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ELEMENTARY
STATISTICS
Section 2-4
Measures of Center
MARIO F. TRIOLA
EIGHTH
EDITION
1
Objectives Day 1
• Given a data set, determine
the mean, median, and
mode.
2
Measures of Center
a value at the
center or middle
of a data set
3
Definitions
Mean
(Arithmetic Mean)
AVERAGE
the number obtained by adding the
values and dividing the total by the
number of values
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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
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Notation
x is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x
x =
n
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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
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Definitions
 Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
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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’)
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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
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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
2
MEDIAN is 5.02
11
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
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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
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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
2 and 6
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Definitions
 Midrange
the value midway between the highest
and lowest values in the original data set
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Definitions
 Midrange
the value midway between the highest
and lowest values in the original data set
Midrange =
highest score + lowest score
2
16
Round-off Rule for
Measures of Center
Carry one more decimal place than is
present in the original set of values
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Mean from a Frequency Table
use class midpoint of classes for variable x
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Mean from a Frequency Table
use class midpoint of classes for variable x
 (f • x)
x =
f
Formula 2-2
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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
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Example Qwerty Keyboard Word
Ratings
Word
Ratings
Interval
Midpoints
Frequency
x
f
0-2
1
20
20
3-5
6-8
4
7
14
15
56
105
9-11
12-14
10
13
2
1
20
13
f
x
 52
x
f
 ( x  f )  214
214
 4.11  4.1 points
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• Pages 65-66 3,5,9,11
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Homework Solutions
pg 65 #3
• Mean
x

x
n
972
x
12
x  81.0 seconds
• Median
35 46 55 65 74 83 88 93 99 107 108 119
Occurs between the 6th and 7th data values
Mode- none83  88
x
 85.5 seconds
2
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Homework Solutions
pg 65 #5
• Mean
x  7.15 minutes
• Median
x  7.20
• Mode
7.70
Jefferson Valley
Providence
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Homework Solutions
pg 65 #9
midpts freq
x*f
40-49
50-59
60-69
70-79
80-89
90-99
14870
x
 74.35  74.4 minutes
200
100-109
f
 200
x  f   14870
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Homework Solutions
pg 65 #11
• Mean
Midpts
x
Freq
f
x*f
2339
x
 46.78  46.8 mph
50
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Objectives Day 2
• Given a data set where the scores vary
in importance, compute a weighted
mean.
• Determine how extreme values affect
measures of center.
• Understand the relationship between
the shape of a distribution and the
relative location of the mean and
median.
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Weighted Mean – used when
scores vary in importance
Formula
 w x
x
w
where x represents the scores and w the corresponding weights
w1 x1  w2 x2  ...  wn xn
x
w1  w2  ... wn
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Example Weighted Mean
Note A straight percentage grade based on all tests being 100 points
• The final grade computation for a freshman statistics
course is based
329 on weighted components.
 .8225  82%
Tests 20% each
400
Final 40%
If the Test scores are 83%, 73%, 82% , and a final exam score of
91% … What is the final grade?
xw
20 83   20  73   20 82    40  91


xw 
20  20  20  40
8400
 84
100
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Weighted Mean common error
The weights do not need to sum to 100
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Example Weighted Mean
• Two algebra classes had the following average test
scores. Period 2 had a mean score of 40 with 24
students and period 8 had a mean score of 34 with 16
students. What is the mean of the two classes
combined?
xw
24  40   16  34 


xw 
24  16
1504
 37.6 points
40
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Example Weighted Mean
What is your GPA?
• Your first semester grades at college are as follows:
Subject
Grade
Credits
3A 4    43 3   3 3  1 2    3 4 

GPA 
Biology
Calculus
College
GPA 
Writing
B
4
3  4  3 1 3
47
B  3.3573
14
Archery
C
1
Chemistry
A
3
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Best Measure of Center
Advantages - Disadvantages
Table 2-13
33
Extreme Values and Measures of
Center
Consider the following data set of salaries at a small
Now change
the $125,000 to $250,000
shipping
company.
30,000 30,000 30,000 30,000 30,000 40,000 45,000 125,000
What is the mean? The median? The mode?
x
360000
 $45, 000
8
x  $30, 000 mode  $30, 000
What is the mean? The median? The mode?
x
485, 000
 $60, 625
8
x  $30, 000 mode  $30, 000
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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.
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Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
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Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-13 (a)
SKEWED LEFT
(negatively)
37
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)
Figure
2-13 (c)
38
Review of Concepts
• The mean of a data set is the balance point of the
values. Think of the mean as a give and take
• The median is the middle value of an ordered data
set.
• The mode is the data value that occurs most
frequently. There may be no mode, one mode, or
more than one mode.
• If a distribution has few values or it is skewed, then
the measures of center may not actually be near the
center of the distribution. You must make
appropriate decisions as to use which measure of
center is most appropriate.
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Page 68 #20, 21, 24
Page 107 # 2, 3
Handout data analysis of mean
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