1.2 Correlation: Meaning and Interpretation 1.2.1 Scatter Diagram: Graphical Presentation of Relationship 1.2.2 Correlation: Linear and Non-Linear Relationship 1.2.3 Direction of Correlation: Positive and Negative 1.2.4 Correlation: The Strength of Relationship 1.2.5 Measurements of Correlation 1.2.6 Correlation and Causality 1.3 Pearson’s Product Moment Coefficient of Correlation 1.3.1 Variance and Covariance: Building Blocks of Correlations 1.3.2 Equations for Pearson’s Product Moment Coefficient of Correlation 1.3.3 Numerical Example 1.3.4 Significance Testing of Pearson’s Correlation Coefficient 1.3.5 Adjusted r 1.3.6 Assumptions for Significance Testing 1.3.7 Ramifications in the Interpretation of Pearson’s r 1.3.8 Restricted Range 1.4 Unreliability of Measurement 1.4.1 Outliers 1.4.2 Curvilinearity 1.5 Using Raw Score Method for Calculating r 1.5.1 Formulas for Raw Score 1.5.2 Solved Numerical for Raw Score Formula

16. 𝑦 = 𝑚𝑥 + 𝑏
𝑦 𝑖𝑠 𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑥 𝑖𝑠 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑚 𝑖𝑠 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑙𝑖𝑛𝑒
𝑏 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

35. 𝑟 = +0.37

41. 𝝆𝒓

43. 𝝆

44. 𝝆

45. 𝝆

55. 𝑋 𝑋
𝑖=1
𝑛
𝑋
𝑛

56. 𝑋 =
𝑖=1
𝑛
𝑋
𝑛

57. 𝑋
𝑆 𝑥
2
𝑋
𝑋 ( (𝑋 − 𝑋)2)
(𝑛)

58. (𝑋 − 𝑋)2

59. 𝑆 𝑥
2
=
(𝑋 − 𝑋)2
𝑛
… for 𝑥 variable
𝑆 𝑦
2
=
(𝑌 − 𝑌)2
𝑛
… for 𝑦 variable

60. 𝑋 𝑎𝑛𝑑 𝑌 𝐶𝑜𝑣 𝑋𝑌 𝑜𝑟 𝑆 𝑋𝑌
𝐶𝑜𝑣 𝑋𝑌 =
(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛

61. (𝑋 − 𝑋)(𝑌 − 𝑌)

63. 𝑟 =
𝐶𝑜𝑣 𝑋𝑌
𝑆 𝑋. 𝑆 𝑌
𝑊ℎ𝑒𝑟𝑒,
𝐶𝑜𝑣 𝑋𝑌 = 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑋 𝑎𝑛𝑑 𝑌
𝑃𝑒𝑎𝑟𝑠𝑜𝑛’𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 …
𝑆 𝑋 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑋
𝑆 𝑌 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑌

64. 𝑟 =
𝐶𝑜𝑣 𝑋𝑌
𝑆 𝑋. 𝑆 𝑌
𝒓
𝑪𝒐𝒗 𝑿𝒀
𝑪𝒐𝒗 𝑿𝒀
𝒓
𝑪𝒐𝒗 𝑿𝒀 𝑺 𝑿. 𝑺 𝒀
𝒊𝒔 𝒂𝒍𝒘𝒂𝒚𝒔
𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆

65. 𝑟 =
(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛
𝑆 𝑋. 𝑆 𝑌
𝐵𝑦 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑤𝑒 𝑐𝑎𝑛 𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛

66. 𝑟 =
(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛. 𝑆 𝑋. 𝑆 𝑌

68. 𝒙 𝒚
n = 10
𝒙
=110
𝒚
=120
𝑥 =
𝑥
𝑛
=
110
10
= 11
𝑦 =
𝑦
𝑛
=
120
10
= 12

69. 𝒙 𝒚 𝒙 − 𝒙 𝒚 − 𝒚
n = 10
𝒙
=110
𝒚
=120

70. 𝒙 𝒚 𝒙 − 𝒙 𝒚 − 𝒚 𝒙 − 𝒙) 𝟐 (𝒚 − 𝒚)2
n = 10
𝒙
=110
𝒚
=120
𝒙 − 𝒙) 𝟐
=156
𝒚 − 𝒚) 𝟐
=100

71. 𝑆 𝑥 =
(𝑥 − 𝑥)2
𝑛
=
156
10
= 4.16
𝑆 𝑦 =
(𝑦 − 𝑦)2
𝑛
=
100
10
= 3.33

72. 𝒙 𝒚 𝒙 − 𝒙 𝒚 − 𝒚 𝒙 − 𝒙) 𝟐 (𝒚 − 𝒚)2 (𝒙 − 𝒙).(𝒚 − 𝒚)
n = 10
𝒙
=110
𝒚
=120
𝒙 − 𝒙) 𝟐
=156
𝒚 − 𝒚) 𝟐
=100
(𝒙 − 𝒙).(𝒚 − 𝒚)
=117

73. 𝑟 =
(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛. 𝑆 𝑋. 𝑆 𝑌
𝑟 =
117
(10)(4.16)(3.33)
𝑟 = +0.937

77. 𝝆

78. (df)
n – 2
n

80. 𝝆
𝝆

81. 𝝆
𝝆

82. 𝝆
𝝆

83. ≤

84. >

87. >
>

90. 𝑟𝑎𝑑𝑗 = 1 −
(1 − 𝑟2)(𝑛 − 1)
𝑛 − 2

91. 𝑟𝑎𝑑𝑗 = 1 −
(1 − 0.9372)(10 − 1)
10 − 2
𝑟𝑎𝑑𝑗 = 1 −
(0.1220)(9)
8
𝑟𝑎𝑑𝑗 = 1 − 0.1373
𝑟𝑎𝑑𝑗 = 0.929

108. 𝑟 =
(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛. 𝑆 𝑋. 𝑆 𝑌
𝑛. 𝑆 𝑋. 𝑆 𝑌 = (𝑥 − 𝑥)2(𝑦 − 𝑦)2
𝑊ℎ𝑒𝑟𝑒,

109. (𝑥 − 𝑥)2 = 𝑥2
−
𝑥 2
𝑛
(𝑦 − 𝑦)2 = 𝑦2
−
𝑦 2
𝑛
(𝑥 − 𝑥)(𝑦 − 𝑦) = 𝑥𝑦 −
𝑥 𝑦
𝑛

110. 𝑟 =
𝑥𝑦 −
𝑥 𝑦
𝑛
𝑥2 −
𝑥 2
𝑛
. 𝑦2 −
𝑦 2
𝑛

111. 𝒙 𝒚 𝒙2 𝒚2 𝒙𝒚
n = 10 𝒙 =110 𝒚 =120 𝒙2 =1366 𝒚2=1540 𝒙y =1437
𝒙 = 𝟏𝟏 𝒚 = 𝟏𝟐

112. 𝑟 =
𝑥𝑦 −
𝑥 𝑦
𝑛
𝑥2 −
𝑥 2
𝑛
. 𝑦2 −
𝑦 2
𝑛
𝑟 =
1437 −
11 𝑥 12
10
(1366 −
1102
10
)(1540 −
1202
10
)
𝑟 =
117
156 𝑥 100
= +0.937