Measures of dispersion (ranges, quartile deviation, variance and standard deviation) by abdul ghafoor sajjad
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Measures of Dispersion Ranges, Quartile Deviation, Variance Standard Deviation
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Measures of dispersion (ranges, quartile deviation, variance and standard deviation) by abdul ghafoor sajjad
- 3. RBST 6012 ADVANCED BIOSTATISTICS AND DATA ANALYSIS SECOND SEMESTER 03CH Dr Abdul Ghafoor Sajjad 3
- 4. PREVIOUS LECTURE Descriptive Statistics Measures of Central tendency (Median , Mode) Dr Abdul Ghafoor Sajjad 4
- 5. LECTURE NO 03 Measures of Dispersion (Ranges, Quartile Deviation, Variance and Standard Deviation) Dr Abdul Ghafoor Sajjad 5
- 6. PreparedBy: Dr Abdul Ghafoor Sajjad Associate Professor /HOD Department of Rehabilitation Sciences Shifa Tameer e Millat university Dr Abdul Ghafoor Sajjad 6
- 14. MEASURES OF DISPERSION Dr Abdul Ghafoor Sajjad 14 Dispersion is the state of getting dispersed or spread. Statistical dispersion means the extent to which a numerical data is likely to vary about an average value. In other words, dispersion helps to understand the distribution of the data.
- 17. TYPES OF DISPERSION There are two main types of dispersion methods in statistics which are: 1. Absolute Measure of Dispersion 2. Relative Measure of Dispersion Dr Abdul Ghafoor Sajjad 17
- 19. TYPES OF DISPERSION Absolute Measure of Dispersion 1. An absolute measure of dispersion contains the same unit as the original data set. 2. Absolute dispersion method expresses the variations in terms of the average of deviations of observations like standard or means deviations. 3. It includes range, quartile deviation, Mean Deviation and Standard Deviation. Dr Abdul Ghafoor Sajjad 19
- 20. Dr Abdul Ghafoor Sajjad 20 TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation
- 22. TYPES OF DISPERSION Relative Measure of Dispersion The relative measures of dispersion are used to compare the distribution of two or more data sets. This measure compares values without units. Common relative dispersion methods include: 1. Co-efficient of Range/Dispersion 2. Co-efficient of Quartile Deviation 3. Co-efficient of Mean Deviation 4. Co-efficient of Standard Deviation Dr Abdul Ghafoor Sajjad 22
- 23. Dr Abdul Ghafoor Sajjad 23 TYPES OF DISPERSION Absolute Relative Co-efficient of dispersion Co-efficient of Quartile Range Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation
- 24. Dr Abdul Ghafoor Sajjad 24 TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation Relative Co-efficient of dispersion Co-efficient of Quartile Range Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation
- 26. MEASURESOF DISPERSION - RANGE Dr Abdul Ghafoor Sajjad 26
- 27. MEASURESOF DISPERSION - RANGE 45 32 37 46 39 36 41 48 36 Dr Abdul Ghafoor Sajjad 27
- 28. Calculatethe RangeofFlexion ROMofevery visit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 28
- 29. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatethe Rangeof ExtensionROM ofeveryvisit Dr Abdul Ghafoor Sajjad 29
- 30. TYPES OFDISPERSION Co-efficient of Range/Dispersion TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation Relative Co-efficient of dispersion Co-efficient of Quartile Deviatione Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation Dr Abdul Ghafoor Sajjad 30
- 31. MEASURESOF DISPERSION Co-efficientof Range/Dispersion 45 32 37 46 39 36 41 48 36 Dr Abdul Ghafoor Sajjad 31
- 32. CalculatetheCo- efficientof Dispersionof FlexionROMof everyvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 32
- 33. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 CalculatetheCo- efficientof Dispersionof ExtensionROM ofeveryvisit Dr Abdul Ghafoor Sajjad 33
- 35. MEASURESOF DISPERSION QUARTILEDEVIATION • The quartile deviation is half of the distance between the third and the first quartile. • Also Called Semi Interquartile Range (SIQR) • The quartile deviation has an attractive feature that the range “Median ± Q.D” • Median ± Q.D contains approximately 50% of the data Dr Abdul Ghafoor Sajjad 35
- 37. MEASURESOF DISPERSION QUARTILEDEVIATION 1 2 3 4 5 6 7 8 9 45 32 37 46 39 36 41 48 36 32 36 36 37 39 41 45 46 48 Q1 Q3 Dr Abdul Ghafoor Sajjad 37
- 39. Calculatethe Quartile Deviation& Writethe “Median±Q.D” ofFlexionROMof everyvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 39
- 40. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatethe Quartile Deviation& Writethe “Median±Q.D” ofExtension ROMofevery visit Dr Abdul Ghafoor Sajjad 40
- 41. TYPES OFDISPERSION Co-efficient of Quartile Deviation TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation Relative Co-efficient of dispersion Co-efficient of Quartile Range Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation Dr Abdul Ghafoor Sajjad 41
- 43. MEASURESOF DISPERSION Co-efficientof QuartileDeviation 1 2 3 4 5 6 7 8 9 45 32 37 46 39 36 41 48 36 32 36 36 37 39 41 45 46 48 Q1 Q3 Dr Abdul Ghafoor Sajjad 43
- 45. CalculatetheCo- efficientof Quartile Deviation ofFlexionROMof everyvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 45
- 46. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 CalculatetheCo- efficientof Quartile Deviationof ExtensionROM ofeveryvisit Dr Abdul Ghafoor Sajjad 46
- 48. MEASURESOF DISPERSION MEAN DEVIATION The arithmetic mean of the absolute deviations of the observations from a measure of central tendency is known as the mean deviation (also called mean absolute deviation). Dr Abdul Ghafoor Sajjad 48
- 49. MEASURESOF DISPERSION MEAN DEVIATION 1. For Sample Data M.D = ∑[xi - x̄] n 2. For Population Data M.D = ∑[xi - u] N Dr Abdul Ghafoor Sajjad 49
- 50. MEASURESOF DISPERSION MEAN DEVIATION 1 2 3 4 5 6 7 8 9 45 32 37 46 39 36 41 48 36 32 36 36 37 39 41 45 46 48 Dr Abdul Ghafoor Sajjad 50
- 51. MEASURESOF DISPERSION MEAN DEVIATION xi Xi - x̄ [Xi - x̄] 32 -8 8 36 -4 4 36 -4 4 37 -3 3 39 -1 1 41 1 1 45 5 5 46 6 6 48 8 8 ∑360 ∑0 ∑40 Dr Abdul Ghafoor Sajjad 51
- 52. MEASURESOF DISPERSION MEAN DEVIATION M.D = ∑[xi - x̄] = 40 = 4.4 n 9 M.D (from mean) = 4.4 Dr Abdul Ghafoor Sajjad 52
- 53. MEASURESOF DISPERSION MEAN DEVIATION xi Xi - x̄ [Xi - x̄] Xi - median 32 -8 8 7 36 -4 4 3 36 -4 4 3 37 -3 3 2 39 -1 1 0 41 1 1 2 45 5 5 6 46 6 6 7 48 8 8 9 ∑360 ∑0 ∑40 ∑39 Dr Abdul Ghafoor Sajjad 53
- 54. MEASURESOF DISPERSION MEAN DEVIATION M.D = ∑[xi - median] = 39 = 4.3 n 9 M.D (From Median) = 4.3 Dr Abdul Ghafoor Sajjad 54
- 55. Calculatethe MeanDeviation frommeanfrom theFlexionROM ofeveryvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 55
- 56. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatethe MeanDeviation frommeanfrom theExtension ROMofevery visit Dr Abdul Ghafoor Sajjad 56
- 57. MEASURESOF DISPERSION MEAN DEVIATION 1. For Group Data M.D = ∑ fi [xi - x̄] n Dr Abdul Ghafoor Sajjad 57
- 58. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) Marks 30-39 40-49 50-59 60-69 70-79 80-89 90-99 No of Students 8 87 190 304 211 85 20 Dr Abdul Ghafoor Sajjad 58
- 59. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) S. No Marks xi Frequency (No of Students) fixi 1 30-39 34.5 8 276 2 40-49 44.5 87 3871.5 3 50-59 55.5 190 10545 4 60-69 66.5 304 20216 5 70-79 77.5 211 16352.5 6 80-89 88.5 85 7522.5 7 90-99 99.5 20 1990 ∑905 ∑60773.5 Dr Abdul Ghafoor Sajjad 59
- 60. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) S. No Marks xi Freque ncy (No of Student s) fixi 1 30-39 34.5 8 276 2 40-49 44.5 87 3871.5 3 50-59 55.5 190 10545 4 60-69 66.5 304 20216 5 70-79 77.5 211 16352.5 6 80-89 88.5 85 7522.5 7 90-99 99.5 20 1990 ∑905 ∑60773. 5 Mean = ∑fixi n Mean = 60773.5/905 Mean = 61.15 Dr Abdul Ghafoor Sajjad 60
- 61. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) S. No Marks xi Frequency (No of Students) fixi Xi - x̄ F[Xi - x̄] 1 30-39 34.5 8 276 - 25.65 205.2 2 40-49 44.5 87 3871.5 - 16.65 1448.55 3 50-59 55.5 190 10545 - 5.65 1056.4 4 60-69 66.5 304 20216 5.35 1626.4 5 70-79 77.5 211 16352.5 16.35 3449.85 6 80-89 88.5 85 7522.5 27.35 2324.75 7 90-99 99.5 20 1990 38.35 767 ∑905 ∑60773.5 ∑10878.15 Dr Abdul Ghafoor Sajjad 61
- 62. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) S. No Marks xi Freque ncy (No of Studen ts) fixi Xi - x̄ F[Xi - x̄] 1 30-39 34.5 8 276 - 25.65 205.2 2 40-49 44.5 87 3871.5 - 16.65 1448.5 5 3 50-59 55.5 190 10545 - 5.65 1056.4 4 60-69 66.5 304 20216 5.35 1626.4 5 70-79 77.5 211 16352. 5 16.35 3449.8 5 6 80-89 88.5 85 7522.5 27.35 2324.7 5 7 90-99 99.5 20 1990 38.35 767 ∑905 ∑60773 .5 ∑10878 .15 M.D = ∑ 10878.15 905 M.D = 12.02 Dr Abdul Ghafoor Sajjad 62
- 63. MEASURESOF DISPERSION MEAN DEVIATION( From Group Data ) Dr Abdul Ghafoor Sajjad 63
- 64. TYPES OFDISPERSION Co-efficient of Mean Deviation TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation Relative Co-efficient of dispersion Co-efficient of Quartile Range Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation Dr Abdul Ghafoor Sajjad 64
- 66. Calculatetheco- efficientofMean Deviationfrom FlexionROMof everyvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 66
- 67. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatetheco- efficientofMean Deviationfrom ExtensionROM ofeveryvisit Dr Abdul Ghafoor Sajjad 67
- 70. MEASURESOF DISPERSION VARIENCE Deduct the mean from each data in the set then squaring each of them and adding each square and finally dividing them by the total no of values in the data set is the variance. Dr Abdul Ghafoor Sajjad 70
- 71. MEASURESOF DISPERSION VARIENCE Variance (σ2)=∑(X−μ)2/N (for population data) (σ2)=∑(X−x̄)2/n (for sample data) Dr Abdul Ghafoor Sajjad 71
- 73. MEASURESOF DISPERSION VARIENCE (σ2) = ∑(X−x̄)2/n = 75.5/7 = 10.48 Dr Abdul Ghafoor Sajjad 73
- 74. Calculatethe Variancefromthe FlexionROMof everyvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 74
- 75. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatethe Variancefromthe ExtensionROM ofeveryvisit Dr Abdul Ghafoor Sajjad 75
- 77. MEASURESOF DISPERSION STANDARDDEVIATION •The square root of the variance is known as the standard deviation i.e. S.D. = √σ. •The most useful measure of dispersion Dr Abdul Ghafoor Sajjad 77
- 78. MEASURESOF DISPERSION STANDARDDEVIATION • The standard deviation can never be a negative number, due to the way it’s calculated and the fact that it measures a distance (distances are never negative numbers). • The smallest possible value for the standard deviation is 0, and that happens only in contrived situations where every single number in the data set is exactly the same (no deviation). • The standard deviation is affected by outliers (extremely low or extremely high numbers in the data set). That’s because the standard deviation is based on the distance from the mean. And remember, the mean is also affected by outliers. • The standard deviation has the same units as the original data. Dr Abdul Ghafoor Sajjad 78
- 79. MEASURESOF DISPERSION STANDARDDEVIATION σ = √∑(X−μ)2/N (for population data) S =√∑(X−x̄)2/n (for sample data) Dr Abdul Ghafoor Sajjad 79
- 80. MEASURESOF DISPERSION STANDARDDEVIATION S.D. = √σ2 = √∑(X−x̄)2/n = √75.5/7 = √10.78 = 3.28 Dr Abdul Ghafoor Sajjad 80
- 81. Calculatethe Standard Deviationfrom theFlexionROM ofeveryvisit S. NO Lumbar Spine ROM Flexion (1st Visit) Lumbar Spine ROM Flexion (4th Visit) Lumbar Spine ROM Flexion (8th Visit) 1 5 45 50 2 10 60 62 3 13 40 45 4 15 48 55 5 18 28 30 6 20 40 52 7 22 50 54 8 24 36 42 9 25 20 25 10 27 17 25 11 40 52 60 Dr Abdul Ghafoor Sajjad 81
- 82. S. NO Lumbar Spine ROM Extension (1st Visit) Lumbar Spine ROM Extension (4th Visit) Lumbar Spine ROM Extension (8th Visit) 1 10 15 18 2 20 22 25 3 10 12 15 4 20 25 28 5 20 22 25 6 10 10 15 7 10 10 15 8 5 5 12 9 15 10 17 10 10 12 18 Calculatethe Standard Deviationfrom theExtension ROMofevery visit Dr Abdul Ghafoor Sajjad 82
- 83. MEASURESOF DISPERSION STANDARDDEVIATION For Group Data S = √ ∑ fi [xi - x̄]² n Dr Abdul Ghafoor Sajjad 83
- 86. MEASURESOF DISPERSION STANDARDDEVIATION S = √ ∑ fi [xi - x̄]²/n = √ 1374/30 = √ 45.8 = 6.76 Dr Abdul Ghafoor Sajjad 86
- 88. TYPES OFDISPERSION Co-efficient of Standard Deviation TYPES OF DISPERSION Absolute Range Quartile Range Mean/Average Deviation Standard Deviation Relative Co-efficient of dispersion Co-efficient of Quartile Range Co-efficient of Mean/Average Deviation Co-efficient of Standard Deviation Dr Abdul Ghafoor Sajjad 88
- 89. MEASURESOF DISPERSION Co-efficientof StandardDeviation S.D. = √σ2 = √∑(X−x̄)2/n = √75.5/7 = √10.78 = 3.28 Dr Abdul Ghafoor Sajjad 89
- 90. MEASURESOFDISPERSION INTERPRETATIONOFSTANDARD DEVIATION • More precisely, it is a measure of the average distance between the values of the data in the set and the mean. A low standard deviation indicates that the data points tend to be very close to the mean; a high standard deviation indicates that the data points are spread out over a large range of values. Dr Abdul Ghafoor Sajjad 90
- 91. MEASURESOFDISPERSION INTERPRETATIONOFSTANDARD DEVIATION •x̄ ± 2S present ¾ (75%) of the Data •x̄ ± 23 present 8/9 (88.8%) of the Data Dr Abdul Ghafoor Sajjad 91
- 93. LECTURE NO 04 • Testing of Normal Distribution • Skewness • Kurtosis Dr Abdul Ghafoor Sajjad 93