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Measure of Dispersion in statistics

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Measure of dispersion has two types Absolute measure and Graphical measure. There are other different types in there.
In this slide the discussed points are:
1. Dispersion & it's types
2. Definition
3. Use
4. Merits
5. Demerits
6. Formula & math
7. Graph and pictures
8. Real life application.

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Measure of Dispersion in statistics

  1. 1. Measure of Dispersion
  2. 2. We are Group 4 Tasnim Ansari Hridi (ID-09) Md. Mehedi Hassan Bappy (ID-21) Debanik Chakraborty (ID-25) Syed Ishtiak Uddin Ahmed (ID-31) Devasish Kaiser (ID-49)
  3. 3. Definition of Measure of Dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.
  4. 4. Example Centre: Same Variation: Different Year 2000: Close Dispersion Year 2015: Wide Dispersion Better Quality Data:Data ofYear 2000
  5. 5. Why Measure of Dispersion Serve as a basis for the control of the variability To compare the variability of two or more series
  6. 6. Facilitate the use of other statistical measures. Reliable Determine the reliability of an average Why Measure of Dispersion
  7. 7. Characteristics of an Ideal Measure of Dispersion  Must be based on all observations of the data.  It should be rigidly defined  It should be easy to understand and calculate.
  8. 8.  Must be least affected by the sampling fluctuation.  Must be easily subjected to further mathematical operations Characteristics of an Ideal Measure of Dispersion  It should not be unduly affected by the extreme values.
  9. 9. Types of Measures of Dispersion
  10. 10. Classification of Measures of dispersion in Statistics
  11. 11. Measures of Dispersion Algebraic Absolute Relative Graphical
  12. 12. Algebraic Measure of Dispersion × Mathematical way to calculate the measure of dispersion. Example: Calculation of Standard Deviation or Co-efficient of Variance by using numbers and formulas.
  13. 13. Characteristics of Algebraic Measure of Dispersion • Mathematical Way • Algebraic Variables are used • Numerical Figures are used here • Formulas & Equations are used
  14. 14. Graphical Measure of Dispersion × The way to calculate the measure of dispersion by figures and graphs. Example: Calculation of Dispersion among the heights of the students of a class from the average height using a graph.
  15. 15. Characteristics of Graphical Measure of Dispersion • It is a visual way of measuring dispersion • Graphs, figures are used • Sometimes, it cannot give the actual result • It helps the reader to have an idea about the dispersion practically at a glance
  16. 16. Absolute Measure of Dispersion Absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measures the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data set.
  17. 17. Classification of Algebraic Measure of Dispersion
  18. 18. Absolute Measure of Dispersion Absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measures the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data set.
  19. 19. Relative Measure of Dispersion These measures are a sort of ratio and are called coefficients. Each absolute measure of dispersion can be converted into its relative measure. It can be used to compare two or more data sets
  20. 20. Difference Between Absolute and Relative Measure of Dispersion 3 This is calculated from original data These measure are calculated absolute measures 2 It is not expressed in terms of percentage It is expressed in terms of percentage 1 It has the variable unit It has no unit Absolute Measure Relative Measure
  21. 21. 6 There is no change in variables and with the absolute measures. There is changes in variables with relative measures. 5 These measure cannot be used to compare the variation of two or more series These measure can be used to compare the variation of two or more series. 4 No use of ratio Use of ratio Absolute Measure Relative Measures
  22. 22. Absolute measures of Dispersion
  23. 23. Classification of Absolute measure Mean Deviation Quartile Deviation Standard Deviation Range
  24. 24. “ Range
  25. 25. Range The difference between the maximum and minimum observations in the data set. R= H-L
  26. 26. 5, 10 , 15 , 20, 7, 9, 12 , 17 , 13 , 6 , 10 , 11 , 17 , 16 Range = H- L = 20- 5 = 15
  27. 27. Merits and Demerits of Range Gives a quick answer Cannot be calculated in open ended distributions Affected by sampling fluctuations Changes from one sample to the next in population Gives a rough answer and is not based on all observationSimple and easy to understand
  28. 28. “ Mean deviation
  29. 29. Mean deviation The average of the absolute values of deviation from the mean(median or mode) is called mean deviation.  = 𝒇 | 𝒙 − 𝒙 | 𝑵
  30. 30. Merits of Mean deviation Simplifies calculations Can be calculated by mean, median and mode Is not affected by extreme measures Used to make healthy comparisons
  31. 31. Demerits of Mean Deviation Not reliable Mathematically illogical to assume all negatives as positives Not suitable for comparing series
  32. 32. “ Quartile Deviation
  33. 33. Quartile Deviation The half distance between 75th percentile i.e. 3rd quartile (Q1) and 25th percentile i.e. 1st quartile (Q3) is Quartile deviation or Interquartile range. Q.D = Q3 – Q1 𝟐
  34. 34. Has better result than range mode. Is not affected by extreme items Merits of Quartile Deviation
  35. 35. Demerits of Quartile Deviation It is completelydependent on thecentral items. All the items of the frequencydistribution are not given equal importance in finding the values of Q1 and Q3 Because it does not take into accountall the items of the series, considered to be inaccurate.
  36. 36. “ Standard Deviation
  37. 37. Standard Deviation Standard deviation is calculated as the square root of average of squared deviations taken from actual mean. It is also called root mean square deviation.  = √ 𝒙− 𝒙 𝟐 𝒏
  38. 38. 68.2% 95.4% 99.7%
  39. 39. Merits of standard deviation It takes intoaccount all the items and is capableof future algebraic treatment andstatistical analysis. It is possible to calculatestandard deviationfor two or more series This measure is most suitable for makingcomparisonsamong two or more series about variability.
  40. 40. Demerits of Standard Deviation It is difficult to compute. It assigns more weights to extreme itemsand less weights to items that are nearer to mean.
  41. 41. Classifications of Relative Measures of Dispersion
  42. 42. Chart of classification Relative Measure Coefficient of Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation
  43. 43. Coefficient of Range
  44. 44. Coefficient of Range The measure of the distribution based on range is the coefficient of range also known as range coefficient of dispersion. Formula: Coefficient of Range= 𝑅𝑎𝑛𝑔𝑒 𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑉𝑎𝑙𝑢𝑒+𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 × 100
  45. 45. “Coefficient of Quartile Deviation
  46. 46. Coefficient of Quartile Deviation A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation. Formula: Coefficient of Quartile Deviation = 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑑𝑖𝑎𝑛 × 100 = Q3 – Q1 Q3 + Q1 × 100 [By Simplification]
  47. 47. Merits & Demerits of Coefficient of Quartile Deviation Merits 1. Easily understood 2. Not much Mathematical Difficulties 3. Better Result than Coefficient of Range  Sampling fluctuation  Ignorance of last 25% of data sets.  Values being irregular Demerits
  48. 48. Coefficient of Mean Deviation
  49. 49. Coefficient of Mean Deviation The relative measure of dispersion we get by dividing Mean Deviation by Mean or Median, is called Coefficient of Mean Deviation. Formula: Coefficient of MD= 𝑀𝑒𝑎𝑛 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑑𝑖𝑎𝑛 𝑜𝑟 𝑀𝑒𝑎𝑛 × 100
  50. 50. Merits & Demerits of Coefficient of Mean Deviation Merits 1. Better Result than Range & Quartile Coefficient. 2. Least sampling fluctuation. 3. Rigidly defined.  Fractional Average.  Cannot be used for sociological studies  Less reliable than Coefficient of Variation Demerits
  51. 51. Coefficient of Variation
  52. 52. Coefficient of Variation Coefficient of Variation is a measure of spread that describes the amount of variability relative to the mean. Formula: Coefficient of Variation= 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑎𝑛 × 100
  53. 53. Merits & Demerits of Coefficient of Variation Merits 1. Best one 2. Most appropriate one 3. Based on Mean and Standard Deviation 4. COV is dimensionless or non- unitized  It is impossible to calculate if Mean is 0  It is difficult to calculate if the values are both positive and negative numbers & if the mean is close to 0. Demerits
  54. 54. Practical Uses of Coefficient of Variance INVESTMENT ANALYSIS STOCK MARKET RISK EVALUATION COMBINED STANDARD DEVIATION OF SEVERAL GROUPS PERFORMANCES OF TWO PLAYERS INDUSTRIES & FACTORIES
  55. 55. Mathematical Application
  56. 56. Coefficient of range Let 1,2,4,6,7 is a set of values of a distribution. Here, Highest Value, XH=7 & Lowest Value, XL=1 So, Range, R= 7-1 = 6 Now, Coefficient of Range = 𝐑 XH + XL × 100 = 𝟔 𝟕+𝟏 × 100 =75%
  57. 57. Coefficient of Quartile deviationLet the number of students in 5 classes are 110, 150, 180, 190, 240 is a set of values. Here, Q1= size of 𝐍+𝟏 𝟒 th item = 130 And, Q3 = size of 𝟑(𝐍+𝟏) 𝟒 th item = 215 So, Coefficient of Quartile Deviation =Q3 – Q1 Q3 + Q1 × 100 = 215−130 215+130 × 100= 24.64 %
  58. 58. Coefficient of Mean Deviation Let the ages of 5 boys in a class is 12, 14, 14, 15, 18. So their Mean, 𝐱 = 𝟏𝟐+𝟏𝟒+𝟏𝟒+𝟏𝟓+𝟏𝟖 𝟓 = 14.6 Mean Deviation, MD = | 𝒙 − 𝒙 | 𝑵 =|12−14.6| + |14−14.6| + |14− 14.6|+ |15−14.6| + |18−14.6| 𝟓 = 1.52 Now, the Coefficient of MD= 𝐌𝐃 𝐱 × 𝟏𝟎𝟎 = 𝟏.𝟓𝟐 𝟏𝟒.𝟔 × 𝟏𝟎𝟎 = 10.41%
  59. 59. Coefficient of VariationSuppose the returns on an investment for 4 years is Tk.1000, Tk.3000, Tk.4500 & Tk.5000. So, Mean, 𝐱 = 3375 Standard Deviation, SD = 1796.99 So, Coefficient of Variation, CV= 𝐒𝐃 𝐱 × 100 = 𝟏𝟕𝟗𝟔.𝟗𝟗 𝟑𝟑𝟕𝟓 × 100 = 53.24%
  60. 60. The daily sale of sugar in a certain grocery shop is given below : Monday Tuesday Wednesday Thursday Friday Saturday 75 kg 120 kg 12 kg 50 kg 70.5 kg 140.5 kg respectively.
  61. 61. “ No of Days sale of sugar Monday 60 Tuesday 120 Wednesday 10 Thursday 50 Friday 70 Saturday 140 𝜮 𝒐𝒇 𝑫𝒂𝒚𝒔 = 𝟔 𝜮𝒙 = 𝟒𝟓𝟎 Mean, 𝑥 = 𝑥 𝑛 = 4𝟓𝟎 6 = 7𝟓
  62. 62. “ x 𝒙 𝟐 60 3600 120 14400 10 100 50 2500 70 4900 140 19600 𝜮𝒙 = 𝟒𝟓𝟎 𝜮𝒙 𝟐 = 45100 Standard deviation: 𝝈 = 𝜮𝒙 𝟐 𝒏 − 𝜮𝒙 𝒏 𝟐 = 𝟒𝟓𝟏𝟎𝟎 𝟔 − 𝟒𝟓𝟎 𝟔 𝟐 = 𝟕𝟓𝟏𝟔. 𝟔𝟔 − 𝟓𝟔𝟐𝟓 = 𝟒𝟑. 𝟒𝟗
  63. 63. Quartile Deviation The marks of 7 students in Mathematics result are given below : 70, 85, 92,68, 75, 96, 84 Find out- • First Quartile Deviation • Third Quartile Deviation
  64. 64. Quartile deviation × First quartile 𝐐 𝟏 = 𝐬𝐢𝐳𝐞 𝐨𝐟 𝐧 + 𝟏 𝟒 𝐭𝐡 𝐢𝐭𝐞𝐦 = size of 𝟕+𝟏 𝟒 𝐭𝐡 𝐢𝐭𝐞𝐦 = size of 2nd item. = 70 ×Third Quartile 𝑸 𝟑 = 𝒔𝒊𝒛𝒆 𝒐𝒇 𝟑 𝒏 + 𝟏 𝒕𝒉 𝟒 𝒊𝒕𝒆𝒎 = size of 𝟑 𝟕+𝟏 𝒕𝒉 𝟒 𝒊𝒕𝒆𝒎 = size of 6th item =92 Arranging the data in ascending order we get, 68,70,75,84,85,92,96
  65. 65. “ Thank you

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