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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
 is not affected by an extreme value
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
7
3
4
5
3
6
3
3
4
5
6
7
(even number of values)
no exact middle -- shared by two numbers
4+5
2
MEDIAN is 4.5
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
7
3
4
5
3
6
3
3
4
5
6
7
(even number of values)
no exact middle -- shared by two numbers
4+5
MEDIAN is 4.5
2
7
3
4
5
3
6
8
3
3
4
5
6
7
8
exact middle -- odd number of values
MEDIAN is 5
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Definitions
 Mode
the score that occurs most frequently
Bimodal – Two Modes
Multimodal – Many Modes
No Mode
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
10
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
11
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
12
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
13
Best Measure of Center
Measure of
Center
Affected by
Extreme Values?
Advantages and Disadvantages
Mean
Yes
Commonly Used, Works with other
statistical methods,
Median
No
Good Choice when there are extreme
values
Mode
No
Good Choice when nominal data is
used
Midrange
Yes
Rarely Used, Sensitive to Extreme
Values
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
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
15
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
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
17
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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