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Chapter 4
Measures of Dispersion, Skewness and Kurtosis
I Range (R)
A. Noninclusive Range
R X j(largest score) X j(smallest score)
B. Inclusive Range
R Xul(largest score) X ll(smallest score)
2
II Semi-Interquartile Range (Q)
A. Q (Q3 Q1) / 2
1. Third quartile (Q3)
2. First quartile (Q1)
i
bll f
fniXQ
433
i
bll f
fniXQ
41
3
74 173 172 071 270 7 2469 8 1768 5 967 2 466 1 265 1 1
n = 28
Table 1. Taylor Manifest Anxiety Score
X j
f j Cum f up
69.51
28(3 / 4) 17
7
70.071
67.51
28 / 4) 4
5
68.100
Q
70.071 68.100
21.97
(1) (2) (3)
i
bll f
fniXQ
433
i
bll f
fniXQ
41
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III Another Median-like Statistic
A. Percentile Point (P%)
where PR denotes a percentile rank
1. P.25 Q1
2. P.50 Q2 median
3. P.75 Q3
i
bRll f
fPniXP
100/%
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IV Standard Deviation
A. Sample Standard Deviation (S)
B. Population Standard Deviation () S
( X i X )2
i1
n
n
( X i )2
i1
n
n
where denotes the population mean
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C. Standard Deviation Formula for Data in a Frequency Distribution
S
f j ( X j X )2
j1
k
n
1. fj denotes the frequency in the jth class
interval; Xj denotes the midpoint of the
jth class interval.
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74 1 74 1(23.5918)73 1 73 1(14.8776)72 0 0 0(8.1633)71 2 142 2(3.4490)70 7 490 7(0.7347)69 8 552 8(0.0204)68 5 340 5(1.3061)67 2 134 2(4.5918)66 1 66 1(9.8776)65 1 65 1(17.1633)
n = 28 1,936 93.4286
Table 2. Taylor Manifest Anxiety Scores
X j
f j f j X j
f j ( X j X )2
S
f j ( X j X )2
j1
k
n
93.4286
281.827
1936
2869.14286
X
f j X jj1
k
n (1) (2) (3) (4)
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V Index of Dispersion (D)
A. D
DP
DPmax
1. DP = no. of distinguishable pairs of observationsin c = 2 to k categories
2. DPmax max. no. of distinguishable pairs of
observations in c categories
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3. Example with c = 2 categories: category A represents one man (a1); category B represents five women (b1, . . . , b5)
B. Range of D is 0–1
1. D = 0 represents no dispersion (no distinguishable pairs); all n observations are in the same category
2. D = 1 represents maximum dispersion (observations are distributed equally over the c categories.
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4. (a) Observed data; (b) Example of maximum dispersion
a1 a1b1
a2a2
b2
b3 a3b4
b1b2
b3
a. b.
a1
a1b1
a2
b2b3
a3
b4
b1 b2b3
Category ACategory BCategory A Category B
b5
D
DP
DPmax
5
9.56
a1b1 a1b2 a1b3 a1b4 a1b5
DP
a1b1 a1b2 a1b3 a2b1 a2b2 a2b3 a3b1 a3b2 a3b3
DPmax
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5. (a) Observed data; (b) Example of maximum dispersion
D
DP
DPmax
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9.89
a1 a1b1
a2a2
b2
b3 a3b4
b1b2
b3
a. b.
a1a1
b1 a2a2
b2b3
a3b4
b1 b2b3
Category ACategory BCategory A Category B
a1b1 a1b2 a1b3 a1b4 a2b1 a2b2 a2b3 a2b4
DP
a1b1 a1b2 a1b3 a2b1 a2b2 a2b3 a3b1 a3b2 a3b3
DPmax
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C. Alternative Computational Formula for D
c = number of categories
n = total number of observations
nj = number of observations in the jth category
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1
22
cn
nnc
D
c
jj
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D. Computational Example with c = 2 Categories
a1 b1
a2
b2
b3 b4
a1b1
a2
b2b3 b4
CategoryBCategory A
2 (6)2 (2)2 (4)2
(6)2(2 1).89
12
1
22
cn
nnc
D
c
jj
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E. Computational Example with c = 5 Categories
Table 3. Admission Data for Students
Applied for Race Admission (AA) Admitted (A)
n % n %
White 268 82.2 179 78.9Black 36 11.0 29 12.8Mex/Amer. 16 4.9 18 7.9Other 3 0.9 1 0.4Unknown 3 0.9 0 0
n = 326 n = 227
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DAA 5 (326)2 (268)2 (36)2 (16)2 (3)2 (3)2
(326)2(5 1).39
DA 5 (227)2 (179)2 (29)2 (18)2 (1)2 (0)2
(227)2(5 1).44
1. Dispersion for students admitted is greater than that for students who applied for admission.
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1
22
cn
nnc
D
c
jj
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VI Relative Merits of the Four Measures of Dispersion
VII Minimum and Maximum Values of S
A. Maximum Value of S
1. Example using the Taylor Manifest Anxiety
data in Table 2
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B. Minimum Value of S for Data in Table 2
Smin R
2n
10
2(28)1.3
Smax
R
2
10
25.0
2. For these data, R = 74.5 – 64.5 = 10 and
n = 28.
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VIII Dispersion and the Normal Distribution
Xf( )
X
ŠMdn Q Mdn Q+
50%
68.27%
100%R
MdnXŠ
ŠSŠX
Š+X S
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IX Detecting Outliers
A. Two Criteria Based on the Mean and Median(Taylor Manifest Data from Tables 1 & 2)
1. X 2.5S
2. Mdn 2(Q3 Q1)
69.1432.5(1.827) 73.7 to 64.6
69.125 2(70.071 68.100) 73.1 to 65.2
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B. Criterion Based on a Box Plot
1. Left whisker computation
Q1 1.5(Q3 Q1)
68.100 1.5(70.071 68.100) 65.1
2. Right whisker computation
Q3 1.5(Q3 Q1)
70.0711.5(70.071 68.100) 73.0
64 66 68 70 72 74
*
Taylor Manifest Anxiety
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C. Box Plot
64 66 68 70 72 74
*
Taylor Manifest Anxiety
1. An asterisk (*) identifies one outlier
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X Skewness (Sk)
Sk
( X i X )3
i1
n
nS3
A. Interpretation of Sk
Sk > 0, positively skewed
Sk = 0, symmetrical
Sk < 0, negatively skewed
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B. Computational Example
Table 4. Quiz Scores
2 –4 16 –64 256 4 –2 4 –8 16 7 1 1 1 1 8 2 4 8 16 9 3 9 27 81
30 0 34 –36 370__________________________________________
X i
X i X ( X i X )2
( X i X )3
( X i X )4
X 30 / 5 6
(1) (2) (3) (4) (5)
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1. Standard deviation for data in Table 4
S
( X i X )2
i1
n
n
34
52.61
2. Skewness for data in Table 4
Sk
( X i X )3
i1
n
nS3
Š36
5(2.61)3
Š0.40
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XI Kurtosis (Kur)
Sk
( X i X )4
i1
n
nS4
3
A. Interpretation of Kur
Kur < 0, platykurtic
Kur = 0, mesokurtic
Kur < 0, leptokurtic
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B. Computational Example for Data in Table 4
Kur
( X i X )4
i1
n
nS4
3
4.13)61.2(
5370
4