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Chapter 3
Measures of Central Tendency
I Mode
A. Definition: the Score or Qualitative Category
that Occurs With the Greatest Frequency
1. Mode (Mo) for the following data, number of
required textbooks for Fred’s four classes, is 2.
2
1
2
3
1
Table 1. Taylor Manifest Anxiety Scores
_______________________________
(1)
Xj
(2)
f
_______________________________
74
73
72
71
70
69
68
67
66
65
1
1
0
2
7
8
5
2
1
1
Mo = 69
_______________________________
n = 28
_______________________________
2
II Mean
A. Definition: the Mean Is the Sum of Scores
Divided by the Number of Scores
B. Formula
X
X1  X 2  L  X n
n
1. X denotes the mean, X i denotes a score, and
n denotes the number of scores
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C. Summation Operator, (Greek capitol sigma)
n
 X i  X1  X 2 L  X n
i1
D. Mean Formula for a Frequency
Distribution
k
X
 fjX j
j1

f1 X1  f2 X 2  L  f k X k
n
n
1. k = number of class intervals
2. f j  frequency of the jth class interval
3. X j  midpoint of the jth class interval
4
Table 2. Taylor Manifest
Anxiety Scores
_________________
(1)
(2)
(3)
fjX j
f
_________________
Xj
74
73
72
71
70
69
68
67
66
65
1
1
0
2
7
8
5
2
1
1
74
73
0
142
490
552
340
134
66
65
n = 28
1,936
k
X
 fjX j
j1
n
1,936

 69.14
28
_________________
_________________
5
III Median (Mdn)
A. Definition: the Median Divides Data Into
Two Groups Having Equal Frequency
1. If n is odd and the scores are ordered, the median
is the (n + 1)/2th score from either end of the
number line.
2. If n is even, the median is the midway point
between the n/2th score and the n/2 + 1th
score from either end of the number line.
6
B. Computational Examples
Real limits of score
1
2
5
4
3
Mdn = 8
6
7
8
9
10
11
12
1. Determination of Mdn when n is odd
Mdn = 8.5
2
3
4
5
6
7
8
9
10
11
12
2. Determination of Mdn when n is even
7
Mdn = 8
a.
1
2
3
4
5
6
7
8
9
10
11
12
Mdn = 7.75
7.50 8.00 8.50
7.75 8.25
b.
1
2
3
4
5
6
7
8
9
10
11
12
3. Determination of Mdn when n is even (a) or odd
(b), and the frequency of the middle score value
is greater than 1
8
Mdn = 7.833
7.500 7.833 8.167 8.500
7.667 8 8.333
1
2
3
4
5
6
7
8
9
10
11
12
4. Determination of Mdn when n is even and the
frequency of the middle score value is greater
than 1
9
C. Computation of Mdn for a Frequency
Distribution
1. Formula when scores are cumulated from below
 n / 2   fb 
Mdn  X ll  i 

f


i
Xll = real lower limit of the class interval
containing the median
i = class interval size
n = number of scores
fb = number of scores below Xll
fi = number of scores in the class interval
containing the median
10
2. Formula for the Mdn when scores are cumulated
from above
 n / 2   fa 
Mdn  X ul  i 

f


i
Xul = real upper limit of class interval
containing the median
fa = number of scores above Xul
11
Table 3. Taylor Manifest Anxiety
Scores
 n / 2   fb 
_____________________________ Mdn  X ll  i 

f
(1)
(2)
(3)
Xj
fj
Cum f up
74
73
72
71
70
69
68
67
66
65
1
1
0
2
7
8
5
2
1
1
(4)
Cum f down
__________________________
17
9
4
2
1
1
2
2
4
11
19
__________________________
n = 28
__________________________

i

 28 / 2  9 
 68.5  1

8

 68.5  0.625  69.12
 n / 2   fa 
Mdn  X ul  i 

f


i
 28 / 2  11
 69.5  1

8

 69.5  0.375  69.12
12
IV Relative Merits of the Mean, Median, and
Mode
V Location of the Mean, Median, and Mode
in a Distribution
f
f
X
Mean
Median
Mode
X
Mean
Median
Mode
13
VI Mean of Two or More Means
A. Weighted Mean
XW 
n1 X1  n2 X 2  L  nn X n
n1  n2  L  nn
VII Summation Rules
A. Sum of a Constant (c)
n7
terms
6
4
4
4 48
n
 c  c  c  L  c  nc
i1
14
B. Sum of a Variable (Vi)
n
 Vi  V1  V2  L  Vn
i1
C. Sum of the Product of a Constant and a
Variable
n
n
i1
i1
 cVi  c  Vi
D. Distribution of Summation
n


i1
Vi2
 2cVi  c
2


n
 Vi2
i1
n
 2c  Vi  nc2
i1
15