2. Practice
Page 65
2.1

4. Positive Skew

5. Note
Slides online

9. Histogram

10. Frequency Polygon

11. Stem-and-Leaf Display
Stem-and-leaf display with a bigger data set
Note: The stem-and-leaf is like a histogram turned sideways!

12. Describing Distributions
Bell-shaped distribution

13. Describing Distributions

14. Describing Distributions

15. Kurtosis
The relative concentration of scores in the center of the distribution
Mesokurtic

16. Kurtosis
The relative concentration of scores in the center of the distribution
Platykurtic

17. Kurtosis
The relative concentration of scores in the center of the distribution
Leptokurtic

20. Measures of Central Tendency
Give one value that represents an entire group of scores
Mean
Median
Mode

21. Mean On your tests you get: 70%, 80%, 80%, 90% The mean is 80%
You know how to do this!

22. Mean = the mean = an instruction to add (sigma) “the sum of” = a score
= the number of scores

23. Practice What is the mean of: 5, 8, 6, 3, 2, 2, 9 Mean = 35 / 7 = 5
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
Mean = / 8 = 13.48

24. Characteristics of the mean
If the mean is subtracted from each score and the differences added, the sum will equal zero

25. Characteristics of the mean
= -10
= 0
= 10
 = 0

26. Characteristics of the mean
The mean is the point about which the sum of the squared deviations is minimized

27. Characteristics of the mean
= -102 = 100
= 02 = 0
= 102 = 100
 = 200

28. Population vs. Sample _ x = The mean of a sample
 = The mean of a population
*They are both calculated the same way!

29. Population vs. Sample

30. The Median
The point that divides a distribution of scores into two parts that are equal in size

31. The Median
10, 5, 13, 6, 14, 17, 2, 6, 9
2, 5, 6, 6, 9, 10, 13, 14, 17

32. The Median
10, 5, 13, 6, 14, 17, 2, 6, 9
2, 5, 6, 6, , 13, 14, 17

33. The Median
5, 8, 9, 15, 20, 25, 50

34. The Median
5, 8, 9, , 25, 50

35. The Median
5, 8, 9, 12, 15, 18, 22, 30, 32, 40

36. The Median
5, 8, 9, 12, , 18, , 30, 32, 40

37. The Median
5, 8, 9, 12,15, ,22, 30, 32, 40
16.5

38. Practice What is the median of: 5, 8, 6, 3, 2, 2, 9
2, 2, 3, 5, 6, 8, 9
(7+1) / 2 = 4
Median = 5

39. Practice What is the median of:
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
10.5, 10.5, 11.6, 11.9, 12.9, 14.7, 15.5, 20.2
(8+1) / 2 = 4.5
( ) / 2 = 12.4
Median = 12.4

40. The Mode The most frequently occurring score
5, 6, 8, 9, 10, 10, 10, 12, 14, 14
Mode = 10

41. Practice What is the mode of: 5, 8, 6, 3, 2, 2, 9 Mode = 2
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
Mode = 10.5

42. Determining Skewness

43. Determining Skewness
For example:
Mean = 4
Median = 10
Mean Median

44. Determining Skewness
For example:
Mean = 10
Median = 4
Median Mean

45. Determining Skewness Mean < Median = Negative Skew
Mean > Median = Positive Skew
Mean = Median = No Skew

46. Which should you use?

47. What is the mean, median, and mode?

48. Mean = 492 / 38 = 12.95

49. Mode = 14

50. (38 + 1)/ 2 = 19.5
Median = 14

52. The Test Scores of 3 Students
Joe = 78, 60, 92, 80, 80
Bob = 47, 100, 98, 45, 100
Mary = 78, 79, 77, 78, 78

53. The Test Scores of 3 Students
Joe = 78, 60, 92, 80, 80
Mean = 78
Bob = 47, 100, 98, 45, 100
Mary = 78, 79, 77, 78, 78

54. Variability
Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together

55. Range The highest score minus the lowest score
Joe = 78, 60, 92, 80, 80
Range = = 32

56. Range The highest score minus the lowest score
Bob = 47, 100, 98, 45, 100
Range = = 55

57. Range The highest score minus the lowest score
Mary = 78, 79, 77, 78, 78
Range = = 2

58. The Test Scores of 3 Students
Joe = 78, 60, 92, 80, 80
Mean = 78 Range = 32
Bob = 47, 100, 98, 45, 100
Mean = 78 Range = 55
Mary = 78, 79, 77, 78, 78
Mean = 78 Range = 2

59. Range In general - the larger the range score, the more variance
Pro: Easy to calculate
Con: The range only depends on two extreme scores; can be misleading

60. Range
20, 62, 54, 32, 28, 44, 72, 69, 50
1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99

61. Range
20, 62, 54, 32, 28, 44, 72, 69, 50
Range = 49
1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99
Range = 98!!

62. Interquartile Range
The range of scores that make up the middle 50 percent of the distribution
Need to find the 25th percentile score and the 75th percentile score

63. Interquartile Range
50%

64. Interquartile Range
.25 (N) = The location of the 25th percentile score counting from the bottom
.25 (N) = The location of the 75th percentile score counting from the top
N = the number of cases
*If the answer is not even simply average
*Similar to how you found the median!!

65. Interquartile Range
IQR = 75th percentile - 25th percentile

66. Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

67. Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
.25 (12) = 3
Counting 3 from the bottom the 25th percentile score = 6

68. Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
.25 (12) = 3
Counting 3 from the top the 75th percentile score = 62

69. Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
IQR = 75th percentile - 25th percentile
56 =

70. Practice
N = 40

71. 40(.25) = 10
= 2
N = 40

72. Practice Find the range for:
8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
102.58, 51.25, 58.00, 96.34, 54.43

73. Practice Find the range for:
8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
Range = = 72
8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
Range = = 75.4
102.58, 51.25, 58.00, 96.34, 54.43
Range = = 51.33

74. Practice Find the interquartile range for:
8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
102.58, 51.25, 58.00, 96.34, 54.43

75. Practice Find the interquartile range for:
8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
4, 4, 8, 10, 15, 22, 25, 43, 56, 56, 64, 76
(12) .25 = 3
= 48

76. Practice Find the interquartile range for:
8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
3.2, 8.5, 12.9, 59.5, 68.2, 75.2, 78.3, 78.6
(8).25 = 2
= 69.8

77. Practice Find the interquartile range for:
102.58, 51.25, 58.00, 96.34, 54.43
51.25, 54.43, 58.00, 96.34,
(5).25 = 1.25
( )/2 = 52.84
( )/2 = 99.46
= 46.62

79. Boxplots
The boxplot graphically displays three different characteristics of the distribution
Extreme scores
Interquartile range
Median

80. Boxplot

81. Boxplot
Interquartile range
25th - 75th percentile

82. Boxplot
Extreme Scores

83. Boxplot
Median

84. Boxplot
Skew -- Look at the “whiskers” to determine if the distribution is skewed

85. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

86. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median =
25th =
75th =
Lowest =
Highest =

87. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median = 22.5
25th = 6
75th = 62
Lowest = 2
Highest = 99

89. Neuroticism

90. Extraversion

91. Conscientiousness

92. Which distribution has a positive skew?

93. Which distribution has a negative skew?

94. Which distribution is most compact?E A B D

95. Which distribution has a median close to 25?

96. Which distribution is most symmetrical?

97. Which distribution has has the largest range?

99. Review Measures of variability Measures of central tendency Mean
Median
Mode
Measures of variability
Range
IQR

100. Variability Range IQR Problem with range and IQR
variability is still measured with only two numbers!

101. Deviation Score Formula
Deviation scores

107. = 16

108. = 16

109. Sample 1 vs. Sample 2 Sample 1: Raw scores: 15, 12, 17, 20 Sample 2:

110. Sample 1 vs. Sample 2 Sample 1: Raw scores: 15, 12, 17, 20
Deviation scores: -1, -4, 1, 4
Sample 2:
Raw scores: 26, 6, 1, 31
Deviation scores: 10, -10, -15, 15

111. Deviation Scores
As variability increases the absolute value of the deviation scores also goes up!
How can we use this information to create a measure of variability?

112. How about?

113. How about?

114. Formula

115. Formula
- 1

116. Why “ – 1” ?
Without it the answer is biased -- its answer tends to be too small
Page 53 – 56 explain
Don’t worry about why -- unless you want too!!

117. ( )

118. ( )
= 6

119. ( )
= 6

120. ( )
= 6

121. ( )
= 6
 = 38

122. Formula
- 1

123. Formula 38
- 1

124. Formula 38 5
- 1

125. Formula 38
9.5 5
- 1

126. Formula 38
9.5
3.08 5
- 1

127. Practice
For the sample data below calculate s
6, 8, 4, 3, 4, 5

128. = 5 ( )
 = 16

129. -1

130. 16
1.79 6 -1

131. Practice
2.34
2.35
2.46