1

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

6

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

8

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.

12

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