Descriptive statistics

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Descriptive statistics - measures of location, dispersion, skewness and kurtosis...

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Descriptive statistics

  1. 1. Descriptive Statistics Rajesh Gunesh http://pages.intnet.mu/cueboy ConsulDesign
  2. 2. DESCRIPTIVE (SUMMARY) STATISTICShttp://pages.intnet.mu/cueboy Dispersion Skewness B C Location A D Kurtosis Characteristics of a distribution © Rajesh Gunesh – Jul 2008
  3. 3. MOMENTS OF A DISTRIBUTIONhttp://pages.intnet.mu/cueboy Fourth moment KURTOSIS Third moment SKEWNESS Second moment DISPERSION First moment LOCATION © Rajesh Gunesh – Jul 2008
  4. 4. IN PLAIN AND SIMPLE ENGLISHhttp://pages.intnet.mu/cueboy Location = Central tendency Dispersion = Spread Skewness = Symmetry Kurtosis = Peakedness © Rajesh Gunesh – Jul 2008
  5. 5. LOCATIONhttp://pages.intnet.mu/cueboy Location ThemeGallery Location is a Design Digital A measure of location, Content & Contents otherwise known as mall developed by central tendency, is a Guild Design Inc. point in a distribution that corresponds to a typical, representative or middle score in that distribution. © Rajesh Gunesh – Jul 2008
  6. 6. MEASURES OF LOCATIONhttp://pages.intnet.mu/cueboy  A measure of location, otherwise known as central tendency, is a point in a distribution that corresponds to a typical, representative or middle score in that distribution.  The most common measures of location are the mean (arithmetic), median and mode. © Rajesh Gunesh – Jul 2008
  7. 7. MEASURES OF LOCATIONhttp://pages.intnet.mu/cueboy Mode Location Median Mean © Rajesh Gunesh – Jul 2008
  8. 8. MEASURES OF LOCATIONhttp://pages.intnet.mu/cueboy 1 The mean © Rajesh Gunesh – Jul 2008
  9. 9. LOCATION – THE MEANhttp://pages.intnet.mu/cueboy  The arithmetic mean is the most common form of average. For a given set of data, it is defined as the sum of the values of all the observations divided by the total number of observations. The mean is denoted by x for a sample and by µ for a population. Other existing means are the geometric, harmonic and weighted means. © Rajesh Gunesh – Jul 2008
  10. 10. LOCATION – THE MEANhttp://pages.intnet.mu/cueboy Arithmetic Geometric Mean Harmonic Weighted © Rajesh Gunesh – Jul 2008
  11. 11. THE ARITHMETIC MEANhttp://pages.intnet.mu/cueboy Mean Definition Formula The sum of all ∑ n observations 1 divided by the x= xi total number of n Arithmetic i =1 observations Text © Rajesh Gunesh – Jul 2008
  12. 12. THE GEOMETRIC MEANhttp://pages.intnet.mu/cueboy Mean Definition Formula The nth root of n the product of n observations Geometric GM = n ∏ i =1 xi Text © Rajesh Gunesh – Jul 2008
  13. 13. THE HARMONIC MEANhttp://pages.intnet.mu/cueboy Mean Definition Formula The reciprocal of 1 HM = the mean of the ∑ n reciprocals of all 1 1 observations n xi Harmonic i =1 Text © Rajesh Gunesh – Jul 2008
  14. 14. THE WEIGHTED MEANhttp://pages.intnet.mu/cueboy Mean Definition Formula ∑ n The mean of a set of numbers wi xi that have been weighted Weighted x= i =1 ∑ n (multiplied by their relative wi importance or × i =1 of occurrence). Text © Rajesh Gunesh – Jul 2008
  15. 15. MEASURES OF LOCATIONhttp://pages.intnet.mu/cueboy 2 The median © Rajesh Gunesh – Jul 2008
  16. 16. LOCATION – THE MEDIANhttp://pages.intnet.mu/cueboy  The median is the middle observation of a distribution and can only be determined after arranging numerical data in ascending (or descending) order. If n is the total number of observations, then the rank of the median is given by (n+1)/2. For ungrouped data, if n is odd, the median is simply the middle observation but, if n is even, then the median is the mean of the two middle observations. © Rajesh Gunesh – Jul 2008
  17. 17. THE MEDIAN (UNGROUPED DATA)http://pages.intnet.mu/cueboy Median Definition Formula The median is the middle 1 observation of a Q2 = (n + 1) distribution of 2 Ungrouped arranged numerical data Text © Rajesh Gunesh – Jul 2008
  18. 18. THE MEDIAN (GROUPED DATA)http://pages.intnet.mu/cueboy Median Definition Formula The median is the middle  n +1 − cf  observation of a Q2 = LCB +  2 ÷(c)  f  distribution of Grouped arranged numerical data © Rajesh Gunesh – Jul 2008
  19. 19. THE MEDIAN (UNGROUPED DATA)http://pages.intnet.mu/cueboy Median Definition Formula When n is odd, the median is the 1 middle observation. Q2 = (n + 1) 2 When n is even, Ungrouped the median is the average or midpoint of the two middle observations. Text © Rajesh Gunesh – Jul 2008
  20. 20. THE MEDIAN (UNGROUPED DATA)http://pages.intnet.mu/cueboy Median (n odd) Example 1: Ungrouped Find the median of 27 13 62 5 44 29 16 1 Solution: Q2 = (n + 1) 2 First re-arrange the numbers in ascending order: 5 13 16 27 29 44 62 (n = 7) Rank of median = (7 + 1)/2 = 4 Median = 27 © Rajesh Gunesh – Jul 2008
  21. 21. THE MEDIAN (UNGROUPED DATA)http://pages.intnet.mu/cueboy Median (n even) Example 2: Ungrouped Find the median of 5 13 16 27 29 44 Solution: First re-arrange the numbers in ascending order: 5 13 16 27 29 44 (n = 6) Rank of median = (6 + 1)/2 = 3.5 1 Median = (16 + 27) = 21.5 2 © Rajesh Gunesh – Jul 2008
  22. 22. MEASURES OF LOCATIONhttp://pages.intnet.mu/cueboy 3 The mode © Rajesh Gunesh – Jul 2008
  23. 23. LOCATION – THE MODEhttp://pages.intnet.mu/cueboy  The mode is the observation which occurs the most or with the highest frequency. For ungrouped data, it may easily be detected by inspection. If there is more than one observation with the same highest frequency, then we either say that there is no mode or that the distribution is multimodal. © Rajesh Gunesh – Jul 2008
  24. 24. THE MODE (GROUPED DATA)http://pages.intnet.mu/cueboy Mode Definition Formula The mode is the observation  f1  which occurs the x = LCB +  ˆ ÷(c) most or with the  f1 + f 2  Grouped highest frequency. Text © Rajesh Gunesh – Jul 2008
  25. 25. DISPERSIONhttp://pages.intnet.mu/cueboy Dispersion ThemeGallery Dispersion is a Design Digital A measure of Content & Contents dispersion shows the mall developed by amount of variation or Guild Design Inc. spread in the scores (values of observations) of a variable. © Rajesh Gunesh – Jul 2008
  26. 26. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy  A measure of dispersion shows the amount of variation or spread in the scores (values of observations) of a variable. When the dispersion is large, the values are widely scattered whereas, when it is small, they are tightly clustered.  The two most well-known measures of dispersion are the range and standard deviation. © Rajesh Gunesh – Jul 2008
  27. 27. MEASURES OF DISPERSION http://pages.intnet.mu/cueboy Standard deviation B Mean absolute Range A C deviation DispersionCoefficient of variation E D Quartile deviation © Rajesh Gunesh – Jul 2008
  28. 28. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy 1 The range © Rajesh Gunesh – Jul 2008
  29. 29. DISPERSION – THE RANGEhttp://pages.intnet.mu/cueboy  The range is the difference between the values of the maximum and minimum observations of a distribution. It can only measure the extent to which the distribution spreads between its endpoints. © Rajesh Gunesh – Jul 2008
  30. 30. THE RANGEhttp://pages.intnet.mu/cueboy Dispersion Definition Formula The difference between the values of the R = xmax − xmin maximum and Range minimum observations of a distribution Text © Rajesh Gunesh – Jul 2008
  31. 31. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy 2 The standard deviation © Rajesh Gunesh – Jul 2008
  32. 32. DISPERSION – STANDARD DEVIATIONhttp://pages.intnet.mu/cueboy  The standard deviation is defined as the positive square root of variance or the square root of the average of the squared distances of the observations of a distribution from its mean. We also use the term standard error in the case of an estimate. © Rajesh Gunesh – Jul 2008
  33. 33. THE STANDARD DEVIATIONhttp://pages.intnet.mu/cueboy Dispersion Definition Formula The square root of the average of the squared 1 n distances of the Standard s= ∑ (x − x ) 2 observations deviation n i =1 from the mean. Text © Rajesh Gunesh – Jul 2008
  34. 34. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy 3 The mean absolute deviation © Rajesh Gunesh – Jul 2008
  35. 35. DISPERSION – MEAN ABSOLUTE DEVIATIONhttp://pages.intnet.mu/cueboy  The mean absolute deviation (MAD) is defined as the average of the distances of the observations of a distribution from its mean. © Rajesh Gunesh – Jul 2008
  36. 36. THE MEAN ABSOLUTE DEVIATIONhttp://pages.intnet.mu/cueboy Dispersion Definition Formula The average of the distances of the observations 1 n of a distribution MAD = ∑ xi − x from its mean. MAD n i =1 Text © Rajesh Gunesh – Jul 2008
  37. 37. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy 4 The quartile deviation © Rajesh Gunesh – Jul 2008
  38. 38. DISPERSION – QUARTILE DEVIATIONhttp://pages.intnet.mu/cueboy  The quartile deviation is equal to half the difference between the lower and upper quartiles and is sometimes called the semi inter-quartile range. © Rajesh Gunesh – Jul 2008
  39. 39. THE QUARTILE DEVIATIONhttp://pages.intnet.mu/cueboy Dispersion Definition Formula It is half the difference between the Q3 − Q1 lower and upper Quartile QD = quartiles deviation 2 Text © Rajesh Gunesh – Jul 2008
  40. 40. MEASURES OF DISPERSIONhttp://pages.intnet.mu/cueboy 5 The coefficient of variation © Rajesh Gunesh – Jul 2008
  41. 41. DISPERSION – COEFFICIENT OF VARIATIONhttp://pages.intnet.mu/cueboy  The coefficient of variation (CV) is mainly used to compare the dispersion of two distributions that have different means and standard deviations; it is thus considered to be a relative measure of dispersion. © Rajesh Gunesh – Jul 2008
  42. 42. COEFFICIENT OF VARIATIONhttp://pages.intnet.mu/cueboy Dispersion Definition Formula It is used to compare two distributions s when they have Coefficient of CV = × 100 the same mean variation x but different standard deviations Text © Rajesh Gunesh – Jul 2008
  43. 43. SKEWNESShttp://pages.intnet.mu/cueboy Skewness ThemeGallery Skewness is a Design Digital Skewness is a Content & Contents measure of symmetry mall developed by – it determines whether Guild Design Inc. there is a concentration of observations somewhere in particular in a distribution. © Rajesh Gunesh – Jul 2008
  44. 44. MEASURES OF SKEWNESShttp://pages.intnet.mu/cueboy Skewness Quartile coefficient Pearson’s coefficient © Rajesh Gunesh – Jul 2008
  45. 45. SKEWNESShttp://pages.intnet.mu/cueboy  Skewness is a measure of symmetry – it determines whether there is a concentration of observations somewhere in particular in a distribution. If most observations lie at the lower end of the distribution, the distribution is said to be positively skewed. If the concentration of observations is towards the upper end of the distribution, then it is said to display negative skewness. A symmetrical distribution is said to have zero skewness. © Rajesh Gunesh – Jul 2008
  46. 46. SKEWNESShttp://pages.intnet.mu/cueboyPositively skewed Symmetrical Negatively skewed  The vertical bars on each diagram indicate the respective positions of the mean (bold), median (dashed) and mode (normal). In the case of a symmetrical distribution, the mean, median and mode are all equal in values (for example, the normal distribution). © Rajesh Gunesh – Jul 2008
  47. 47. MEASURES OF SKEWNESShttp://pages.intnet.mu/cueboy 1 Pearson’s coefficient © Rajesh Gunesh – Jul 2008
  48. 48. SKEWNESS – PEARSON’S COEFFICIENThttp://pages.intnet.mu/cueboy  This is the most accurate measure of skewness since its formula contains two of the most reliable statistics, the mean and standard deviation. © Rajesh Gunesh – Jul 2008
  49. 49. PEARSON’S COEFFICIENThttp://pages.intnet.mu/cueboy Skewness Definition Formula The most accurate measure of 3( x − Q2 ) skewness since its formula contains α= the two most Pearson’s s reliable statistics: the mean and standard deviation Text © Rajesh Gunesh – Jul 2008
  50. 50. MEASURES OF SKEWNESShttp://pages.intnet.mu/cueboy 2 Quartile coefficient © Rajesh Gunesh – Jul 2008
  51. 51. SKEWNESS – QUARTILE COEFFICIENThttp://pages.intnet.mu/cueboy  A less accurate but relatively quicker way of estimating skewness is by the use of quartiles of a distribution © Rajesh Gunesh – Jul 2008
  52. 52. QUARTILE COEFFICIENThttp://pages.intnet.mu/cueboy Skewness Definition Formula A less accurate but relatively Q1 + Q3 − 2Q2 quicker way of α= estimating Quartile Q3 − Q1 skewness is by the use of quartiles of a distribution Text © Rajesh Gunesh – Jul 2008
  53. 53. KURTOSIShttp://pages.intnet.mu/cueboy Kurtosis ThemeGallery Kurtosis is a Design Digital Kurtosis may be Content & Contents considered as a mall developed by measure of the relative Guild Design Inc. concentration of observations in the centre, upper and lower ends and the shoulders of a distribution. © Rajesh Gunesh – Jul 2008
  54. 54. KURTOSIShttp://pages.intnet.mu/cueboy  Kurtosis indicates the degree of peakedness of a unimodal frequency distribution. It may be also considered as a measure of the relative concentration of observations in the centre, upper and lower ends and the shoulders of a distribution. Kurtosis usually indicates to which extent a curve (distribution) departs from the bell-shaped or normal curve. © Rajesh Gunesh – Jul 2008
  55. 55. KURTOSIShttp://pages.intnet.mu/cueboy  It is customary to subtract 3 from the coefficient of kurtosis for the sake of reference to the normal distribution. A negative value would indicate a platykurtic curve whereas a positive coefficient of kurtosis indicates a leptokurtic distribution. A value close to 0 means that the distribution is mesokurtic, that is, close to the normal. © Rajesh Gunesh – Jul 2008
  56. 56. KURTOSIShttp://pages.intnet.mu/cueboy Platykurtic Mesokurtic Leptokurtic © Rajesh Gunesh – Jul 2008
  57. 57. A MEASURE OF KURTOSIShttp://pages.intnet.mu/cueboy Kurtosis Coefficient of kurtosis © Rajesh Gunesh – Jul 2008
  58. 58. MEASURES OF KURTOSIShttp://pages.intnet.mu/cueboy 1 Coefficient of kurtosis © Rajesh Gunesh – Jul 2008
  59. 59. COEFFICIENT OF KURTOSIShttp://pages.intnet.mu/cueboy Kurtosis Definition Formula It is customary to ∑ subtract 3 from for the sake of ( x − x )4 reference to the normal Coefficient β= distribution ns 4 Text © Rajesh Gunesh – Jul 2008
  60. 60. SUMMARY OF DESCRIPTIVE STATISTICShttp://pages.intnet.mu/cueboy Distribution Distribution Characteristics Location Dispersion Skewness Kurtosis © Rajesh Gunesh – Jul 2008
  61. 61. http://pages.intnet.mu/cueboy

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