Download Definitions

Measures of Center
Section 33 -2
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Definitions
Mean
1
(Arithmetic Mean)
AVERAGE
the number obtained by adding the
values and dividing the total by the
number of values
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Mean as a Balance Point
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Mean as a Balance Point
Mean
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Notation
Σ
denotes the addition 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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5
Notation
x is pronounced ‘x-bar ’ and denotes the mean of a set
of sample values
Σx
x =
n
µ
is pronounced ‘mu’ and denotes the mean of all values
in a population
µ =
Σx
N
Calculators can calculate the mean of data
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Definitions
v Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
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Definitions
v Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
v often denoted by x~
3
(pronounced ‘x -tilde
tilde’’)
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Definitions
v Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
v often denoted by x~
(pronounced ‘x -tilde
tilde’’)
v is not affected by an extreme value
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6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
(even number of values)
no exact middle -- shared by two numbers
3.60 + 6.44
MEDIAN is 5.02
2
Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pearson
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6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
10
(even number of values)
no exact middle -- shared by two numbers
3.60 + 6.44
4
MEDIAN is 5.02
2
6.72
3.46
3.60
6.44
26.70
3.46
3.60
6.44
6.72
26.70
(in order -
odd number of values)
exact middle
MEDIAN is 6.44
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Definitions
v 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 nominal data
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Examples
a. 5 5 5 3 1 5 1 4 3 5
ïMode is 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
ïBimodal -
c. 1 2 3 6 7 8 9 10
ïNo Mode
2 and 6
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Definitions
v Midrange
5
the value midway between the highest
and lowest values in the original data
set
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Definitions
v Midrange
the value midway between the highest
and lowest values in the original data
set
Midrange =
highest score + lowest score
2
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Round-off Rule for
RoundMeasures of Center
Carry one more decimal place than is
present in the original set of values
2 4 7 8 13 15
x = 8.166…
8.166… = 8.2
2.1 3.5 6.4 8.1
x = 5.025
= 5.03
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Measures of Center
Mean
6
Median
Mode
Midrange
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Mean from a Frequency Distribution
use class midpoint of classes for variable x
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Mean from a Frequency Distribution
use class midpoint of classes for variable x
x =
Σ (f • x)
Σf
Formula 3-2
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Mean from a Frequency Distribution
use class midpoint of classes for variable x
x =
Σ (f • x)
Σf
7
Formula 3-2
x = class midpoint
f = frequency
Σ f=n
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Weighted Mean
x =
Σ (w • x)
Σw
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Mean for a
Frequency Distribution
Quiz
Scores
Frequency
0-4
2
5-9
5
10-14
8
15-19
11
20-24
7
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Mean for a
Frequency Distribution
Quiz
Scores
Midpoints
Frequency
0-4
2
2
5-9
7
5
10-14
12
8
15-19
17
11
20-24
22
7
8
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Mean for a
Frequency Distribution
Quiz
Scores
0-4
Midpoints
Frequency
2
2
5
5-9
7
10-14
12
8
15-19
17
11
20-24
22
7
x=
Σ (f • x)
Σf
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Calculator Basics for
Statistical Data
1. Put calculator into statistical mode
2. Clear previous data
3. Enter data (and frequency)
4. Select key(s) that calculate x
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Mean for a Frequency Table
Quiz
Scores
Midpoints
Frequency
0-4
2
2
5-9
7
5
10-14
12
8
15-19
17
11
20-24
22
7
9
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Mean for a Frequency Table
x = 14.4
( rounded to one more
decimal place
than data )
Quiz
Scores
Midpoints
0-4
2
2
5-9
7
5
10-14
12
8
15-19
17
11
20-24
22
7
Frequency
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Quiz Scores
12
Frequency
10
8
6
4
2
0
2
7
12
Quiz Scores
( Midpoints)
17
22
14.4
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Definitions
v Symmetric
Data is symmetric if the left half of its
histogram is roughly a mirror of its
right half.
10
v Skewed
Data is skewed if it is not symmetric
and if it extends more to one side than
the other.
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Skewness
Figure
3-2 (b)
Mode
=
Mean
=
Median
SYMMETRIC
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Skewness
Figure
3-2 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Figure
3-2 (a)
Mean
Mode
Median
SKEWED LEFT
(negatively)
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Skewness
Figure
3-2 (b)
Mode
=
Mean
=
11
Median
SYMMETRIC
Figure
3-2 (a)
Mean
Mode
Mean
Mode
Median
Median
SKEWED LEFT
Figure
3-2 (c)
SKEWED RIGHT
(positively)
(negatively)
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Important Distributions
Normal
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Important Distributions
Normal
Uniform
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Important Distributions
12
Normal
Uniform
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Important Distributions
Normal
Uniform
Skewed Right
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Important Distributions
Normal
Uniform
Skewed Right
Skewed Left
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13
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