Download Measures of Center

ELEMENTARY
STATISTICS
Section 2-4
Measures of Center
EIGHTH
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Measures of Center
a value at the
center or middle
of a data set
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Definitions
Mean
(Arithmetic Mean)
AVERAGE
the number obtained by adding the
values and dividing the total by the
number of values
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Notation

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
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Notation
x is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x
x =
n
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Notation
x is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x
x =
n
µ
is pronounced ‘mu’ and denotes the mean of all values
in a population
µ =
x
N
Calculators can calculate the mean of data
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Definitions
 Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Definitions
 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’)
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Definitions
 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
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
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
2
MEDIAN is 5.02
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
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
6.72
3.46
3.60
6.44
26.70
3.46
3.60
6.44
6.72
26.70
(in order -
exact middle
odd number of values)
MEDIAN is 6.44
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Definitions
 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
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Examples
a. 5 5 5 3 1 5 1 4 3 5
Mode is 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
Bimodal -
c. 1 2 3 6 7 8 9 10
No Mode
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2 and 6
13
Definitions
 Midrange
the value midway between the
highest
and lowest values in the
original data set
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Definitions
 Midrange
the value midway between the
highest
and lowest values in the
original data set
Midrange =
highest score + lowest score
2
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Round-off Rule for
Measures of Center
Carry one more decimal place than is
present in the original set of values
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Mean from a Frequency Table
use class midpoint of classes for variable x
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Mean from a Frequency Table
use class midpoint of classes for variable x
 (f • x)
x =
f
Formula 2-2
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Mean from a Frequency Table
use class midpoint of classes for variable x
 (f • x)
x =
f
Formula 2-2
x = class midpoint
f = frequency
f=n
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Weighted Mean
 (w • x)
x =
w
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Best Measure of Center
Advantages - Disadvantages
Table 2-13
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Definitions
 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.
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-13 (a)
SKEWED LEFT
(negatively)
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-13 (a)
SKEWED LEFT
(negatively)
Mean
Mode
Median
SKEWED RIGHT
(positively)
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Figure
2-13 (c)
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