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3 IIMean A.Definition: the Mean Is the Sum of Scores Divided by the Number of Scores B.Formula
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1
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
2
Table 1. Taylor Manifest Anxiety Scores_______________________________
(1) (2)
X j f_______________________________
74 173 172 071 270 7 Mo = 6969 868 5
67 2 66 1 65 1_______________________________
n = 28_______________________________
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II Mean
A. Definition: the Mean Is the Sum of Scores Divided by the Number of Scores
B. Formula
1. X denotes the mean, X i denotes a score, and
n denotes the number of scores
nXXXX n
21
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C. Summation Operator, (Greek capitol sigma)
D. Mean Formula for a Frequency Distribution
1. k = number of class intervals
2. f j frequency of the jth class interval
3. X j midpoint of the jth class interval
n
n
ii XXXX
21
1
nXfXfXf
n
Xf
X kk
k
jjj
22111
5
Table 2. Taylor Manifest Anxiety Scores
_________________(1) (2) (3)
X j f
_________________74 17473 17372 00
71 2142
70 7490
69 8552
68 5340 67 2134 66 166
65 165
_________________n = 28
1,936_________________
f j X j
X f j X j
j1
k
n
1,936
2869.14
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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 medianis 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 + 1thscore from either end of the number line.
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B. Computational Examples
1. Determination of Mdn when n is odd
2. Determination of Mdn when n is even
Real limits of scoreMdn = 8
1 2 3 4 5 6 7 8 9 10 11 12
112 3 4 5 6 7 8 9 10 12
Mdn = 8.5
8
3. Determination of Mdn when n is even (a) or odd (b), and the frequency of the middle score value is greater than 1
1 2 3 4 5 6 7 8 9 10 11 12
Mdn = 8
a.
1 2 3 4 5 6 7 8 9 10 11 12
8.257.758.00 8.50
b.
Mdn = 7.75
7.50
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4. Determination of Mdn when n is even and the frequency of the middle score value is greater than 1
Mdn = 7.833
1 2 3 4 5 6 7 8 9 10 11 12
7.833 8.167 8.5007.5007.667 8 8.333
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C. Computation of Mdn for a Frequency Distribution1. Formula when scores are cumulated from below
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
Mdn X ll i
n / 2 fbfi
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2. Formula for the Mdn when scores are cumulated from above
Xul = real upper limit of class interval
containing the median
fa = number of scores above Xul
Mdn Xul i
n / 2 fafi
12
__________________________74 1 173 1 272 0 271 2 470 7 1169 8 17 1968 5 967 2 466 1 265 1 1__________________________ n = 28__________________________
Table 3. Taylor Manifest Anxiety Scores_____________________________
X j
f j Cum f up Cum f down
68.5 0.625 69.12
69.5 0.375 69.12
(2) (3) (4) (1)
i
bll f
fniXMdn
2
i
aul f
fniXMdn
2
8
922815.68
8
1122815.69
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IV Relative Merits of the Mean, Median, and Mode
V Location of the Mean, Median, and Modein a Distribution
MeanMedian
Mode
f
X
f
MeanMedian
Mode
X
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VI Mean of Two or More Means
A. Weighted Mean
VII Summation Rules
A. Sum of a Constant (c)
n
nnW nnn
XnXnXnX
21
2211
n
i
nccccc1
n terms
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B. Sum of a Variable (Vi)
C. Sum of the Product of a Constant and a Variable
cVi c Vi
i1
n
i1
n
D. Distribution of Summation
n
n
ii VVVV
211
2
11 1
221
2 22 ncVcVccVVn
ii
n
i
n
iii