Measures of DispersionAre measures of scatter ( spread) about an averagei.e. extent to which individual items vary

Measures of Dispersion Absolute Measures – measure value in same units – ageRelative Measures - % or coefficient of absolute measures

Measures of Dispersion

1. Range2. Inter-quartile range 3. Quartile deviation 4. Mean deviation5. Standard deviation

A. 1. Range = Xmax – X min = L-S Xmax – X min L-S2. Coefficient of Range = ---------------- = ------ Xmax + X min L+S

Measures of DispersionQ1. Calculate range & co-efficient of range from following information 480,562,570,322,435,497,675,732,375,482,791,820,275

B. Quartiles1.Inter quartile range = Q3 – Q12. Quartile déviation or semi inter quartile range = ( Q3 – Q1)/2a. In a normal distributionQ1 < Q2 < Q3 Q2 = Mb. In a symmetrical distributionQ2 + Quartile Déviation = Q3 Q2 - Quartile Déviation = Q1

Q1 = first quartile or lower quartileQ2 = second / middle Quartile or medianQ3 = third quartile or upper quartile Q3 – Q1Coefficient of Quartile deviation = ----------- Q3 + Q1

Coefficient of Quartile DeviationDeviation by Quartiles =---------------------- x

100 Median

Calculation of Quartile deviation under continuous series

1. If inclusive class intervals , convert to exclusive class intervals

2. Size of class intervals should be equal throughout distribution

3. L2 of first class interval should be equal to L1 of next class interval

4. If mid values are given , it is necessary to determine class intervals

5. If it is open end type of frequency distribution , coefficient of variation is suitable measure

Calculation of Quartile deviation N+1Q1=size of (---------) th item of the

series 4 3(N+1)

Q3=size of ---------) th item of the series

4

Q2. Calculate quartile deviation & its co-efficient for the data given below

168147 150 169 170 154 156 171 162 159 174 173 166 164 172

Q3. Compute quartile deviation & its coefficient for following data

X 10 12 14 16 18 20 22 24 28 30 34 36 38

F 3 6 10 15 20 24 30 22 18 14 10 6 6

Soln. calculate cumulative frequencycalculate Q1=N+1/4 &Q3=3(N+1)/4 th observation

Procedure:Compute cumulative frequency NFind out Q1 & Q3 classes by m (Q1)=--------- 4 3N

& m(Q3)= --------- 4

After locating l1, l2 , f & c substitute values in l2-l1 NQ1= l1+ --------- ( m-c) where m =--------- f 4 N/4 - C Q1 =l1+ --------- (l2-l1) f l1= lower limit of quartile classl2 = upper limit of quartile class f =frequency of quartile classc =cumulative frequency before quartile classM = quartile position

After locating l1, l2 , f & c substitute values in l2-l1 3NQ3= l1+ --------- ( m-c) where m =--------- f 4 3N/4 - C Q3 =l1+ --------- (l2-l1) f

Q4. Compute quartile deviation & its s coefficient for marks of 215 student

Marks 0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100

Students 10 15 28 32 40 35 26 14 10 5

Soln. condition if class interval inclusive convert into exclusive, class size equal calculate cumulative frequencycalculate m (Q1)=N/4 &m(Q3)=3N/4 th observationinter quartile range = (Q3-Q1)quartile deviation = (Q3-Q1)/2

X 10 12 14 16 18 20 22 24 28 30 34 36 38

F 3 6 10 15 20 24 30 22 18 14 10 6 6

cf 3 9 19 34 54 78 108

130 148 162 172 178 184

Soln. calculate cumulative frequencycalculate Q1=N+1/4 &Q3=3(N+1)/4 th observation

inter quartile range = (Q3-Q1)quartile deviation = (Q3-Q1)/2 Q3-Q1coefficient of quartile deiation =

-----------------

Q3+Q1

l2-l1 N

Q1 = l1+--------------* (m-c) m= ------------

m 4l1- lower limit of Q1 class , l2= upper limit of Q1 classf = frequency of Q1 class , c= cumulative frequency before Q1 class

l2-l1 3NQ3 = l1+--------------* (m-c) m= ------------- f 4l1- lower limit of Q3 class , l2= upper limit of Q3 classf = frequency of Q3 class , c= cumulative frequency before Q3 class

Inter quartile range = (Q3-Q1)Quartile deviation = (Q3-Q1)/2 Q3-q1 Coefficient of quartile deiation = ----------------- q3+q1

Mean Deviation = sum of absolute deviations from an average divided by total number of itemsCoefficient of Mean Deviation = mean Deviation / Mean

Σ f(x-a)mod Σ f dmod Mean deviation = ------------- = --------------- Σ fx N

Q5A. calculate mean deviation & coefficient of mean for the following two series

A105 112 110 125 138 149 161 175 185 190

B 22 24 26 28 30 32 34 40 44 50

Standard deviation of a series is the square root of the average of the squared deviations from the mean ( Average – Arithmatic mean)

Standard deviation σ – positive square root of arithmetic mean of squares of deviations Σ dx2 Σ fdx2 σ = √ (-------) = (--------) N NFor frequencies of a value σ Coefficient of Standard deviation = ------------------

average

σ Coefficient of variation = --------------- x 100 average

Q5Calculate standard deviation & coefficient of variation

X 65 67 68 68 69 71 72 72

Q6.Calculate standard deviation & coefficient of variation

X 95 100

105

115

125

130

135

140

150

160

170

f 5 8 12 15 35 40 30 20 10 10 10

Q6.Calculate standard deviation & coefficient of variation

X 95 100 105 115 125 130 135 140 150 160 170

f 5 8 12 15 35 40 30 20 10 10 10

dx=(x-130)

-35 -30 -25 -15 -05 0 5 10 20 30 40

Standard deviation σ – positive square root of arithmetic mean of squares of deviations Σ dx Σ dx2 Σ dxσ = √ (-------)2 = √ ------- - (------------) 2

N N N

Σ fdx Σ fdx2 Σ fdx σ = √ (-------)2 = √ ------- - (------------) 2 Σ f Σ f Σ f

A 158 160 163 165 167 170 172 175 177 181

B 163 158 167 170 160 180 170 175 172 175

By using standard deviation find out which series is more variable

A 158

160

163

165

167

170

172

175

177

181

1688 168.8

B 163

158

167

170

160

180

170

175

172

175

1690 169

dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2

A 158

160

163

165

167

170

172

175

177

181

1688 168.8

B 163

158

167

170

160

180

170

175

172

175

1690 169

dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2

dx2A

144

100

49 25 9 0 4 25 49 121

526

A 158 160 163 165 167 170 172 175 177 181 1688 168.8

B 163 158 167 170 160 180 170 175 172 175 1690 169

dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2

dx2A 144 100 49 25 9 0 4 25 49 121 526

dxB -7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2

A 158 160

163

165

167

170

172

175

177

181 1688 168.8

B 163 158

167

170

160

180

170

175

172

175 1690 169

dxA

-12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2

dx2A

144 100

49 25 9 0 4 25 49 121 526

dxB

-7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2

dx2B

49 144

9 0 100

100

0 25 4 25 456

Σ dx Σ dx2 Σ dx 526 12σ = √ (-------)2 = √ ------- - (------------) 2 =√

[ ----------- - ( ------) 2 N N

10 10 =√ [ 52.6- 1.2*1.2] =√ 52.6-1.44 =√ 51.46 =7.2Coefficient of variation = σ / x bar = 7.2*100/

166.8 = 4.26%

Σ dx Σ dx2 Σ dxΣ dx Σ dx2 Σ dx 456 10σ = √ (-------)2 = √ ------- - (------------) 2 =√

[ ----------- - ( ------) 2 N N

10 10 =√ [ 45.6- 1] =√ 44.6 = 6.7

Coefficient of variation = σ / x bar = 6.7*100 / 169 = 3.96%

Σ fd’x Σ fd’x2 Σ fd’x σ = √ (-------)2 x i = [√ ------- - (------------) 2

]xi Σ f Σ f Σ fi= class interval

Σ fd’x Σ fd’x2 Σ fd’x σ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xi Σ f Σ f Σ fi= class interval

class 80-84

75-79

70-74

65-69

60-64

55-59

50-54

45-49

40-44

35-39

30-34

25-29

Mid v 82 77 72 67 62 57 52 47 42 37 32 27

frequency

1 1 1 4 4 7 6 6 6 3 0 1

dx=x-52 30 25 20 15 10 5 0 -5 -10 -15 -20 -25

d’x=(x-52/5)

6 5 4 3 2 1 0 -1 -2 -3 -4 -5

fd’x 6 5 4 12 8 7 0 -6 -12 -9 0 -5

d’x2 36 25 16 9 4 1 0 1 4 9 16 25

fd’x2 36 25 16 36 16 7 0 6 24 27 0 25

σ = i* σA = 5* σ Σ dx Σ fd’x2 Σ fd’x 218

10σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ --------- - ( ------) 2

N N N 40 40

=√ [ 5.45- 0.25 ] =√ 5.20 = 2.32σ = i* σA = 5* σ = 5*2.32 = 11.60

Standard deviation σ – positive square root of arithmetic mean of squares of deviationsPropertiesStandard deviation σ is independent of change of origin but not of scaleIf dx = x-A σx=σd

x-A If d’x =------ σx= i.σd

i

Standard deviation σ – positive square root of arithmetic mean of squares of deviations N1σ1

2 + N2σ22 + N3σ3

2 + …….Nnσn2

σ12...n = √ -------------------------------------------- N1 + N2+ N3 +……….. Nn

compute coefficient of variation & comment which factory profits are more consistent

Particulars

Factory A Factory B

Average profits

19.7 21

Standard deviation

6.5 8.64

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