INTRODUCTION-STATISTICS_BASICS_DESCRIPTIVES.ppt

Statistics

1. 1Dept. of AEE, FFA, The UWI, St.Augustine
KATHIRAVAN Gopalan
INTRODUCTION TO
STATISTICS

2. 2Dept. of AEE, FFA, The UWI, St.Augustine
Statistics?
Three meanings…
Specific numbers
Method of analysis
Statistical measure of sample

3. 3Dept. of AEE, FFA, The UWI, St.Augustine
Specific number
numerical measurement determined by a
set of data
Example: Twenty-three percent of people
polled believed that there are
too many polls.
…Statistics…

4. 4Dept. of AEE, FFA, The UWI, St.Augustine
Method of analysis
a collection of methods for planning
experiments, obtaining data, and then
then organizing, summarizing, presenting,
analyzing, interpreting, and drawing
conclusions based on the data
…Statistics…

5. 5Dept. of AEE, FFA, The UWI, St.Augustine
…Statistics
Sample -Statistics
any statistical measure relating to sample
Example: Mean of sample.

6. 6Dept. of AEE, FFA, The UWI, St.Augustine
Definitions…
Population
the complete collection of all
elements (scores, people,
measurements, and so on) to be
studied. The collection is complete
in the sense that it includes all
subjects to be studied.

7. 7Dept. of AEE, FFA, The UWI, St.Augustine
Census
the collection of data from every
element in a population
Sample
a subcollection of elements drawn
from a population
…Definitions…

8. 8Dept. of AEE, FFA, The UWI, St.Augustine
Parameter
a numerical measurement describing
some characteristic of a population
…Definitions…

9. 9Dept. of AEE, FFA, The UWI, St.Augustine
Parameter
a numerical measurement describing
some characteristic of a population
population
parameter
…Definitions…

10. 10Dept. of AEE, FFA, The UWI, St.Augustine
Statistic
a numerical measurement describing
some characteristic of a sample
…Definitions…

11. 11Dept. of AEE, FFA, The UWI, St.Augustine
Statistic
a numerical measurement describing
some characteristic of a sample
sample
statistic
…Definitions…

12. 12Dept. of AEE, FFA, The UWI, St.Augustine
Quantitative data
numbers representing counts or
measurements
…Definitions…

13. 13Dept. of AEE, FFA, The UWI, St.Augustine
Quantitative data
numbers representing counts or
measurements
Qualitative (or categorical or
attribute) data
can be separated into different categories
that are distinguished by some nonnumeric
characteristics
…Definitions…

14. 14Dept. of AEE, FFA, The UWI, St.Augustine
Quantitative data
the incomes of college graduates
…Definitions…

15. 15Dept. of AEE, FFA, The UWI, St.Augustine
Quantitative data
the incomes of college graduates
Qualitative (or categorical or
attribute) data
the genders (male/female) of college
graduates
…Definitions…

16. 16Dept. of AEE, FFA, The UWI, St.Augustine
Discrete
data result when the number of possible values is
either a finite number or a ‘countable’ number of
possible values
0, 1, 2, 3, . . .
…Definitions…

17. 17Dept. of AEE, FFA, The UWI, St.Augustine
Discrete
data result when the number of possible values is
either a finite number or a ‘countable’ number of
possible values
0, 1, 2, 3, . . .
Continuous
(numerical) data result from infinitely many possible
values that correspond to some continuous scale
that covers a range of values without gaps,
interruptions, or jumps
2 3
…Definitions…

18. 18Dept. of AEE, FFA, The UWI, St.Augustine
Discrete
The number of eggs that hens lay; for
example, 3 eggs a day.
…Definitions…

19. 19Dept. of AEE, FFA, The UWI, St.Augustine
Discrete
The number of eggs that hens lay; for
example, 3 eggs a day.
Continuous
The amounts of milk that cows produce;
for example, 2.343115 gallons a day.
…Definitions…

20. 20Dept. of AEE, FFA, The UWI, St.Augustine
nominal level of measurement
characterized by data that consist of names,
labels, or categories only. The data cannotbe
arranged in an ordering scheme (such as low
to high)
Example: survey responses yes, no, undecided
…Definitions…

21. 21Dept. of AEE, FFA, The UWI, St.Augustine
ordinallevelofmeasurement
involves data that may be arranged in some
order, but differences between data values
either cannot be determined or are meaningless
Example: Course grades A, B, C, D, or F
…Definitions…

22. 22Dept. of AEE, FFA, The UWI, St.Augustine
intervallevelofmeasurement
like the ordinal level, with the additional property
that the difference between any two data values is
meaningful. However, there is no natural zero
starting point (where noneof the quantity is
present)
Example: Years 1000, 2000, 1776, and 1492
…Definitions…

23. 23Dept. of AEE, FFA, The UWI, St.Augustine
ratiolevelofmeasurement
the interval level modified to include the natural
zero starting point (where zero indicates that
noneof the quantity is present). For values at
this level, differences and ratios are meaningful.
Example: Prices of college textbooks
…Definitions…

24. 24Dept. of AEE, FFA, The UWI, St.Augustine
Levels of Measurement
Nominal-categories only
Ordinal-categories with some order
Interval-differences but no natural
starting point
Ratio-differences anda natural starting
point

25. 25Dept. of AEE, FFA, The UWI, St.Augustine
KATHIRAVAN Gopalan
STATISTICS
Measures of Averages

26. 26Dept. of AEE, FFA, The UWI, St.Augustine
a value at the
center or middle
of a data set
Measures of Center

27. 27Dept. of AEE, FFA, The UWI, St.Augustine
Mean
(Arithmetic Mean)
AVERAGE
the number obtained by adding the
values and dividing the total by the
number of values
Definitions

28. 28Dept. of AEE, FFA, The UWI, St.Augustine
Notations
denotes the additionof a set of values
xis the variableusually used to represent the individual
data values
nrepresents the number of data values in a sample
Nrepresents the number of data values in a population

29. 29Dept. of AEE, FFA, The UWI, St.Augustine
Notations
is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x=
n
x
x

30. 30Dept. of AEE, FFA, The UWI, St.Augustine
Notation
µis pronounced ‘mu’ and denotes themean of all values
in a population
is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
Calculators can calculate the mean of data
x=
n
x
x
N
µ=
x

31. 31Dept. of AEE, FFA, The UWI, St.Augustine
…Definitions…
Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude

32. 32Dept. of AEE, FFA, The UWI, St.Augustine
Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
often denoted by x(pronounced ‘x-tilde’)
~
…Definitions…

33. 33Dept. of AEE, FFA, The UWI, St.Augustine
Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
often denoted by x(pronounced ‘x-tilde’)
is not affected by an extreme value
~
…Definitions…

34. 34Dept. of AEE, FFA, The UWI, St.Augustine
6.72 3.46 3.606.44
3.463.60 6.446.72
no exact middle --shared by two numbers
3.60 + 6.44
2
(even number of values)
MEDIAN is 5.02

35. 35Dept. of AEE, FFA, The UWI, St.Augustine
6.72 3.46 3.606.44 26.70
3.463.60 6.446.72 26.70
(in order - odd number of values)
exact middle MEDIANis 6.44
6.72 3.46 3.606.44
3.463.60 6.446.72
no exact middle --shared by two numbers
3.60 + 6.44
2
(even number of values)
MEDIAN is 5.02

36. 36Dept. of AEE, FFA, The UWI, St.Augustine
Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
denoted by M
the only measure of central tendency that can be
used with nominaldata
…Definitions…

37. 37Dept. of AEE, FFA, The UWI, St.Augustine
a.5 5 5 3 1 5 1 4 3 5
b.1 2 2 2 3 4 5 6 6 6 7 9
c.1 2 3 6 7 8 9 10
Examples
Mode is 5
Bimodal -2 and 6
No Mode

38. 38Dept. of AEE, FFA, The UWI, St.Augustine
Midrange
the value midway between the highest and
lowest values in the original data set
…Definitions…

39. 39Dept. of AEE, FFA, The UWI, St.Augustine
Midrange
the value midway between the highest and
lowest values in the original data set
Midrange=
highest score + lowest score
2
…Definitions…

40. 40Dept. of AEE, FFA, The UWI, St.Augustine
Carry one more decimal place than is
present in the original set of values
Round-off Rule for
Measures of Center

41. 41Dept. of AEE, FFA, The UWI, St.Augustine
use class midpoint of classes for variable x
Mean from a Frequency Table

42. 42Dept. of AEE, FFA, The UWI, St.Augustine
use class midpoint of classes for variable x
Mean from a Frequency Table
x=
f
(f • x)


43. 43Dept. of AEE, FFA, The UWI, St.Augustine
use class midpoint of classes for variable x
Mean from a Frequency Table
x=class midpoint
f= frequency
f=n
x=
f
(f • x)


44. 44Dept. of AEE, FFA, The UWI, St.Augustine
Weighted Mean
x=
w
(w • x)


45. 45Dept. of AEE, FFA, The UWI, St.Augustine
Advantages -Disadvantages
Best Measure of Center

46. 46Dept. of AEE, FFA, The UWI, St.Augustine
KATHIRAVAN Gopalan
STATISTICS
Measures of Variation

47. 47Dept. of AEE, FFA, The UWI, St.Augustine
Milk Yield of Cows in
Two Different Dairy Farms
in litres
Farm -A
Farm -B
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0

48. 48Dept. of AEE, FFA, The UWI, St.Augustine
Farm -A
Farm -B
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0
Farm -A
7.15
7.20
7.7
7.10
Farm -B
7.15
7.20
7.7
7.10
Mean
Median
Mode
Midrange
Milk Yield of Cows in
Two Different Dairy Farms
in litres

49. 49Dept. of AEE, FFA, The UWI, St.Augustine
Dotplots of Milk Yiels

50. 50Dept. of AEE, FFA, The UWI, St.Augustine
Measures of Variation…

51. 51Dept. of AEE, FFA, The UWI, St.Augustine
Range
value
highest lowest
value
…Measures of Variation…

52. 52Dept. of AEE, FFA, The UWI, St.Augustine
a measure of variation of the scores
about the mean
(average deviation from the mean)
Standard Deviation
…Measures of Variation…

53. 53Dept. of AEE, FFA, The UWI, St.Augustine
Sample Standard Deviation
Formula

54. 54Dept. of AEE, FFA, The UWI, St.Augustine
Sample Standard Deviation
Formula
calculators can compute the
sample standard deviation of data
(x-x)
2
n -1
S=

55. 55Dept. of AEE, FFA, The UWI, St.Augustine
Sample Standard Deviation
Shortcut Formula
n (n -1)
s =
n (x
2
)-(x)
2
calculators can compute the
sample standard deviation of data

56. 56Dept. of AEE, FFA, The UWI, St.Augustine
Mean Deviation Formula
(absolute deviation)

57. 57Dept. of AEE, FFA, The UWI, St.Augustine
x -x
Mean Absolute Deviation
Formula
n

58. 58Dept. of AEE, FFA, The UWI, St.Augustine
Population Standard Deviation
calculators can compute the
population standard deviation
of data
2
(x-µ)
N
=

59. 59Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation

60. 60Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample

61. 61Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s

62. 62Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s
Textbook

63. 63Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s
Sx
Textbook

64. 64Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s
Sx
Textbook
Some graphics
calculators

65. 65Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s
Sx
x
n-1
Textbook
Some graphics
calculators

66. 66Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample
s
Sx
x
n-1
Textbook
Some graphics
calculators
Some
non-graphics
calculators

67. 67Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample Population
s
Sx
x
n-1
Textbook
Some graphics
calculators
Some
non-graphics
calculators

68. 68Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample Population

x
x
n
s
Sx
x
n-1
Textbook
Some graphics
calculators
Some
non-graphics
calculators

69. 69Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample Population

x
x
n
s
Sx
x
n-1
Book
Some graphics
calculators
Some
non-graphics
calculators
Textbook
Some graphics
calculators
Some
non-graphics
calculators

70. 70Dept. of AEE, FFA, The UWI, St.Augustine
Symbols
for Standard Deviation
Sample Population

x
x
n
s
Sx
x
n-1
Book
Some graphics
calculators
Some
non-graphics
calculators
Textbook
Some graphics
calculators
Some
non-graphics
calculators
Articles in professional journals and reports often use SD for standard
deviation and VAR for variance.

71. 71Dept. of AEE, FFA, The UWI, St.Augustine
Variance
…Measures of Variation…

72. 72Dept. of AEE, FFA, The UWI, St.Augustine
Measures of Variation
Variance
standard deviation squared

73. 73Dept. of AEE, FFA, The UWI, St.Augustine
Measures of Variation
Variance
standard deviation squared
s

2
2
}
use square key
on calculator
Notation

74. 74Dept. of AEE, FFA, The UWI, St.Augustine
Sample
Variance
Population
Variance
Variance
(x-x )
2
n -1
s
2
=
(x-µ)
2
N

2
=

75. 75Dept. of AEE, FFA, The UWI, St.Augustine
Round-off Rule
for measures of variation
Carry one more decimal place than
is present in the original set of
values.
Round only the final answer, never in the
middle of a calculation.

76. 76Dept. of AEE, FFA, The UWI, St.Augustine
Standard Deviation from a
Frequency Table
Use the class midpoints as the x values
Calculators can compute the standard deviation for
frequency table
n(n -1)S=
n[(f • x
2
)]-[(f•x)]
2

77. 77Dept. of AEE, FFA, The UWI, St.Augustine
Estimation of Standard Deviation
Range Rule of Thumb
x-
2s
x
x+ 2s
Range 4s
or
(minimum
usual value)
(maximum
usual value)

78. 78Dept. of AEE, FFA, The UWI, St.Augustine
Estimation of Standard Deviation
Range Rule of Thumb
x-
2s
x
x+ 2s
Range 4s
or
(minimum
usual value)
(maximum
usual value)
Range
4
s 

79. 79Dept. of AEE, FFA, The UWI, St.Augustine
Estimation of Standard Deviation
Range Rule of Thumb
x-
2s
x
x+ 2s
Range 4s
or
(minimum
usual value)
(maximum
usual value)
Range
4
s 
=
highest value -lowest value
4

80. 80Dept. of AEE, FFA, The UWI, St.Augustine
Usual Sample Values

81. 81Dept. of AEE, FFA, The UWI, St.Augustine
minimum ‘usual’ value (mean) -2 (standard deviation)
minimumx -2(s)
Usual Sample Values

82. 82Dept. of AEE, FFA, The UWI, St.Augustine
minimum ‘usual’ value (mean) -2 (standard deviation)
minimumx -2(s)
maximum ‘usual’ value (mean) + 2 (standard deviation)
maximumx + 2(s)
Usual Sample Values

83. 83Dept. of AEE, FFA, The UWI, St.Augustine
x
The Empirical Rule
(applies to bell-shaped distributions)

84. 84Dept. of AEE, FFA, The UWI, St.Augustine
x-s x x+s
68% within
1 standard deviation
34% 34%
The Empirical Rule
(applies to bell-shaped distributions)

85. 85Dept. of AEE, FFA, The UWI, St.Augustine
x-2s x-s x x+2sx+s
68% within
1 standard deviation
34% 34%
95% within
2 standard deviations
The Empirical Rule
(applies to bell-shaped distributions)
13.5% 13.5%

86. 86Dept. of AEE, FFA, The UWI, St.Augustine
x-3s x-2s x-s x x+2s x+3sx+s
68% within
1 standard deviation
34% 34%
95% within
2 standard deviations
99.7% of data are within 3 standard deviations of the mean
The Empirical Rule
(applies to bell-shaped distributions)
0.1%
0.1%
2.4% 2.4%
13.5% 13.5%

87. 87Dept. of AEE, FFA, The UWI, St.Augustine
Measures of Variation Summary
For typical data sets, it is unusualfor a
score to differ from the mean by more than
2 or 3 standard deviations.

88. 88Dept. of AEE, FFA, The UWI, St.Augustine
Skewness …

89. 89Dept. of AEE, FFA, The UWI, St.Augustine
Symmetric
Data is symmetric if the left half of its
histogram is roughly a mirror of its
right half.
Skewed
Data is skewed if it is not symmetric
and if it extends more to one side than
the other.
Definitions

90. 90Dept. of AEE, FFA, The UWI, St.Augustine
Skewness
Mode = Mean = Median
SYMMETRIC

91. 91Dept. of AEE, FFA, The UWI, St.Augustine
Skewness
Mode = Mean = Median
SKEWED LEFT
(negatively)
SYMMETRIC
Mean Mode
Median

92. 92Dept. of AEE, FFA, The UWI, St.Augustine
Skewness
Mode = Mean = Median
SKEWED LEFT
(negatively)
SYMMETRIC
Mean Mode
Median
SKEWED RIGHT
(positively)
Mean Mode
Median