Dispersion uwsb

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Dispersion uwsb

  1. 1. Measures of DispersionAre measures of scatter ( spread) about anaveragei.e. extent to which individual items varyMeasures of DispersionAbsolute Measures – measure value in sameunits – ageRelative Measures - % or coefficient of absolutemeasures
  2. 2. Measures of Dispersion1. Range2. Inter-quartile range3. Quartile deviation4. Mean deviation5. Standard deviation
  3. 3. A. 1. Range = Xmax – X min = L-SXmax – X min L-S2. Coefficient of Range = ---------------- = ------Xmax + X min L+S
  4. 4. Measures of DispersionQ1. Calculate range & co-efficient of range fromfollowing information480,562,570,322,435,497,675,732,375,482,791,820,275
  5. 5. B. Quartiles1.Inter quartile range = Q3 – Q12. Quartile deviationor semi inter quartile range = ( Q3 – Q1)/2a. In a normal distributionQ1 < Q2 < Q3Q2 = Mb.In a symmetrical distributionQ2 + Quartile Deviation = Q3Q2 - Quartile Deviation = Q1
  6. 6. 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
  7. 7. 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
  8. 8. Calculation of Quartile deviationN+1Q1=size of (---------) th item of the series43(N+1)Q3=size of ---------) th item of the series4
  9. 9. Q2. Calculate quartile deviation & its co-efficient for thedata given below168147 150 169 170 154 156 171 162 159 174 173 166 164 172
  10. 10. Q3. Compute quartile deviation & itscoefficient for 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
  11. 11. ProcedureCompute cumulative frequencyNFind out Q1 & Q3 classes by m (Q1)=---------43N& m(Q3)= ---------4
  12. 12. 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
  13. 13. After locating l1, l2 , f & c substitute values inl2-l1 3NQ3= l1+ --------- ( m-c) where m =---------f 43N/4 - CQ3 =l1+ --------- (l2-l1)f
  14. 14. 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 convertinto exclusive, class size equalcalculate cumulative frequencycalculate m (Q1)=N/4 &m(Q3)=3N/4 thobservationinter quartile range = (Q3-Q1)quartile deviation = (Q3-Q1)/2
  15. 15. 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 =-----------------
  16. 16. l2-l1 NQ1 = l1+--------------* (m-c) m=------------m 4l1- lower limit of Q1 class , l2= upper limit of Q1classf = frequency of Q1 class , c= cumulativefrequency before Q1 class
  17. 17. l2-l1 3NQ3 = l1+--------------* (m-c) m=-------------f 4l1- lower limit of Q3 class , l2= upper limit of Q3classf = frequency of Q3 class , c= cumulative frequencybefore Q3 class
  18. 18. Inter quartile range = (Q3-Q1)Quartile deviation = (Q3-Q1)/2Q3-q1Coefficient of quartile deiation = -----------------q3+q1
  19. 19. Mean Deviation = sum of absolute deviationsfrom an average divided by total number ofitemsCoefficient of Mean Deviation = meanDeviation / Mean
  20. 20. Σ f(x-a)mod Σ f dmodMean deviation = ------------- = ---------------Σ fx N
  21. 21. Q5A. calculate mean deviation & coefficient of mean forthe following two seriesA105 112 110 125 138 149 161 175 185 190B 22 24 26 28 30 32 34 40 44 50
  22. 22. Standard deviation of a series is the squareroot of the average of the squared deviationsfrom the mean ( Average – Arithmatic mean)
  23. 23. arithmetic mean of squares of deviationsΣ dx2Σ fdx2σ = √ (-------) = (--------)N NFor frequencies of a valueσCoefficient of Standard deviation = ------------------averageσCoefficient of variation = --------------- x 100
  24. 24. Q5Calculate standard deviation & coefficientof variationX 65 67 68 68 69 71 72 72
  25. 25. Q6.Calculate standard deviation & coefficient ofvariationX 95 100105115125130135140150160170f 5 8 12 15 35 40 30 20 10 10 10
  26. 26. 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
  27. 27. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsΣ dx Σ dx2Σ dxσ = √ (-------)2= √ ------- - (------------) 2N N NΣ fdx Σ fdx2Σ fdxσ = √ (-------)2= √ ------- - (------------) 2Σ f Σ f Σ f
  28. 28. 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 whichseries is more variable
  29. 29. A 1581601631651671701721751771811688 168.8B 1631581671701601801701751721751690 169dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
  30. 30. 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)2dx2A144 100 49 25 9 0 4 25 49 121 526
  31. 31. A 158 160163165167170172175177 181 1688 168.8B 163 158167170160180170175172 175 1690 169dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2dx2A 144 10049 25 9 0 4 25 49 121 526dxB -7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2
  32. 32. 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
  33. 33. Σ dx Σ dx2 Σ dx 526 12σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------)2N 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%
  34. 34. 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%
  35. 35. Σ fd’x Σ fd’x2Σ fd’xσ = √ (-------)2x i = [√ ------- - (------------) 2]xiΣ f Σ f Σ fi= class interval
  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. 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’x2Σ 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
  38. 38. 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
  39. 39. Standard deviation σ – positive square root ofarithmetic mean of squares of deviationsN1σ12+ N2σ22+ N3σ32+ …….Nnσn2σ12...n = √ --------------------------------------------N1 + N2+ N3 +……….. Nn
  40. 40. compute coefficient of variation & comment which factoryprofits are more consistentParticularsFactory A Factory BAverageprofits19.7 21Standarddeviation6.5 8.64

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