, because half of the deviations are positive and the other half are negative, if you take the average it will come to zero 1 2 3 4 1 2 1 2 1 3 4 So to avoid this from happening, we square each deviation ( -4 2 = 16 , -3 2 = 9, - 2 2 = 4, -1 2 = 1, 1 2 = 1, 2 2 = 4 , 3 2 = 9, 4 2 = 16). Then we add up all of the squared deviations and divide that number by the number of observations 4 units above the mean squared = 16 2 units above the mean squared = 4 1 unit below the mean squared = 1 4 units above the mean squared = 16 Now we sum all of the squared deviations 16 + 9 + 4 + 4 + 1 + 1 + 1 + 0 + 0 + 0 + 0 + 1+ 1 + 1 + 4 + 4 + 9 + 16 = 116 Remember – these values represent the distance or deviation from the mean, squared. This is Sum of Squared Deviation The average of the sum of squared deviations = = 116 SS /18 observations = 6.4 Variance

93. The “one way” in one way analysis of variance indicates that we have only one dependent variable

94. The “one way” in one way analysis of variance indicates that we have only one dependent variable Dependent Variable: amount of pizza eaten

95. The “one way” in one way analysis of variance indicates that we have only one dependent variable and only one independent variable

96. The “one way” in one way analysis of variance indicates that we have only one dependent variable and only one independent variable Independent Variable: athletes

97. The “one way” in one way analysis of variance indicates that we have only one dependent variable and only one independent variable with at least two levels.

98. The “one way” in one way analysis of variance indicates that we have only one dependent variable and only one independent variable with at least two levels. Level one: football player

99. Level two: basketball player The “one way” in one way analysis of variance indicates that we have only one dependent variable and only one independent variable with at least two levels. Level one: football player

100. Note that an independent sample t -test could be run as well . The results will be the same, but one-way ANOVA can also analyze at least three levels.

101. Note that an independent sample t -test could be run as well . The results will be the same, but one-way ANOVA can also analyze at least three levels. Level one: football player

102. Note that an independent sample t -test could be run as well . The results will be the same, but one-way ANOVA can also analyze at least three levels. Level two: basketball player Level one: football player

103. Note that an independent sample t -test could be run as well . The results will be the same, but one-way ANOVA can also analyze at least three levels. Level two: basketball player Level one: football player Level three: soccer player

104. We begin with a question:

105. We begin with a question: Who eats more slices of pizza in one sitting, football players, basketball players or soccer players ?

106. We begin with a question: Who eats more slices of pizza in one sitting, football players, basketball players or soccer players? How would you convert this question into a null hypothesis ?

107. N ull hypothesis:

108. N ull hypothesis: There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players .

109. N ull hypothesis: There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players . To test whether the null hypothesis will be retained or rejected we compare a value called the F statistic or F ratio with what we call the F critical value .

110. We will compute the F ratio from the following data set:

111. We will compute the F ratio from the following data set: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

112. We will compute the F ratio from the following data set: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

113. We will compute the F ratio from the following data set : And then check to see if it is larger than the F-critical. If it is, then we will reject our null hypothesis: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

114. We will compute the F ratio from the following data set : And then check to see if it is larger than the F-critical. If it is, then we will reject our null hypothesis: There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players . football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

115. We will compute the F ratio from the following data set : And then check to see if it is larger than the F-critical. If it is, then we will reject our null hypothesis: There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players . football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

116. Otherwise, we will retain the null hypothesis:

117. Otherwise, we will retain the null hypothesis: There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players .

118. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ).

119. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ). We would then compare it with the F critical value.

120. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ). We would then compare it with the F critical value. If the F ratio is larger than the F critical value then we would consider it to be a rare occurrence and reject the null hypothesis .

121. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ). We would then compare it with the F critical value. If the F ratio is larger than the F critical value then we would consider it to be a rare occurrence and reject the null hypothesis . So, let’s see if it is

122. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ). We would then compare it with the F critical value. If the F ratio is larger than the F critical value then we would consider it to be a rare occurrence and reject the null hypothesis . So, let’s see if it is by looking at what is called the Table of Probabilities for the F Distribution .

123. So, let’s say after doing our Analysis of Variance calculation we have an F ratio or F value of 22.17 (nice round number, right? ). We would then compare it with the F critical value. If the F ratio is larger than the F critical value then we would consider it to be a rare occurrence and reject the null hypothesis . So, let’s see if it is by looking at what is called the Table of Probabilities for the F Distribution . This table will help us locate the F critical for our data set.

124. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later).

125. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later).

126. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later).

127. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later) . So, with an F critical of 5.14 and an F ratio of 22.17, (we will show you how to calculate this F ratio from the data set below ),

128. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later) . So, with an F critical of 5.14 and an F ratio of 22.17, (we will show you how to calculate this F ratio from the data set below ), football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

129. Let’s say that the F critical value turns out to be 5.14 (we’ll show you how we got that value later) . So, with an F critical of 5.14 and an F ratio of 22.17, (we will show you how to calculate this F ratio from the data set below ), we would have to say that that is a rare occurrence at the .05 alpha level and we would reject the null hypothesis. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

130. Obviously, if the critical F value had been, say, 9.55,

131. Obviously, if the critical F value had been, say, 9.55,

132. Obviously, if the critical F value had been, say, 9.55,

133. Obviously, if the critical F value had been, say, 9.55, we would have to say that that is a not a rare occurrence at the .05 alpha level and we would retain the null hypothesis.

134. Obviously, if the critical F value had been, say, 9.55, we would have to say that that is a not a rare occurrence at the .05 alpha level and we would retain the null hypothesis. So how did we calculate the 22.17 F ratio in the first place ? And how was the F critical determined so we could reject or retain our null- hypothesis?

135. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you )

136. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you ) Step 1 – calculate the sums of squares between groups

137. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you ) Step 1 – calculate the sums of squares between groups Step 2 – calculate the sums of squares within groups

138. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you ) Step 1 – calculate the sums of squares between groups Step 2 – calculate the sums of squares within groups Step 3 – place the sums of squares values in an ANOVA table

139. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you ) Step 1 – calculate the sums of squares between groups Step 2 – calculate the sums of squares within groups Step 3 – place the sums of squares values in an ANOVA table Step 4 - determine the degrees of freedom

140. Here are the steps we follow to determine to reject or retain the null-hypothesis: (Note – there will be new terminology. Don’t be too concerned about it. At a certain point each concept below will be explained to you ) Step 1 – calculate the sums of squares between groups Step 2 – calculate the sums of squares within groups Step 3 – place the sums of squares values in an ANOVA table Step 4 - determine the degrees of freedom Step 5 – divide the between and the within sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups.

141. Step 6 – Divide the between groups means square by the within groups mean square to get the F-ratio

142. Step 6 – Divide the between groups means square by the within groups mean square to get the F-ratio Step 7 – locate the F critical on the F Distribution Table

143. Step 6 – Divide the between groups means square by the within groups mean square to get the F-ratio Step 7 – locate the F critical on the F Distribution Table Step 8 – determine which is bigger

144. Step 6 – Divide the between groups means square by the within groups mean square to get the F-ratio Step 7 – locate the F critical on the F Distribution Table Step 8 – determine which is bigger Step 9 – retain or reject the null hypothesis

145. Step 6 – Divide the between groups means square by the within groups mean square to get the F-ratio Step 7 – locate the F critical on the F Distribution Table Step 8 – determine which is bigger Step 9 – retain or reject the null hypothesis Step 10 – if the null is rejected conduct a posthoc test to see where the differences lie (this will be shown in another presentation)

146. Step 1 - calculate the sums of squares between groups

147. Step 1 - calculate the sums of squares between groups Let’s illustrate this statement with a data set :

148. Step 1 - calculate the sums of squares between groups Let’s illustrate this statement with a data set : football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

149. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

150. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average

151. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5

152. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means

153. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5

154. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 average

155. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 average c c c

156. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means Another way to state Average of Averages is Mean of Means. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 average

157. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means Another way to state Average of Averages is Mean of Means. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 mean of means 10.7

158. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means Another way to state Average of Averages is Mean of Means . Average of Averages or Mean of Means is also called the Grand Mean . football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 mean of means 10.7

159. Step 1 - calculate the sums of squares between groups First we calculate the mean for each group . Then we create a new column of group means Another way to state Average of Averages is Mean of Means . Average of Averages or Mean of Means is also called the Grand Mean . football player pizza eaten 1 17 slices 2 18 slices 3 19 slices average 18 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices average 9 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices average 5 player group means football 18 b-ball 9 soccer 5 Grand Mean 10.7

160. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean .

161. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean . player group means football 18 b-ball 9 soccer 5 Grand Mean 10.7

162. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean . player group means football 18 b-ball 9 soccer 5 Grand Mean 10.7

163. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean . player group means football 18 b-ball 9 soccer 5 Grand Mean 10.7

164. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean .

165. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean . group means 18 9 5 football b-ball soccer

166. Step 1 - calculate the sums of squares between groups Now we will compute the sum of squared deviations like we’ve done before with variance but this time between group means and the grand mean . group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer

167. Step 1 - calculate the sums of squares between groups So, here is the deviation between each of the group means and grand mean. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer

168. Step 1 - calculate the sums of squares between groups So, here is the deviation between each of the group means and grand mean. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = =

169. Step 1 - calculate the sums of squares between groups Now we square the deviations between groups and grand mean. If we didn’t, when we try to sum the deviations they will come to zero. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = =

170. Step 1 - calculate the sums of squares between groups Now we square the deviations between groups and grand mean. If we didn’t, when we try to sum the deviations they will come to zero. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 =

171. Step 1 - calculate the sums of squares between groups Finally, we multiply these squared deviations by the number of students in each group. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 =

172. Step 1 - calculate the sums of squares between groups Finally, we multiply these squared deviations by the number of students in each group. group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = # in each group 3 3 3 x x x

173. Step 1 - calculate the sums of squares between groups Why do we do this?

174. Step 1 - calculate the sums of squares between groups Why do we do this ? We do this so as to provide greater weight to those groups with more students.

175. Step 1 - calculate the sums of squares between groups Why do we do this ? We do this so as to provide greater weight to those groups with more students. As an example, if there were 6 soccer players, then we would do the following :

176. Step 1 - calculate the sums of squares between groups Why do we do this ? We do this so as to provide greater weight to those groups with more students. As an example, if there were 6 soccer players, then we would do the following : group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = # in each group 3 3 6 x x x

177. Step 1 - calculate the sums of squares between groups Why do we do this ? We do this so as to provide greater weight to those groups with more students. As an example, if there were 6 soccer players, then we would do the following : group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 6

178. Step 1 - calculate the sums of squares between groups Let’s go back to our original data set:

179. Step 1 - calculate the sums of squares between groups Let’s go back to our original data set: group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3

180. Step 1 - calculate the sums of squares between groups Let’s go back to our original data set: group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 161.3 8.3 96.3 = = =

181. Step 1 - calculate the sums of squares between groups So, the Sum of the Weighted Squared Deviations between groups and grand mean is: group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 161.3 8.3 96.3 = = =

182. Step 1 - calculate the sums of squares between groups So, the Sum of the Weighted Squared Deviations between groups and grand mean is: group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 161.3 8.3 96.3 266.0 = = = sum

183. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer.

184. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices

185. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 football b-ball soccer

186. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 5.3 football b-ball soccer grand mean

187. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer

188. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer deviation - 1.3 - 0.3 1.7 = = =

189. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer deviation - 1.3 - 0.3 1.7 = = = squared deviation 1.8 0.1 2.8 2 = 2 = 2 =

190. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer deviation - 1.3 - 0.3 1.7 = = = squared deviation 1.8 0.1 2.8 2 = 2 = 2 = x x x # in each group 3 3 3

191. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer deviation - 1.3 - 0.3 1.7 = = = squared deviation 1.8 0.1 2.8 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 5.3 0.3 8.3 = = =

192. Watch what happens to the sum of squared deviations between group and grand mean when the group means are closer. So let’s say: Football players ate 4 slices; B-ball players ate 5 slices; Soccer players ate 7 slices group means 4 5 7 grand mean 5.3 5.3 5.3 – – – football b-ball soccer deviation - 1.3 - 0.3 1.7 = = = squared deviation 1.8 0.1 2.8 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 5.3 0.3 8.3 14.0 = = = sum

193. Notice how when the group means are closer, their sum of squares between groups is much smaller than when they are further apart like our original analysis. (See below)

194. Notice how when the group means are closer, their sum of squares between groups is much smaller than when they are further apart like our original analysis. (See below) group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 161.3 8.3 96.3 266.0 = = = sum

195. The weighted sums of squares between the groups represents how spread apart the levels

196. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players )

197. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable

198. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable ( athletes )

199. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable ( athletes ) are from one another.

200. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable ( athletes ) are from one another . We believe we know the source of the differences among the groups.

201. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable ( athletes ) are from one another . We believe we know the source of the differences among the groups. In this case we believe that the variance in the amount of pizza eaten is a function of whether an athlete plays football, basketball or soccer. This is our alternative hypothesis .

202. The weighted sums of squares between the groups represents how spread apart the levels ( football, basketball, and soccer players ) of the independent variable ( athletes ) are from one another . We believe we know the source of the differences among the groups. In this case we believe that the variance in the amount of pizza eaten is a function of whether an athlete plays football, basketball or soccer. This is our alternative hypothesis .

203. Important note: To simplify the phrase “sum of weighted squares deviations between the group means and grand mean” we simply state: Between Groups Sum of Squares.

204. Important note: To simplify the phrase “sum of weighted squares deviations between the group means and grand mean” we simply state: Between Groups Sum of Squares .

205. Important note: To simplify the phrase “sum of weighted squares deviations between the group means and grand mean” we simply state: Between Groups Sum of Squares . Now on to Step 2 – calculate the sums of squares within groups

206. Step 2 – calculate the sums of squares within groups

207. Step 2 – calculate the sums of squares within groups On the other hand, the sum of squares within the groups represents variance for which we have not accounted. We don’t know the source of the variance within the groups.

208. Step 2 – calculate the sums of squares within groups On the other hand, the sum of squares within the groups represents variance for which we have not accounted. We don’t know the source of the variance within the groups. So, we compute the within groups sum of squares.

209. Step 2 – calculate the sums of squares within groups On the other hand, the sum of squares within the groups represents variance for which we have not accounted. We don’t know the source of the variance within the groups. So, we compute the within groups sum of squares. Let’s think about why do we do this .

210. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group.

211. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this :

212. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : mean = 5 mean = 9 mean = 18

213. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : mean = 5 mean = 9 mean = 18 grand mean = 10.67

214. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : (Hint : if this were the case then there most likely would be a significant difference between the means .) mean = 5 mean = 9 mean = 18 grand mean = 10.67

215. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : Or, they could be clustered around their means like this (very spread out and overlapping) :

216. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : Or, they could be clustered around their means like this (very spread out and overlapping) : mean = 5 mean = 9 mean = 18

217. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : Or, they could be clustered around their means like this (very spread out and overlapping) : mean = 5 mean = 9 mean = 18 grand mean = 10.67

218. Step 2 – calculate the sums of squares within groups Based on the group means alone (2, 5, 9) we still do not know how much variability there is within each group. For all we know, the scores are clustered around their means like this : Or, they could be clustered around their means like this (very spread out and overlapping) : (Hint: if this were the case then there would most likely NOT be a significant difference between the means, because there is too much overlap between the distributions .) mean = 5 mean = 9 mean = 18 grand mean = 10.67

219. Step 2 – calculate the sums of squares within groups By computing the within groups sums of squares we will be able to consider how narrow or wide these sample distributions are.

220. Step 2 – calculate the sums of squares within groups By computing the within groups sums of squares we will be able to consider how narrow or wide these sample distributions are.

221. Step 2 – calculate the sums of squares within groups By computing the within groups sums of squares we will be able to consider how narrow or wide these sample distributions are.

222. Step 2 – calculate the sums of squares within groups As you recall, our between groups sums of squares value is 266 :

223. Step 2 – calculate the sums of squares within groups As you recall, our between groups sums of squares value is 266 : group means 18 9 5 grand mean 10.7 10.7 10.7 – – – football b-ball soccer deviation 7.3 - 1.7 - 5.7 = = = squared deviation 53.8 2.8 32.1 2 = 2 = 2 = x x x # in each group 3 3 3 w eighted sq. dev . 161.3 8.3 96.3 266.0 = = = sum

224. Step 2 – calculate the sums of squares within groups As you recall, our between groups sums of squares value is 266 : There are two ways to compute the within groups sums of squares :

225. Step 2 – calculate the sums of squares within groups As you recall, our between groups sums of squares value is 266 : There are two ways to compute the within groups sums of squares : The short way

226. Step 2 – calculate the sums of squares within groups As you recall, our between groups sums of squares value is 266 : There are two ways to compute the within groups sums of squares : The short way The long way

227. We will begin with the long way.

228. We will begin with the long way. The long way is calculated by computing the within sums of squares for football players plus the sums of squares for basketball players plus the sums of squares with soccer players.

229. We will begin with the long way. The long way is calculated by computing the within sums of squares for football players plus the sums of squares for basketball players plus the sums of squares with soccer players.

230. Now, we compute the sums of squares within football players:

231. Now, we compute the sums of squares within football players: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices

232. Now, we compute the sums of squares within football players : then within basketball players: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices

233. Now, we compute the sums of squares within football players : then within basketball players: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices

234. Now, we compute the sums of squares within football players : then within basketball players: finally, within soccer players: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices

235. Now, we compute the sums of squares within football players : then within basketball players: finally, within soccer players: football player pizza eaten 1 17 slices 2 18 slices 3 19 slices basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

236. We begin with our football players.

237. We begin with our football players. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices

238. We begin with our football players. Compute the mean. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices

239. We begin with our football players. Compute the mean. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

240. We begin with our football players. Compute the mean. football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

241. We begin with our football players. Compute the mean. sample mean 18 18 18 football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

242. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. sample mean 18 18 18 – – – = = = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

243. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. sample mean 18 18 18 – – – deviation - 1 1 = = = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

244. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. Square each deviation score. sample mean 18 18 18 – – – deviation - 1 1 = = = 2 = 2 = 2 = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

245. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. Square each deviation score. sample mean 18 18 18 – – – deviation - 1 1 = = = squared deviation 1 1 2 = 2 = 2 = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

246. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. Square each deviation score . The sum of the squared deviations for the football player group … sample mean 18 18 18 – – – deviation - 1 1 = = = squared deviation 1 1 2 = 2 = 2 = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices

247. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. Square each deviation score . The sum of the squared deviations for the football player group … sample mean 18 18 18 – – – deviation - 1 1 = = = squared deviation 1 1 2 = 2 = 2 = = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices sum

248. We begin with our football players. Compute the mean. Subtract the mean from each player’s slices. Square each deviation score . The sum of the squared deviations for the football player group is 2 . sample mean 18 18 18 – – – deviation - 1 1 = = = squared deviation 1 1 2 2 = 2 = 2 = = football player pizza eaten 1 17 slices 2 18 slices 3 19 slices mean 18 slices sum

249. Now we will do the same for the basketball player group.

250. Now we will do the same for the basketball player group. basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices

251. Now we will do the same for the basketball player group. basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

252. Now we will do the same for the basketball player group. sample mean 9 9 9 basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

253. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – = = = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

254. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – deviation - 1 1 = = = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

255. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – deviation - 1 1 = = = 2 = 2 = 2 = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

256. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – deviation - 1 1 = = = squared deviation 1 1 2 = 2 = 2 = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices

257. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – deviation - 1 1 = = = squared deviation 1 1 2 = 2 = 2 = = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices sum

258. Now we will do the same for the basketball player group. sample mean 9 9 9 – – – deviation - 1 1 = = = squared deviation 1 1 2 2 = 2 = 2 = = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices sum

259. Now we will do the same for the basketball player group. So the sum of squares within the basketball player group is 2 . sample mean 9 9 9 – – – deviation - 1 1 = = = squared deviation 1 1 2 2 = 2 = 2 = = basketball player pizza eaten 1 8 slices 2 9 slices 3 10 slices mean 9 slices sum

260. Now we will do the same for the soccer player group.

261. Now we will do the same for the soccer player group. soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices

262. Now we will do the same for the soccer player group. soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

263. Now we will do the same for the soccer player group. sample mean 5 5 5 soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

264. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – = = = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

265. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – deviation - 4 4 = = = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

266. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – deviation - 4 4 = = = 2 = 2 = 2 = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

267. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – deviation - 4 4 = = = squared deviation 16 16 2 = 2 = 2 = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices

268. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – deviation - 4 4 = = = squared deviation 16 16 2 = 2 = 2 = = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices sum

269. Now we will do the same for the soccer player group. sample mean 5 5 5 – – – deviation - 4 4 = = = squared deviation 16 16 32 2 = 2 = 2 = = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices sum

270. Now we will do the same for the soccer player group. So the sum of squares within the soccer player group is 32 . sample mean 5 5 5 – – – deviation - 4 4 = = = squared deviation 16 16 32 2 = 2 = 2 = = soccer player pizza eaten 1 1 slices 2 5 slices 3 9 slices mean 5 slices sum

271. Let’s summarize the sum of squares within groups:

272. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 .

273. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 .

274. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 .

275. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 . Now lets’ sum up these within groups sum of squares

276. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 . Now lets’ sum up these within groups sum of squares Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 .

277. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 . Now lets’ sum up these within groups sum of squares Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 .

278. Let’s summarize the sum of squares within groups: Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 . Now lets’ sum up these within groups sum of squares Sum of squares within the football player group = 2 . Sum of squares within the basketball player group = 2 . Sum of squares within the soccer player group = 32 . 36 .

279. Another way to calculate the total sums of squares is to put all of the scores in a column:

280. Another way to calculate the total sums of squares is to put all of the scores in a column: players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9

281. Calculate the grand mean: players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9

282. Calculate the grand mean: players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7

283. And subtract players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7

284. And subtract the grand mean from them: players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – –

285. Which equals players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – –

286. Which equals the deviation players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – – deviation 6.3 7.3 8.3 - 2.7 - 1.7 - 0.7 - 9.7 - 5.7 - 1.7 = = = = = = = = =

287. Then square those deviations players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – – deviation 6.3 7.3 8.3 - 2.7 - 1.7 - 0.7 - 9.7 - 5.7 - 1.7 = = = = = = = = =

288. Then square those deviations players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – – deviation 6.3 7.3 8.3 - 2.7 - 1.7 - 0.7 - 9.7 - 5.7 - 1.7 = = = = = = = = = deviation 39.7 53.3 68.9 7.3 2.9 0.5 94.1 32.5 2.9 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 =

289. Then sum up the squared deviations from the mean and you get the total sums of squares players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – – deviation 6.3 7.3 8.3 - 2.7 - 1.7 - 0.7 - 9.7 - 5.7 - 1.7 = = = = = = = = = deviation 39.7 53.3 68.9 7.3 2.9 0.5 94.1 32.5 2.9 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 =

290. Then sum up the squared deviations from the mean and you get the total sums of squares players slices F1 17 F2 18 F3 19 B1 8 B2 9 B3 10 S1 1 S2 5 S3 9 grand mean 10.7 grand mean 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 – – – – – – – – – deviation 6.3 7.3 8.3 - 2.7 - 1.7 - 0.7 - 9.7 - 5.7 - 1.7 = = = = = = = = = deviation 39.7 53.3 68.9 7.3 2.9 0.5 94.1 32.5 2.9 302.0 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 = 2 = sum

291. You have probably figured out the short way to compute the within groups sums of squares:

292. You have probably figured out the short way to compute the within groups sums of squares:   Compute the between group sum of squares (266.0)

293. You have probably figured out the short way to compute the within groups sums of squares:   Compute the between group sum of squares (266.0) Subtract it from the total sums of squares (302.0 – 266.0 ), which equals the within group sums of squares (36.0)

294. Step 3 – place the sums of squares values in an ANOVA table As you recall

295. Step 3 – place the sums of squares values in an ANOVA table To calculate total sums of squares we simply add the between groups and within groups sums of squares.

296. Step 3 – place the sums of squares values in an ANOVA table To calculate total sums of squares we simply add the between groups and within groups sums of squares. Group Sum of Squares Between groups 266.0 Within groups 36.0 total 302.0

297. Step 3 – place the sums of squares values in an ANOVA table To calculate total sums of squares we simply add the between groups and within groups sums of squares. Group Sum of Squares Between groups 266.0 Within groups 36.0 total 302.0

298. Step 4 – determine the degrees of freedom To calcul

299. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups.

300. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom :

301. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom # of groups minus one equals

302. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals # of persons minus # of groups equals

303. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 # of persons minus # of groups equals 3

304. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – # of persons minus # of groups equals 3

305. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 # of persons minus # of groups equals 3

306. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = # of persons minus # of groups equals 3

307. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = 2 # of persons minus # of groups equals 3

308. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = 2 # of persons minus # of groups equals 9 3

309. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = 2 # of persons minus # of groups equals 9 – 3

310. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = 2 # of persons minus # of groups equals 9 – 3 =

311. Step 4 – determine the degrees of freedom We then divide the sum of squares by their degrees of freedom to get the mean square value for both groups . Here is how we calculate the number of degrees of freedom : Sums of Squares BETWEEN Groups Degrees of Freedom Sums of Squares WITHIN Groups Degrees of Freedom # of groups minus one equals 3 – 1 = 2 # of persons minus # of groups equals 9 – 3 = 6

312. Step 4 – determine the degrees of freedom Back to the table:

313. Step 4 – determine the degrees of freedom Back to the table: Group Sum of Squares Deg. of F. Between groups 266.0 Within groups 36.0

314. Step 4 – determine the degrees of freedom Back to the table: Group Sum of Squares Deg. of F. Between groups 266.0 2 Within groups 36.0 6

315. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. Back to the table:

316. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. The mean square is like an average. So dividing the sum of squares by the degrees freedom is like dividing a data set total (1, 2, 3, 4, 5 = 15) by the number of data points (5). (15/5 = 3). But in this case, the data points are the squared deviations. Back to the table:

317. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. For example, you may remember that the weighted squared deviations between groups were: 161.3 8.3 96.3 Back to the table:

318. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. For example, you may remember that the weighted squared deviations between groups were: 161.3 8.3 96.3 To take the average of this you would divide it by three; however, in this case we divide it by the degrees of freedom (why we do this is explained in another presentation) .

319. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. So 161.3 + 8.3 + 96.3 divided by 2 degrees of freedom =

320. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. So 161.3 + 8.3 + 96.3 divided by 2 degrees of freedom = 132.95

321. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. So 161.3 + 8.3 + 96.3 divided by 2 degrees of freedom = 132.95 As you may also remember, the variance is the average of the sums of squared deviations. Well, guess what? The mean square is essentially the variance of the between group.

322. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. Group Sum of Squares Deg. of F. Mean Square Between groups 266.0 2 Within groups 36.0 6

323. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. Group Sum of Squares Deg. of F. Mean Square Between groups 266.0 2 133.0 Within groups 36.0 6

324. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. Group Sum of Squares Deg. of F. Mean Square Between groups 266.0 2 133.0 Within groups 36.0 6 6.0

325. Step 5 – divide the between and the within groups sums of squares by their corresponding degrees of freedom to get the means square values for both the between and within groups. Group Sum of Squares Deg. of F. Mean Square Between groups 266.0 2 133.0 Within groups 36.0 6 6.0 Also known as the variance

326. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio

327. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio Finally, we divide the Between Groups Mean Square (133.0) by the Within Groups Mean Square (6.0) to get the F-ratio.

328. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio Finally, we divide the Between Groups Mean Square (133.0) by the Within Groups Mean Square (6.0) to get the F-ratio. Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 266.0 2 133.0 Within groups 36.0 6 6.0

329. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio Finally, we divide the Between Groups Mean Square (133.0) by the Within Groups Mean Square (6.0) to get the F-ratio. Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 266.0 2 133.0 Within groups 36.0 6 6.0

330. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio Finally, we divide the Between Groups Mean Square (133.0) by the Within Groups Mean Square (6.0) to get the F-ratio. Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 266.0 2 133.0 22.17 Within groups 36.0 6 6.0

331. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio What is the F-ratio?

332. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio What is the F-ratio? As just explained, the ratio of the mean sum of squares between groups to the mean sum of squares within groups generates an F-statistic.

333. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio What is the F-ratio? As just explained, the ratio of the mean sum of squares between groups to the mean sum of squares within groups generates an F-statistic. It is this F-statistic that we will use to test our null hypothesis :

334. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio What is the F-ratio? As just explained, the ratio of the mean sum of squares between groups to the mean sum of squares within groups generates an F-statistic. It is this F-statistic that we will use to test our null hypothesis : There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players . If

335. Step 6 – divide the between groups means square by the within groups mean square to get the F-ratio What is the F-ratio? As just explained, the ratio of the mean sum of squares between groups to the mean sum of squares within groups generates an F-statistic. It is this F-statistic that we will use to test our null hypothesis : There is no statistically significant difference in the quantity of pizza consumed in one sitting among football, basketball, and soccer players . If the F critical value is greater than our F-statistic of 22.17

336. Step 7 – locate the F critical on the F Distribution Table value

337. Step 7 – locate the F critical on the F Distribution Table This is done by using the number of degrees of freedom in the denominator (within groups = 6) and the degrees of freedom in the numerator (between groups =2) and determining where these two intersect .

338. Step 7 – locate the F critical on the F Distribution Table This

339. Step 7 – locate the F critical on the F Distribution Table This

340. Step 7 – locate the F critical on the F Distribution Table This

341. Step 7 – locate the F critical on the F Distribution Table This

342. Step 7 – locate the F critical on the F Distribution Table This And we find the F critical value, which is 5.14

343. Step 8 – determine which is bigger This

344. Step 8 – determine which is bigger It just so happens that our F-ratio is 22.17, which means it is bigger than the F critical value.

345. Step 9 – retain or reject the null hypothesis It

346. Step 9 – retain or reject the null hypothesis We will therefore reject the null hypothesis at the .05 alpha level, or in other words, with a probability of Type-I error less than .05.

347. Step 9 – retain or reject the null hypothesis We will therefore reject the null hypothesis at the .05 alpha level, or in other words, with a probability of Type-I error less than .05 . If the F-ratio had been 4.38, then we would fail to reject or in other words, retain the null hypothesis .

348. Do you see what makes a small or large F-ratio value?

349. Do you see what makes a small or large F-ratio value ? Here is what makes the difference:

350. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together,

351. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, mean = 5 mean = 9 mean = 18

352. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller mean = 5 mean = 9 mean = 18

353. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 12.0 2 6.0 Within groups ?

354. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups ),

355. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups ), mean = 5 mean = 9 mean = 18

356. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups), then the within groups mean square will be smaller mean = 5 mean = 9 mean = 18

357. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups), then the within groups mean square will be smaller, Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 12.0 2 6.0 Within groups 120.0 6 20.0

358. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups), then the within groups mean square will be smaller, then the F ratio will be extremely smaller than what we found with the pizza eating athlete groups : Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 12.0 2 6.0 Within groups 120.0 6 20.0

359. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups), then the within groups mean square will be smaller, then the F ratio will be extremely smaller than what we found with the pizza eating athlete groups : Group Sum of Squares Deg. of F. Mean Square F-ratio Between groups 12.0 2 6.0 0.3 Within groups 120.0 6 20.0

360. Do you see what makes a small or large F-ratio value ? Here is what makes the difference : When the groups or their means are closer together, then the between groups mean square will be smaller . And to compound the situation, if the within groups are large (meaning there is a lot of difference or variability within groups), then the within groups mean square will be smaller, then the F ratio will be extremely smaller than what we found with the pizza eating athlete groups : With an F critical still at 5.14 and an F ratio of 0.3, we would retain the null hypothesis .

361. Note: if there are really no differences among three groups in terms of some dependent variable, the mean sum of squares between groups will be very similar to the mean sum of squares within groups.

362. Note: if there are really no differences among three groups in terms of some dependent variable, the mean sum of squares between groups will be very similar to the mean sum of squares within groups. The ratio of such mean squares will be close to 1. As the differences among the groups increases, the ratio of mean sums of squares will increase above 1.

363. Note: if there are really no differences among three groups in terms of some dependent variable, the mean sum of squares between groups will be very similar to the mean sum of squares within groups. The ratio of such mean squares will be close to 1. As the differences among the groups increases, the ratio of mean sums of squares will increase above 1. At some point, the ratio of mean sums of squares (F) will be large enough that we will conclude that there are probably systematic differences among the groups.

364. Note: if there are really no differences among three groups in terms of some dependent variable, the mean sum of squares between groups will be very similar to the mean sum of squares within groups. The ratio of such mean squares will be close to 1. As the differences among the groups increases, the ratio of mean sums of squares will increase above 1. At some point, the ratio of mean sums of squares (F) will be large enough that we will conclude that there are probably systematic differences among the groups. We will reject the null hypothesis of no differences and investigate where the differences occur.

365. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). We will

366. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). Is the difference between football and basketball players or is it between football and soccer players or basketball and soccer players or all three? The post hoc will provide this information. This will be shown in another presentation.

367. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

368. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

369. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

370. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

371. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

372. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

373. Step 10 – if the null is rejected, conduct a post hoc test to see where the differences lie (this will be shown in another presentation ). In order to discover what pattern of differences generated the significant F- statistic, we conduct post hoc tests, which use different logics and calculations, some of which are more conservative than others in how they protect against cumulative Type-I error across multiple tests. In essence, each post hoc test is comparing each group to every other group in a series of two-group tests. The results of the series of two-group tests identifies which of the many possible patterns generated the significant F-statistic and where the differences lie.

374. In summary, One way Analysis of Variance is a method that will help you determine if there is a statistically significant difference between more than two group means. It does not tell you which groups differ, just that they do. Further testing will determine which groups differ.

375. End of Presentation