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Basic Statistical Measures

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Shubham Mehta
Shubham Mehta

The document provides statistical analysis of marks data from 250 students of a class. It includes: 1) Calculation of measures of central tendency like arithmetic mean (39.4), median (41.27), and mode (41.23) from the raw marks data. 2) Construction of a frequency distribution table and associated graphs like bar graph and frequency polygon. 3) Calculation of measures of dispersion like variance (92.93) and standard deviation (9.64). 4) Mention of other statistical measures like skewness (-0.19553) and kurtosis to analyze the shape of the distribution.

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University of Delhi, Delhi-110007
Aim of the Presentation The main aim of this presentation is to show the students, the use of statistics in analyzing, presentation and making inferences from data in our day to day life. The raw data for this project includes the Marks of 250 students of a class.
Acknowledgements We would like to extend our heartiest regard to our mentor and teacher Dr. Narinder Pushkarna or NP Sir who has guided us during the length and breadth of this highly demanding work and always supporting us in every way he could. We would also like to thank our friends of Department of Statistics, Ramjas College who helped us in finishing this extensive deal. Aayush, Poorva, Shubham, Sahiba
Raw Data 32 48 41 41 39 47 53 51 30 18 54 42 31 15 32 32 48 46 37 56 38 62 50 38 35 26 51 40 42 22 44 37 45 37 44 21 47 31 41 18 68 38 30 52 42 41 34 52 60 38 41 41 48 28 42 53 29 21 49 36 30 32 37 29 38 33 49 35 37 40 43 17 24 38 41 32 46 38 22 50 46 38 26 23 25 50 46 15 42 52 41 28 40 40 45 38 31 37 48 30 40 35 42 51 56 33 38 36 42 44 31 45 45 50 50 51 48 41 53 32 40 49 40 34 38 43 28 34 44 58 40 37 19 34 37 45 36 24 47 33 36 38 61 44 50 32 45 30 43 31 46 31 32 44 35 40 48 34 55 39 48 53 43 43 41 50 34 55 39 48 32 43 42 34 33 31 39 34 32 24 40 47 27 34 34 50 42 47 44 33 17 36 57 38 33 42 23 35 17 46 48 31 31 33 26 50 37 58 44 29 47 47 57 41 42 55 43 37 54 45 37 42 47 44 44 52 19 46 50 38 52 48 23 47 42 45 39 41 33 24 48 40 60 38 38 44 48 38 44 43
Frequency Distribution Inclusive type classification Exclusive type classification Marks Number of students Marks Number of students 15-19 9 14.5-19.5 9 20-24 11 19.5-24.5 11 25-29 10 24.5-29.5 10 30-34 44 29.5-34.5 44 35-39 45 34.5-39.5 45 40-44 54 39.5-44.5 54 45-49 37 44.5-49.5 37 50-54 26 49.5-54.5 26 55-59 8 54.5-59.5 8 60-64 5 59.5-64.5 5 65-69 1 64.5-69.5 1
Associated Graphic Representations 9 11 10 44 45 54 37 26 8 5 1 0 10 20 30 40 50 60 17 22 27 32 37 42 47 52 57 62 67 NumberofStudents Class Marks Bar Graph showing the marks of students NO OF STUDENTS
Associated Graphic Representations 9 11 10 44 45 54 37 26 8 5 1 0 10 20 30 40 50 60 17 22 27 32 37 42 47 52 57 62 67 NumberofStudents Class marks ( (u+l)/2 ) Frequency Polygon showing marks of students NO OF STUDENTS
Associated Graphic Representations 1 2 3 4 5 6 7 8 9 10 11 12 Pie Chart showing the marks of students
Arithmetic Mean AM is sum of all observations divided by total number of observations. A.M. = 裡fixi / 裡fi = 9930/250 CLASS MARKS NO OF STUDENTS fixi 17 9 153 22 11 242 27 10 270 32 44 1408 37 45 1665 42 54 2268 47 37 1739 52 26 1352 57 8 456 62 5 310 67 1 67 裡=250 裡=9930
Harmonic Mean HM is sum of all observations divided by total number of observations. H.M. =1/[ (1/N)* 裡fi / xi] =36.93
Calculations MARKS CLASS MARKS NO OF STUDENTS CUMULATIVE FREQUENCY fixi fixi 2 14.5-19.5 17 9 9 153 2601 19.5-24.5 22 11 20 242 5324 24.5-29.5 27 10 30 270 7290 29.5-34.5 32 44 74 1408 45056 34.5-39.5 37 45 119 1665 61605 39.5-44.5 42 54 173 2268 95256 44.5-49.5 47 37 210 1739 81733 49.5-54.5 52 26 236 1352 70304 54.5-59.5 57 8 244 456 25992 59.5-64.5 62 5 249 310 19220 64.5-69.5 67 1 250 67 4489 Total 裡=250 裡=9930 裡=418870
Median class ---- 34.5 39.5 Modal class ---- 39.5-44.5 Median = l + (N/2-CF) * h/f =34.5 + (135-74)*5/45 =34.5 + 6.77 = 41.27 Mode = l + [h*(f1 - f0 )/ (2f1 - f0 f2 )] =39.5 + [5*(54 - 45)/((2*54)-45-37)] =39.5 + 1.73 =41.23
Variance and Standard Deviation Quantitative data vary about a measure of central tendency and these measures of deviations are called measures of dispersion or variation. Variance=(1/N)*裡fixi 2 - [ (1/N)*裡fixi ] 2 = 92.93 Standard Variation = sqt(variance) = 9.64 MARKS CLASS MARKS NO OF STUDENTS fixi fixi 2 14.5-19.5 17 9 153 2601 19.5-24.5 22 11 242 5324 24.5-29.5 27 10 270 7290 29.5-34.5 32 44 1408 45056 34.5-39.5 37 45 1665 61605 39.5-44.5 42 54 2268 95256 44.5-49.5 47 37 1739 81733 49.5-54.5 52 26 1352 70304 54.5-59.5 57 8 456 25992 59.5-64.5 62 5 310 19220 64.5-69.5 67 1 67 4489 Total 裡=250 裡=9930 裡=418870
Measure of Kurtosis and skewness The measures of the direction and degree of asymmetry are called measures of skewness. The measures of the peakedness or flatness of the frequency curves are called measures of kurtosis. Skewness = -0.19553 ( as calculated from MS EXCEL )

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Basic Statistical Measures

  • 1. University of Delhi, Delhi-110007
  • 2. Aim of the Presentation The main aim of this presentation is to show the students, the use of statistics in analyzing, presentation and making inferences from data in our day to day life. The raw data for this project includes the Marks of 250 students of a class.
  • 3. Acknowledgements We would like to extend our heartiest regard to our mentor and teacher Dr. Narinder Pushkarna or NP Sir who has guided us during the length and breadth of this highly demanding work and always supporting us in every way he could. We would also like to thank our friends of Department of Statistics, Ramjas College who helped us in finishing this extensive deal. Aayush, Poorva, Shubham, Sahiba
  • 4. Raw Data 32 48 41 41 39 47 53 51 30 18 54 42 31 15 32 32 48 46 37 56 38 62 50 38 35 26 51 40 42 22 44 37 45 37 44 21 47 31 41 18 68 38 30 52 42 41 34 52 60 38 41 41 48 28 42 53 29 21 49 36 30 32 37 29 38 33 49 35 37 40 43 17 24 38 41 32 46 38 22 50 46 38 26 23 25 50 46 15 42 52 41 28 40 40 45 38 31 37 48 30 40 35 42 51 56 33 38 36 42 44 31 45 45 50 50 51 48 41 53 32 40 49 40 34 38 43 28 34 44 58 40 37 19 34 37 45 36 24 47 33 36 38 61 44 50 32 45 30 43 31 46 31 32 44 35 40 48 34 55 39 48 53 43 43 41 50 34 55 39 48 32 43 42 34 33 31 39 34 32 24 40 47 27 34 34 50 42 47 44 33 17 36 57 38 33 42 23 35 17 46 48 31 31 33 26 50 37 58 44 29 47 47 57 41 42 55 43 37 54 45 37 42 47 44 44 52 19 46 50 38 52 48 23 47 42 45 39 41 33 24 48 40 60 38 38 44 48 38 44 43
  • 5. Frequency Distribution Inclusive type classification Exclusive type classification Marks Number of students Marks Number of students 15-19 9 14.5-19.5 9 20-24 11 19.5-24.5 11 25-29 10 24.5-29.5 10 30-34 44 29.5-34.5 44 35-39 45 34.5-39.5 45 40-44 54 39.5-44.5 54 45-49 37 44.5-49.5 37 50-54 26 49.5-54.5 26 55-59 8 54.5-59.5 8 60-64 5 59.5-64.5 5 65-69 1 64.5-69.5 1
  • 6. Associated Graphic Representations 9 11 10 44 45 54 37 26 8 5 1 0 10 20 30 40 50 60 17 22 27 32 37 42 47 52 57 62 67 NumberofStudents Class Marks Bar Graph showing the marks of students NO OF STUDENTS
  • 7. Associated Graphic Representations 9 11 10 44 45 54 37 26 8 5 1 0 10 20 30 40 50 60 17 22 27 32 37 42 47 52 57 62 67 NumberofStudents Class marks ( (u+l)/2 ) Frequency Polygon showing marks of students NO OF STUDENTS
  • 9. Arithmetic Mean AM is sum of all observations divided by total number of observations. A.M. = 裡fixi / 裡fi = 9930/250 CLASS MARKS NO OF STUDENTS fixi 17 9 153 22 11 242 27 10 270 32 44 1408 37 45 1665 42 54 2268 47 37 1739 52 26 1352 57 8 456 62 5 310 67 1 67 裡=250 裡=9930
  • 10. Harmonic Mean HM is sum of all observations divided by total number of observations. H.M. =1/[ (1/N)* 裡fi / xi] =36.93
  • 11. Calculations MARKS CLASS MARKS NO OF STUDENTS CUMULATIVE FREQUENCY fixi fixi 2 14.5-19.5 17 9 9 153 2601 19.5-24.5 22 11 20 242 5324 24.5-29.5 27 10 30 270 7290 29.5-34.5 32 44 74 1408 45056 34.5-39.5 37 45 119 1665 61605 39.5-44.5 42 54 173 2268 95256 44.5-49.5 47 37 210 1739 81733 49.5-54.5 52 26 236 1352 70304 54.5-59.5 57 8 244 456 25992 59.5-64.5 62 5 249 310 19220 64.5-69.5 67 1 250 67 4489 Total 裡=250 裡=9930 裡=418870
  • 12. Median class ---- 34.5 39.5 Modal class ---- 39.5-44.5 Median = l + (N/2-CF) * h/f =34.5 + (135-74)*5/45 =34.5 + 6.77 = 41.27 Mode = l + [h*(f1 - f0 )/ (2f1 - f0 f2 )] =39.5 + [5*(54 - 45)/((2*54)-45-37)] =39.5 + 1.73 =41.23
  • 13. Variance and Standard Deviation Quantitative data vary about a measure of central tendency and these measures of deviations are called measures of dispersion or variation. Variance=(1/N)*裡fixi 2 - [ (1/N)*裡fixi ] 2 = 92.93 Standard Variation = sqt(variance) = 9.64 MARKS CLASS MARKS NO OF STUDENTS fixi fixi 2 14.5-19.5 17 9 153 2601 19.5-24.5 22 11 242 5324 24.5-29.5 27 10 270 7290 29.5-34.5 32 44 1408 45056 34.5-39.5 37 45 1665 61605 39.5-44.5 42 54 2268 95256 44.5-49.5 47 37 1739 81733 49.5-54.5 52 26 1352 70304 54.5-59.5 57 8 456 25992 59.5-64.5 62 5 310 19220 64.5-69.5 67 1 67 4489 Total 裡=250 裡=9930 裡=418870
  • 14. Measure of Kurtosis and skewness The measures of the direction and degree of asymmetry are called measures of skewness. The measures of the peakedness or flatness of the frequency curves are called measures of kurtosis. Skewness = -0.19553 ( as calculated from MS EXCEL )