1. Numerical Measures

2. Numerical Measures Measures of Central Tendency (Location)
Measures of Non Central Location
Measure of Variability (Dispersion, Spread)
Measures of Shape

3. Measures of Central Tendency (Location)
Mean
Median
Mode
Central Location

4. Measures of Non-central Location
Quartiles, Mid-Hinges
Percentiles

5. Measure of Variability (Dispersion, Spread)
Variance, standard deviation
Range
Inter-Quartile Range
Variability

6. Measures of Shape
Skewness
Kurtosis

7. Summation Notation

8. Summation Notation
Let x1, x2, x3, … xn denote a set of n numbers.
Then the symbol
denotes the sum of these n numbers
x1 + x2 + x3 + …+ xn

9. Example
Let x1, x2, x3, x4, x5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13

10. Then the symbol
denotes the sum of these 5 numbers
x1 + x2 + x3 + x4 + x5 =
= 66

11. Meaning of parts of summation notation
Final value for i
each term of the sum
Quantity changing in each term of the sum
Starting value for i

12. Example
Again let x1, x2, x3, x4, x5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13

13. Then the symbol
denotes the sum of these 3 numbers = =
= 12979

14. Measures of Central Location (Mean)

15. Mean Let x1, x2, x3, … xn denote a set of n numbers.
Then the mean of the n numbers is defined as:

16. Example
Again let x1, x2, x3, x4, x5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13

17. Then the mean of the 5 numbers is:

18. Interpretation of the Mean
Let x1, x2, x3, … xn denote a set of n numbers.
Then the mean, , is the centre of gravity of those the n numbers.
That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point

19. xn x1 x3 x4 x2

20. In the Example 21 7 10 13 15 20 10

21. The mean, , is also approximately the center of gravity of a histogram

22. Measures of Central Location (Median)

23. The Median Let x1, x2, x3, … xn denote a set of n numbers.
Then the median of the n numbers is defined as the number that splits the numbers into two equal parts.
To evaluate the median we arrange the numbers in increasing order.

24. If the number of observations is odd there will be one observation in the middle.
This number is the median.
If the number of observations is even there will be two middle observations.
The median is the average of these two observations

25. Example
Again let x1, x2, x3, x3 , x4, x5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13

26. The numbers arranged in order are:
Unique “Middle” observation – the median

27. Example 2
Let x1, x2, x3 , x4, x5 , x6 denote the 6 denote numbers:
Arranged in increasing order these observations would be:
Two “Middle” observations

28. Median
= average of two “middle” observations =

29. Example The data on N = 23 students Variables Verbal IQ Math IQ
Initial Reading Achievement Score
Final Reading Achievement Score

30. The following table gives data on Verbal IQ, Math IQ,
Data Set #3
The following table gives data on Verbal IQ, Math IQ,
Initial Reading Acheivement Score, and Final Reading Acheivement Score
for 23 students who have recently completed a reading improvement program
Initial Final
Verbal Math Reading Reading
Student IQ IQ Acheivement Acheivement

31. Computing the Median Stem leaf Diagrams
Median = middle observation =12th observation

32. Summary

33. Some Comments
The mean is the centre of gravity of a set of observations. The balancing point.
The median splits the obsevations equally in two parts of approximately 50%

34. The median splits the area under a histogram in two parts of 50%
The mean is the balancing point of a histogram
50%
50%
median

35. For symmetric distributions the mean and the median will be approximately the same value
50%
50%
Median &

36. For Positively skewed distributions the mean exceeds the median
For Negatively skewed distributions the median exceeds the mean
50%
50%
median

37. An outlier is a “wild” observation in the data Outliers occur because
of errors (typographical and computational)
Extreme cases in the population

38. The mean is altered to a significant degree by the presence of outliers
Outliers have little effect on the value of the median
This is a reason for using the median in place of the mean as a measure of central location
Alternatively the mean is the best measure of central location when the data is Normally distributed (Bell-shaped)

39. Review

40. Summarizing Data
Graphical Methods

41. Histogram
Grouped Freq Table
Stem-Leaf Diagram

42. Numerical Measures Measures of Central Tendency (Location)
Measures of Non Central Location
Measure of Variability (Dispersion, Spread)
Measures of Shape
The objective is to reduce the data to a small number of values that completely describe the data and certain aspects of the data.

43. Measures of Central Location (Mean)

44. Mean Let x1, x2, x3, … xn denote a set of n numbers.
Then the mean of the n numbers is defined as:

45. Interpretation of the Mean
Let x1, x2, x3, … xn denote a set of n numbers.
Then the mean, , is the centre of gravity of those the n numbers.
That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point

46. xn x1 x3 x4 x2

47. The mean, , is also approximately the center of gravity of a histogram

48. The Median Let x1, x2, x3, … xn denote a set of n numbers.
Then the median of the n numbers is defined as the number that splits the numbers into two equal parts.
To evaluate the median we arrange the numbers in increasing order.

49. If the number of observations is odd there will be one observation in the middle.
This number is the median.
If the number of observations is even there will be two middle observations.
The median is the average of these two observations

50. Measures of Non-Central Location
Percentiles
Quartiles (Hinges, Mid-hinges)

51. Definition
The P×100 Percentile is a point , xP , underneath a distribution that has a fixed proportion P of the population (or sample) below that value
P×100 % xP

52. Definition (Quartiles)
The first Quartile , Q1 ,is the 25 Percentile , x0.25
25 %
x0.25

53. The second Quartile , Q2 ,is the 50th Percentile , x0.50
50 %
x0.50

54. The second Quartile , Q2 , is also the median and the 50th percentile

55. The third Quartile , Q3 ,is the 75th Percentile , x0.75
75 %
x0.75

56. divide the population into 4 equal parts of 25%.
The Quartiles – Q1, Q2, Q3
divide the population into 4 equal parts of 25%.
25 %
25 %
25 %
25 % Q1 Q2 Q3

57. Computing Percentiles and Quartiles
There are several methods used to compute percentiles and quartiles. Different computer packages will use different methods
Sometimes for small samples these methods will agree (but not always)
For large samples the methods will agree within a certain level of accuracy

58. Computing Percentiles and Quartiles – Method 1
The first step is to order the observations in increasing order.
We then compute the position, k, of the P×100 Percentile.
k = P × (n+1)
Where n = the number of observations

59. Example The data on n = 23 students Variables Verbal IQ Math IQ
Initial Reading Achievement Score
Final Reading Achievement Score
We want to compute the 75th percentile and
the 90th percentile

60. The position, k, of the 75th Percentile.
k = P × (n+1) = .75 × (23+1) = 18
The position, k, of the 90th Percentile.
k = P × (n+1) = .90 × (23+1) = 21.6
When the position k is an integer the percentile is the kth observation (in order of magnitude) in the data set.
For example the 75th percentile is the 18th (in size) observation

61. When the position k is an not an integer but an integer(m) + a fraction(f).
i.e. k = m + f
then the percentile is
xP = (1-f) × (mth observation in size)
+ f × (m+1st observation in size)
In the example the position of the 90th percentile is:
k = 21.6
Then
x.90 = 0.4(21st observation in size)
+ 0.6(22nd observation in size)

62. xP = (1-f) × (mth observation in size)
When the position k is an not an integer but an integer(m) + a fraction(f).
i.e. k = m + f
then the percentile is
xP = (1-f) × (mth observation in size)
+ f × (m+1st observation in size)
mth obs
(m+1)st obs
xp = (1- f) ( mth obs) + f [(m+1)st obs]

63. When the position k is an not an integer but an integer(m) + a fraction(f).
i.e. k = m + f
mth obs
(m+1)st obs
xp = (1- f) ( mth obs) + f [(m+1)st obs]
Thus the position of xp is 100f% through the interval between the mth observation and the (m +1)st observation

64. Example
The data Verbal IQ on n = 23 students arranged in increasing order is:

65. x0.75 = 75th percentile = 18th observation in size =105
(position k = 18)
x0.90 = 90th percentile
= 0.4(21st observation in size)
+ 0.6(22nd observation in size)
= 0.4(111)+ 0.6(118) = 115.2
(position k = 21.6)

66. An Alternative method for computing Quartiles – Method 2
Sometimes this method will result in the same values for the quartiles.
Sometimes this method will result in the different values for the quartiles.
For large samples the two methods will result in approximately the same answer.

67. Let x1, x2, x3, … xn denote a set of n numbers.
The first step in Method 2 is to arrange the numbers in increasing order.
From the arranged numbers we compute the median.
This is also called the Hinge

68. Example
Consider the 5 numbers:
Arranged in increasing order:
The median (or Hinge) splits the observations in half
Median (Hinge)

69. The lower mid-hinge (the first quartile) is the “median” of the lower half of the observations (excluding the median).
The upper mid-hinge (the third quartile) is the “median” of the upper half of the observations (excluding the median).

70. Consider the five number in increasing order:
Lower Half
Upper Half
Median (Hinge) 13
Upper Mid-Hinge
(First Quartile)
(7+10)/2 =8.5
Upper Mid-Hinge
(Third Quartile)
(15+21)/2 = 18

71. Computing the median and the quartile using the first method:
Position of the median: k = 0.5(5+1) = 3
Position of the first Quartile: k = 0.25(5+1) = 1.5
Position of the third Quartile: k = 0.75(5+1) = 4.5
Q3 = 18
Q1 = 8. 5
Q2 = 13

72. Both methods result in the same value
This is not always true.

73. The data Verbal IQ on n = 23 students arranged in increasing order is:
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
Upper Mid-Hinge
(Third Quartile)
105
Lower Mid-Hinge
(First Quartile) 89
Median (Hinge) 96

74. Computing the median and the quartile using the first method:
Position of the median: k = 0.5(23+1) = 12
Position of the first Quartile: k = 0.25(23+1) = 6
Position of the third Quartile: k = 0.75(23+1) = 18
Q3 = 105
Q1 = 89
Q2 = 96

75. Many programs compute percentiles, quartiles etc.
Each may use different methods.
It is important to know which method is being used.
The different methods result in answers that are close when the sample size is large.

76. Box-Plots Box-Whisker Plots
A graphical method of displaying data
An alternative to the histogram and stem-leaf diagram

77. To Draw a Box Plot
Compute the Hinge (Median, Q2) and the Mid-hinges (first & third quartiles – Q1 and Q3 )
We also compute the largest and smallest of the observations – the max and the min
The five number summary
min, Q1, Q2, Q3, max

78. The data Verbal IQ on n = 23 students arranged in increasing order is:
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
Q3 = 105
min = 80
Q1 = 89
Q2 = 96
max = 119

79. The Box Plot is then drawn
Drawing above an axis a “box” from Q1 to Q3.
Drawing vertical line in the box at the median, Q2
Drawing whiskers at the lower and upper ends of the box going down to the min and up to max.

80. Upper Whisker
Lower Whisker
Box Q3
min Q1 Q2
max

81. The data Verbal IQ on n = 23 students arranged in increasing order is:
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
min = 80
Q1 = 89
Q2 = 96
Q3 = 105
max = 119
This is sometimes called the five-number summary

82. Box Plot of Verbal IQ 70 80 90
100
110
120
130

83. Box Plot can also be drawn vertically70 80 90
100
110
120
130
Box Plot can also be drawn vertically

84. Box-Whisker plots (Verbal IQ, Math IQ)

85. Box-Whisker plots (Initial RA, Final RA )

86. Summary Information contained in the box plot
25%
25%
25%
25%
Middle 50% of population

87. Advance Box Plots

88. An outlier is a “wild” observation in the data
Outliers occur because
of errors (typographical and computational)
Extreme cases in the population
We will now consider the drawing of box-plots where outliers are identified

89. To Draw a Box Plot we need to:
Compute the Hinge (Median, Q2) and the Mid-hinges (first & third quartiles – Q1 and Q3 )
The difference Q3– Q1 is called the inter-quartile range (denoted by IQR)
To identify outliers we will compute the inner and outer fences

90. The fences are like the fences at a prison
The fences are like the fences at a prison. We expect the entire population to be within both sets of fences.
If a member of the population is between the inner and outer fences it is a mild outlier.
If a member of the population is outside of the outer fences it is an extreme outlier.

91. Inner fences

92. f1 = Q1 - (1.5)IQR f2 = Q3 + (1.5)IQR Lower inner fence
Upper inner fence
f2 = Q3 + (1.5)IQR

93. Outer fences

94. Lower outer fence F1 = Q1 - (3)IQR Upper outer fence F2 = Q3 + (3)IQR

95. Observations that are between the lower and upper inner fences are considered to be non-outliers.
Observations that are outside the inner fences but not outside the outer fences are considered to be mild outliers.
Observations that are outside outer fences are considered to be extreme outliers.

96. mild outliers are plotted individually in a box-plot using the symbol
extreme outliers are plotted individually in a box-plot using the symbol
non-outliers are represented with the box and whiskers with
Max = largest observation within the fences
Min = smallest observation within the fences

97. Extreme outlier
Box-Whisker plot representing the data that are not outliers
Mild outliers
Inner fences
Outer fence

98. Example Data collected on n = 109 countries in 1995.
Data collected on k = 25 variables.

99. The variables Population Size (in 1000s)
Density = Number of people/Sq kilometer
Urban = percentage of population living in cities
Religion
lifeexpf = Average female life expectancy
lifeexpm = Average male life expectancy

100. literacy = % of population who read
pop_inc = % increase in popn size (1995)
babymort = Infant motality (deaths per 1000)
gdp_cap = Gross domestic product/capita
Region = Region or economic group
calories = Daily calorie intake.
aids = Number of aids cases
birth_rt = Birth rate per 1000 people

101. death_rt = death rate per 1000 people
aids_rt = Number of aids cases/ people
log_gdp = log10(gdp_cap)
log_aidsr = log10(aids_rt)
b_to_d =birth to death ratio
fertility = average number of children in family
log_pop = log10(population)

102. cropgrow = ??
lit_male = % of males who can read
lit_fema = % of females who can read
Climate = predominant climate

103. The data file as it appears in SPSS

104. Consider the data on infant mortality
Stem-Leaf diagram stem = 10s, leaf = unit digit

105. Summary Statistics median = Q2 = 27 Quartiles
Lower quartile = Q1 = the median of lower half
Upper quartile = Q3 = the median of upper half
Interquartile range (IQR)
IQR = Q1 - Q3 = 66.5 – 12 = 54.5

106. The Outer Fences The Inner Fences
lower = Q1 - 3(IQR) = 12 – 3(54.5) =
upper = Q3 = 3(IQR) = (54.5) = 230.0
No observations are outside of the outer fences
The Inner Fences
lower = Q1 – 1.5(IQR) = 12 – 1.5(54.5) =
upper = Q3 = 1.5(IQR) = (54.5) =
Only one observation (168 – Afghanistan) is outside of the inner fences – (mild outlier)

107. Box-Whisker Plot of Infant Mortality

108. Example 2
In this example we are looking at the weight gains (grams) for rats under six diets differing in level of protein (High or Low) and source of protein (Beef, Cereal, or Pork).
Ten test animals for each diet

109. Gains in weight (grams) for rats under six diets
Table
Gains in weight (grams) for rats under six diets
differing in level of protein (High or Low)
and source of protein (Beef, Cereal, or Pork)
Level
 High Protein
Low protein
Source
 Beef
 Cereal
 Pork
Beef
Cereal
Pork
Diet 1 2 3 4 5 6 73 98 94 90
107 49
102 74 79 76 95 82
118 56 96 97
104
111 64 80 86 81 88 51
100
108 72
106 87 77 91 67 70
117
120 89 61 92
105 78 58
Median
103.0
87.0
100.0
82.0
84.5
81.5
Mean
85.9
99.5
79.2
83.9
78.7
IQR
24.0
18.0
11.0
23.0
16.0
PSD
17.78
13.33
8.15
17.04
11.05
Variance
229.11
225.66
119.17
192.84
246.77
273.79
Std. Dev.
15.14
15.02
10.92
13.89
15.71
16.55

110. High Protein
Low Protein
Beef
Cereal
Pork
Cereal
Pork
Beef

111. Conclusions Weight gain is higher for the high protein meat diets
Increasing the level of protein - increases weight gain but only if source of protein is a meat source

112. Next topic: Numerical Measures of Variability