Measures of dispersion unitedworld school of business

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Measures of dispersion unitedworld school of business

  1. 1. Designation: PGDM StudentBatch: 2012-2014.Specialization: Major – MarketingMinor – FinanceFacebook: http://on.fb.me/106D9UMBlog: http://bit.ly/13yMJAtLinkedIn: http://lnkd.in/jFgcBV
  2. 2. Measures of DispersionAre measures of scatter ( spread) about anaveragei.e. extent to which individual items varyMeasures of DispersionAbsolute Measures – measure value in same units– ageRelative Measures - % or coefficient of absolutemeasures
  3. 3. Measures of Dispersion1. Range2. Inter-quartile range3. Quartile deviation4. Mean deviation5. Standard deviation
  4. 4. A. 1. Range = Xmax – X min = L-SXmax – X min L-S2. Coefficient of Range = ---------------- = ------Xmax + X min L+S
  5. 5. Measures of DispersionQ1. Calculate range & co-efficient of range fromfollowing information480,562,570,322,435,497,675,732,375,482,791,820,275
  6. 6. B. Quartiles1.Inter quartile range = Q3 – Q12. Quartile déviationor semi inter quartile range = ( Q3 – Q1)/2a. In a normal distributionQ1 < Q2 < Q3Q2 = Mb. In a symmetrical distributionQ2 + Quartile Déviation = Q3Q2 - Quartile Déviation = Q1
  7. 7. Q1 = first quartile or lower quartileQ2 = second / middle Quartile or medianQ3 = third quartile or upper quartileQ3 – Q1Coefficient of Quartile deviation = -----------Q3 + Q1Coefficient of Quartile DeviationDeviation by Quartiles =---------------------- x 100Median
  8. 8. Calculation of Quartile deviation undercontinuous series1. If inclusive class intervals , convert toexclusive class intervals2. Size of class intervals should be equalthroughout distribution3. L2 of first class interval should be equal toL1 of next class interval4. If mid values are given , it is necessary todetermine class intervals5. If it is open end type of frequencydistribution , coefficient of variation issuitable measure
  9. 9. Calculation of Quartile deviationN+1Q1=size of (---------) th item of the series43(N+1)Q3=size of ---------) th item of the series4
  10. 10. Q2. Calculate quartile deviation & its co-efficientfor the data given below168147 150 169 170 154 156 171 162 159 174 173 166 164 172
  11. 11. Q3. Compute quartile deviation & its coefficientfor following dataX 10 12 14 16 18 20 22 24 28 30 34 36 38F 3 6 10 15 20 24 30 22 18 14 10 6 6Soln. calculate cumulative frequencycalculate Q1=N+1/4 &Q3=3(N+1)/4 th observation
  12. 12. Procedure:Compute cumulative frequencyNFind out Q1 & Q3 classes by m (Q1)=---------43N& m(Q3)= ---------4
  13. 13. After locating l1, l2 , f & c substitute values inl2-l1 NQ1= l1+ --------- ( m-c) where m =---------f 4N/4 - CQ1 =l1+ --------- (l2-l1)fl1= lower limit of quartile classl2 = upper limit of quartile classf =frequency of quartile classc =cumulative frequency before quartile classM = quartile position
  14. 14. After locating l1, l2 , f & c substitute values inl2-l1 3NQ3= l1+ --------- ( m-c) where m =---------f 43N/4 - CQ3 =l1+ --------- (l2-l1)f
  15. 15. Q4. Compute quartile deviation & itscoefficient for marks of 215 studentsMarks 0-1010-2020-3030-4040-5050-6060-7070-8080-9090-100Students 10 15 28 32 40 35 26 14 10 5Soln. condition if class interval inclusive convert intoexclusive, class size equalcalculate cumulative frequencycalculate m (Q1)=N/4 &m(Q3)=3N/4 th observationinter quartile range = (Q3-Q1)quartile deviation = (Q3-Q1)/2
  16. 16. X 10 12 14 16 18 20 22 24 28 30 34 36 38F 3 6 10 15 20 24 30 22 18 14 10 6 6cf 3 9 19 34 54 78 108130 148 162 172 178 184Soln. calculate cumulative frequencycalculate Q1=N+1/4 &Q3=3(N+1)/4 th observationinter quartile range = (Q3-Q1)quartile deviation = (Q3-Q1)/2Q3-Q1coefficient of quartile deiation = -----------------Q3+Q1
  17. 17. l2-l1 NQ1 = l1+--------------* (m-c) m= ------------m 4l1- lower limit of Q1 class , l2= upper limit of Q1 classf = frequency of Q1 class , c= cumulative frequencybefore Q1 class
  18. 18. 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 frequencybefore Q3 class
  19. 19. Inter quartile range = (Q3-Q1)Quartile deviation = (Q3-Q1)/2Q3-q1Coefficient of quartile deiation = -----------------q3+q1
  20. 20. Mean Deviation = sum of absolute deviationsfrom an average divided by total number ofitemsCoefficient of Mean Deviation = meanDeviation / Mean
  21. 21. Σ f(x-a)mod Σ f dmodMean deviation = ------------- = ---------------Σ fx N
  22. 22. Q5A. calculate mean deviation & coefficient ofmean for the following two seriesA105 112 110 125 138 149 161 175 185 190B 22 24 26 28 30 32 34 40 44 50
  23. 23. Standard deviation of a series is the squareroot of the average of the squared deviationsfrom the mean ( Average – Arithmatic mean)
  24. 24. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsΣ dx2 Σ fdx2σ = √ (-------) = (--------)N NFor frequencies of a valueσCoefficient of Standard deviation = ------------------averageσCoefficient of variation = --------------- x 100average
  25. 25. Q5Calculate standard deviation & coefficient ofvariationX 65 67 68 68 69 71 72 72
  26. 26. Q6.Calculate standard deviation & coefficient ofvariationX 95 100105115125130135140150160170f 5 8 12 15 35 40 30 20 10 10 10
  27. 27. Q6.Calculate standard deviation & coefficient ofvariationX 95 100 105 115 125 130 135 140 150 160 170f 5 8 12 15 35 40 30 20 10 10 10dx=(x-130)-35 -30 -25 -15 -05 0 5 10 20 30 40
  28. 28. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsΣ dx Σ dx2 Σ dxσ = √ (-------)2 = √ ------- - (------------) 2N N NΣ fdx Σ fdx2 Σ fdxσ = √ (-------)2 = √ ------- - (------------) 2Σ f Σ f Σ f
  29. 29. A 158 160 163 165 167 170 172 175 177 181B 163 158 167 170 160 180 170 175 172 175By using standard deviation find out which seriesis more variable
  30. 30. A 1581601631651671701721751771811688 168.8B 1631581671701601801701751721751690 169dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
  31. 31. A 1581601631651671701721751771811688 168.8B 1631581671701601801701751721751690 169dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2dx2A14410049 25 9 0 4 25 49 121526
  32. 32. A 158 160 163 165 167 170 172 175 177 181 1688 168.8B 163 158 167 170 160 180 170 175 172 175 1690 169dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2dx2A 144 100 49 25 9 0 4 25 49 121 526dxB -7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2
  33. 33. A 158 160163165167170172175177181 1688 168.8B 163 158167170160180170175172175 1690 169dxA-12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2dx2A144 10049 25 9 0 4 25 49 121 526dxB-7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2dx2B49 1449 0 1001000 25 4 25 456
  34. 34. Σ dx Σ dx2 Σ dx 526 12σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------)2N N 1010=√ [ 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%
  35. 35. Σ dx Σ dx2 Σ dxΣ dx Σ dx2 Σ dx 456 10σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------) 2N N 1010=√ [ 45.6- 1] =√ 44.6 = 6.7Coefficient of variation = σ / x bar = 6.7*100 / 169 = 3.96%
  36. 36. Σ fd’x Σ fd’x2 Σ fd’xσ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xiΣ f Σ f Σ fi= class interval
  37. 37. Σ fd’x Σ fd’x2 Σ fd’xσ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xiΣ f Σ f Σ fi= class interval
  38. 38. class 80-8475-7970-7465-6960-6455-5950-5445-4940-4435-3930-3425-29Mid v 82 77 72 67 62 57 52 47 42 37 32 27frequency1 1 1 4 4 7 6 6 6 3 0 1dx=x-52 30 25 20 15 10 5 0 -5 -10 -15 -20 -25d’x=(x-52/5)6 5 4 3 2 1 0 -1 -2 -3 -4 -5fd’x 6 5 4 12 8 7 0 -6 -12 -9 0 -5d’x2 36 25 16 9 4 1 0 1 4 9 16 25fd’x2 36 25 16 36 16 7 0 6 24 27 0 25σ = i* σA = 5* σΣ dx Σ fd’x2 Σ fd’x 218 10σ = √ (-------)2 = √ ------- - (------------) 2=√ [ --------- - ( ------) 2N N N 40 40=√ [ 5.45- 0.25 ] =√ 5.20 = 2.32σ = i* σA = 5* σ = 5*2.32 = 11.60
  39. 39. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsPropertiesStandard deviation σ is independent of change oforigin but not of scaleIf dx = x-A σx=σdx-AIf d’x =------ σx= i.σdi
  40. 40. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsN1σ12 + N2σ22 + N3σ32 + …….Nnσn2σ12...n = √ --------------------------------------------N1 + N2+ N3 +……….. Nn
  41. 41. compute coefficient of variation & comment whichfactory profits are more consistentParticularsFactory A Factory BAverageprofits19.7 21Standarddeviation6.5 8.64
  42. 42. 907/A Uvarshad,GandhinagarHighway,Ahmedabad – 382422.Ahmedabad KolkataInfinity Benchmark,10th Floor, Plot G1,Block EP & GP,Sector V, Salt-Lake,Kolkata – 700091.MumbaiGoldline Business CentreLinkway Estate,Next to Chincholi FireBrigade, Malad (West),Mumbai – 400 064.

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