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MEASURES OF
CENTER
(Class Lecture in Statistics)
Part 3

3
...
1 - 2

Measures of
Center

3
...


3
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3
...

x is the variable usually used to represent
the data values
...


3
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1 - 7

Consider the following sum:

1 2 3 4 5
2

2

2

This can be
written in sigma
notation as:

2

2
Each of the terms
is in the form of k2,
where k is an
integer from 1 to 5
...
1 - 8

Sigma Notation
n

a
i 1

i

 a1  a2 
...
1 - 9

Determine the sum
4

k  2

 (1  2)  (2  2)  (3  2)  (4  2)

k 1

 3 4  5 6

 18

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1 - 11

Determine the sum
4

k
(

1
)
 (2k  1)
k 0

  1 2(0)  1  1 2(1)  1   1 2(2)  1   1 2(3)  1  1 2(4)  1
0

1

2

3

4

 1 3  5  7  9
5

3
...
 n 

2
k 1
n

nn  12n  1
k  1  2  3 
...
 n 

4
k 1
2

n

3

3

3

3

2

3

3
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1 - 14

Determine the sum

 i
6

i 1

2



1

6

6

i 1

i 1

  i 2  1

 6(7)(13) 

6

 6 

 97
3
...
1 - 16

Example of summation
• If there are n data values that
are denoted as:

x1 , x2 ,  , xn
Then:

 x  x1  x2      xn
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1 - 18

Sample Mean

x
is pronounced ‘x-bar’ and denotes the
mean of a set of sample values

x =

x
n
3
...
875  45
...
1 - 20

Notation
µ Greek letter mu used to denote the
population mean

N represents the number of data values in
a population
...
1 - 21

Population Mean

µ =

x
N

Note: here x represents the data values in
the population

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1 - 23

Mean
 Disadvantage

Is sensitive to every data value, one
extreme value can affect it dramatically; is
not a resistant measure of center

Example:

21,25,32,48,53,62,62,64 →
21,25,32,48,53,62,62,300 →

x  45
...
4
3
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1 - 25

Finding the Median
First sort the values (arrange them in
order), the follow one of these
1
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2
...

3
...
40

1
...
42

0
...
48

1
...
42

0
...
73

1
...
10

5
...
73  1
...
915
2
3
...
40

1
...
42

0
...
48

1
...
73

1
...
10

0
...
42

0
...
66

5
...
73
3
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1 - 29

Mode
Bimodal

two data values occur with the
same greatest frequency
Multimodal
more than two data values
occur with the same greatest
frequency
No Mode no data value is repeated
Mode is the only measure of central
tendency that can be used with nominal
data
3
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40 1
...
42 0
...
48 1
...
10

b) 27 27 27 55 55 55 88 88 99

Bimodal -

c) 1 2 3 6 7 8 9 10

No Mode

27 & 55

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1 - 32

Example of Midrange
• data values:
5
...
13

0
...
49

0
...
86

1
...
41  0
...
915
2

3
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1 - 34

Best Measure of Center

3
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3
...
41

1
...
42

0
...
65

1
...
69

5
...
13  0
...
49  0
...
86  1
...
65

 1
...
664
3
...
8 kg) during their
freshman year? Explain
3
...
9 kg
n

18

• Median
-5 -2 -2 -2 -2 0 0 1 1 2 2 3 3 4 5 7 8 11

1 2
median 
 1
...
1 - 39

Example
• Mode is -2
• Midrange
 5  11
midrange 
 3
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1 - 40

Example
• All of the measures of center
are below 6
...
8
kg) during their freshman year

3
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Think about the method used to
collect the sample data
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1 - 42

Mean/Median with Graphing
Calculator

• First,

enter the list of data values

•Then select 2nd STAT (LIST) and
arrow right to MATH option 3:mean(
or 4: median(

•and input the desired list
3
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x
x =
n

= 149
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1 - 44

Example of Computing the
Mean Using Calculator

Median is
150

3
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1 - 46



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