Download lect2-2

Measures of
Central Tendency
Section 2-4
M A R I O F. T R I O L A
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Longman
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Measure of
Central Tendency
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Measure of
Central Tendency
a value at the
center or middle
of a data set
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Definitions
Mean
Arithmetic Mean
AVERAGE
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Mean as a Balance Point
FIGURE 2-7
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Mean as a Balance Point
Mean
FIGURE 2-7
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Notation
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Notation
S denotes the summation of a set of values
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Notation
S denotes the summation of a set of values
x is the variable usually used to represent the individual
data values
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Notation
S denotes the summation of a set of values
x is the variable usually used to represent the individual
data values
n represents the number of data values in a sample
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Notation
S denotes the summation of a set of values
x is the variable usually used to represent the individual
data values
n represents the number of data values in a sample
N represents the number of data values in a population
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Notation
S denotes the summation of a set of values
x is the variable usually used to represent the individual
data values
n represents the number of data values in a sample
N represents the number of data values in a population
x is pronounced ‘x-bar’ and denotes the mean of a set of
sample values
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Notation
S denotes the summation of a set of values
x is the variable usually used to represent the individual
data values
n represents the number of data values in a sample
N represents the number of data values in a population
x is pronounced ‘x-bar’ and denotes the mean of a set of
sample values
µ
is pronounced ‘mu’ and denotes the mean of all values
in a population
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Definitions
 Mean
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Definitions
 Mean
the value obtained by adding the scores and
dividing the total by the number of scores
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Definitions
 Mean
the value obtained by adding the scores and
dividing the total by the number of scores
Sample
S x
x =
n
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Definitions
 Mean
the value obtained by adding the scores and
dividing the total by the number of scores
Sample
Population
x =
Sx
n
Sx
µ =
N
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Definitions
 Mean
the value obtained by adding the scores and
dividing the total by the number of scores
Sample
Population
x =
Sx
n
Sx
µ =
N
Calculators can calculate the mean of data
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Examples
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Mean from a Frequency Table
use class mark of classes for variable x
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Mean from a Frequency Table
use class mark of classes for variable x
S (f • x)
x =
Sf
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Formula 2-2
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Mean from a Frequency Table
use class mark of classes for variable x
S (f • x)
x =
Formula 2-2
Sf
x = class mark
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Mean from a Frequency Table
use class mark of classes for variable x
S (f • x)
x =
Formula 2-2
Sf
x = class mark
f = frequency
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Mean from a Frequency Table
use class mark of classes for variable x
S (f • x)
x =
Formula 2-2
Sf
x = class mark
f = frequency
Sf=n
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Weighted Mean
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Weighted Mean
S (w • x)
x =
Sw
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Examples
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Definitions
 Median
the middle value when scores are arranged
in (ascending or descending) order
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Definitions
 Median
the middle value when scores are arranged
in (ascending or descending) order
~
often denoted by x (pronounced ‘x-tilde’)
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Definitions
 Median
the middle value when scores are arranged
in (ascending or descending) order
~
often denoted by x (pronounced ‘x-tilde’)
is not affected by an extreme value
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Examples
•
•
5
5
5
3
1
5
1
4
3
5
2
1 1 2
(in order)
3
3
4
5
5
5
5
5
exact middle
MEDIAN is 4
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Examples
•
•
5
5
5
3
1
5
1
4
3
5
2
1 1 2
(in order)
3
3
4
5
5
5
5
5
MEDIAN is 4
exact middle
•
1
1
3
3
4
5
5
5
5
5
no exact middle -- shared by two numbers
4+5
= 4.5
2
MEDIAN is 4.5
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Definitions
 Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
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Examples
a.
5 5 5 3 1 5 1 4 3 5
b.
2 2 2 3 4 5 6 6 6 7 9
c.
2 3 6 7 8 9 10
• Mode is 5
• Bimodal (2 & 6)
• No Mode
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Examples
a.
5 5 5 3 1 5 1 4 3 5
b.
2 2 2 3 4 5 6 6 6 7 9
c.
2 3 6 7 8 9 10
d.
2 2 3 3 3 4
e.
2 2 3 3 4 4 5 5
• Mode is 5
• Bimodal (2 & 6)
• No Mode
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Examples
a.
5 5 5 3 1 5 1 4 3 5
b.
2 2 2 3 4 5 6 6 6 7 9
c.
2 3 6 7 8 9 10
d.
2 2 3 3 3 4
e.
2 2 3 3 4 4 5 5
• Mode is 5
• Bimodal (2 & 6)
• No Mode
• Mode is 3
• No Mode
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Remark: Mode is the only measure of central
tendency that can be used with nominal data
Blood types:
O
35
A
14
B
16
AB 10
• Mode is “O” blood type
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Definitions
 Midrange
the value halfway between the highest
and lowest scores
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Definitions
 Midrange
the value halfway between the highest
and lowest scores
Midrange =
highest score + lowest score
2
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Examples
•
•
5
5
5
3
1
5
1
4
3
5
2
1 1 2
(in order)
3
3
4
5
5
5
5
5
Midrange is (5 + 1)/2 = 3
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Round-off rule for
measures of central tendency
Carry one more decimal place than is present
in the original set of data
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Best Measure
of Central Tendency
Table 2-6
Advantages - Disadvantages
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Table 2-6
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Skewness
Figure
2-8 (b)
Mode
=
Mean
=
Median
SYMMETRIC
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Skewness
Figure
2-8 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-8 (a)
SKEWED LEFT
(negatively)
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Skewness
Figure
2-8 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-8 (a)
SKEWED LEFT
(negatively)
Mean
Mode
Median
SKEWED RIGHT
(positively)
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Figure
2-8 (c)
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