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# Measures of dispersion - united world school of business

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Measures of dispersion - Unitedworld School of Business

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### Measures of dispersion - united world school of business

1. 1. Measures of Dispersion Are measures of scatter ( spread) about an average i.e. extent to which individual items vary Measures of Dispersion Absolute Measures – measure value in same units – age Relative Measures - % or coefficient of absolute measures
2. 2. Measures of Dispersion 1. Range 2. Inter-quartile range 3. Quartile deviation 4. Mean deviation 5. Standard deviation
3. 3. A. 1. Range = Xmax – X min = L-S Xmax – X min L-S 2. Coefficient of Range = ---------------- = ------ Xmax + X min L+S
4. 4. Measures of Dispersion Q1. Calculate range & co-efficient of range from following information 480,562,570,322,435,497,675,732,375,482,791,820,275
5. 5. B. Quartiles 1.Inter quartile range = Q3 – Q1 2. Quartile déviation or semi inter quartile range = ( Q3 – Q1)/2 a. In a normal distribution Q1 < Q2 < Q3 Q2 = M b. In a symmetrical distribution Q2 + Quartile Déviation = Q3 Q2 - Quartile Déviation = Q1
6. 6. Q1 = first quartile or lower quartile Q2 = second / middle Quartile or median Q3 = third quartile or upper quartile Q3 – Q1 Coefficient of Quartile deviation = ----------- Q3 + Q1 Coefficient of Quartile Deviation Deviation by Quartiles =---------------------- x 100 Median
7. 7. 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
8. 8. Calculation of Quartile deviation N+1 Q1=size of (---------) th item of the series 4 3(N+1) Q3=size of ---------) th item of the series 4
9. 9. Q2. Calculate quartile deviation & its co-efficient for the data given below 168 147 150 169 170 154 156 171 162 159 174 173 166 164 172
10. 10. 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 frequency calculate Q1=N+1/4 &Q3=3(N+1)/4 th observation
11. 11. Procedure: Compute cumulative frequency N Find out Q1 & Q3 classes by m (Q1)=--------- 4 3N & m(Q3)= --------- 4
12. 12. After locating l1, l2 , f & c substitute values in l2-l1 N Q1= l1+ --------- ( m-c) where m =--------- f 4 N/4 - C Q1 =l1+ --------- (l2-l1) f l1= lower limit of quartile class l2 = upper limit of quartile class f =frequency of quartile class c =cumulative frequency before quartile class M = quartile position
13. 13. After locating l1, l2 , f & c substitute values in l2-l1 3N Q3= l1+ --------- ( m-c) where m =--------- f 4 3N/4 - C Q3 =l1+ --------- (l2-l1) f
14. 14. 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 frequency calculate m (Q1)=N/4 &m(Q3)=3N/4 th observation inter quartile range = (Q3-Q1) quartile deviation = (Q3-Q1)/2
15. 15. 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 10 8 130 148 162 172 178 184 Soln. calculate cumulative frequency calculate Q1=N+1/4 &Q3=3(N+1)/4 th observation inter quartile range = (Q3-Q1) quartile deviation = (Q3-Q1)/2 Q3-Q1 coefficient of quartile deiation = ----------------- Q3+Q1
16. 16. l2-l1 N Q1 = l1+--------------* (m-c) m= ------------ m 4 l1- lower limit of Q1 class , l2= upper limit of Q1 class f = frequency of Q1 class , c= cumulative frequency before Q1 class
17. 17. l2-l1 3N Q3 = l1+--------------* (m-c) m= ------------- f 4 l1- lower limit of Q3 class , l2= upper limit of Q3 class f = frequency of Q3 class , c= cumulative frequency before Q3 class
18. 18. Inter quartile range = (Q3-Q1) Quartile deviation = (Q3-Q1)/2 Q3-q1 Coefficient of quartile deiation = --------------- -- q3+q1
19. 19. Mean Deviation = sum of absolute deviations from an average divided by total number of items Coefficient of Mean Deviation = mean Deviation / Mean
20. 20. Σ f(x-a)mod Σ f dmod Mean deviation = ------------- = --------------- Σ fx N
21. 21. Q5A. calculate mean deviation & coefficient of mean for the following two series A 105 112 110 125 138 149 161 175 185 190 B 22 24 26 28 30 32 34 40 44 50
22. 22. Standard deviation of a series is the square root of the average of the squared deviations from the mean ( Average – Arithmatic mean)
23. 23. Standard deviation σ – positive square root of arithmetic mean of squares of deviations Σ dx2 Σ fdx2 σ = √ (-------) = (--------) N N For frequencies of a value σ Coefficient of Standard deviation = ------------------ average σ Coefficient of variation = --------------- x 100 average
24. 24. Q5Calculate standard deviation & coefficient of variation X 65 67 68 68 69 71 72 72
25. 25. Q6.Calculate standard deviation & coefficient of variation X 95 10 0 10 5 11 5 12 5 13 0 13 5 14 0 15 0 16 0 170 f 5 8 12 15 35 40 30 20 10 10 10
26. 26. 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
27. 27. Standard deviation σ – positive square root of arithmetic mean of squares of deviations Σ dx Σ dx2 Σ dx σ = √ (-------)2 = √ ------- - (------------) 2 N N N Σ fdx Σ fdx2 Σ fdx σ = √ (-------)2 = √ ------- - (------------) 2 Σ f Σ f Σ f
28. 28. 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
29. 29. A 15 8 16 0 16 3 16 5 16 7 17 0 17 2 17 5 17 7 18 1 1688 168.8 B 16 3 15 8 16 7 17 0 16 0 18 0 17 0 17 5 17 2 17 5 1690 169 dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
30. 30. A 15 8 16 0 16 3 16 5 16 7 17 0 17 2 17 5 17 7 18 1 1688 168.8 B 16 3 15 8 16 7 17 0 16 0 18 0 17 0 17 5 17 2 17 5 1690 169 dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2 dx2 A 14 4 10 0 49 25 9 0 4 25 49 12 1 526
31. 31. 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
32. 32. A 158 16 0 16 3 16 5 16 7 17 0 17 2 17 5 17 7 181 1688 168.8 B 163 15 8 16 7 17 0 16 0 18 0 17 0 17 5 17 2 175 1690 169 dx A -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2 dx2 A 144 10 0 49 25 9 0 4 25 49 121 526 dx B -7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2 dx2 B 49 14 4 9 0 10 0 10 0 0 25 4 25 456
33. 33. Σ dx Σ dx2 Σ dx 526 12 σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------) 2 N N 10 10 =√ [ 52.6- 1.2*1.2] =√ 52.6-1.44 =√ 51.46 =7.2 Coefficient of variation = σ / x bar = 7.2*100/ 166.8 = 4.26%
34. 34. Σ dx Σ dx2 Σ dx Σ dx Σ dx2 Σ dx 456 10 σ = √ (-------)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%
35. 35. Σ fd’x Σ fd’x2 Σ fd’x σ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xi Σ f Σ f Σ f i= class interval
36. 36. Σ fd’x Σ fd’x2 Σ fd’x σ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xi Σ f Σ f Σ f i= class interval
37. 37. 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 frequenc y 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 10 σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ --------- - ( ------) 2 N N N 40 40 =√ [ 5.45- 0.25 ] =√ 5.20 = 2.32 σ = i* σA = 5* σ = 5*2.32 = 11.60
38. 38. Standard deviation σ – positive square root of arithmetic mean of squares of deviations Properties Standard deviation σ is independent of change of origin but not of scale If dx = x-A σx=σd x-A If d’x =------ σx= i.σd i
39. 39. Standard deviation σ – positive square root of arithmetic mean of squares of deviations N1σ1 2 + N2σ2 2 + N3σ3 2 + …….Nnσn 2 σ12...n = √ -------------------------------------------- N1 + N2+ N3 +……….. Nn
40. 40. compute coefficient of variation & comment which factory profits are more consistent Particular s Factory A Factory B Average profits 19.7 21 Standard deviation 6.5 8.64
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