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Measures of Central Tendency

Measures of Central Tendency. Measures of Central Tendency. Definition Measures of Central Tendency (Mean, Median, Mode). Central Tendency. Refers to a characteristic where the frequency of a variable tends to cluster around the ‘center’. Measures of Central Tendency. Arithmetic Mean

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Measures of Central Tendency

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  1. Measures of Central Tendency

  2. Measures of Central Tendency • Definition • Measures of Central Tendency (Mean, Median, Mode)

  3. Central Tendency • Refers to a characteristic where the frequency of a variable tends to cluster around the ‘center’

  4. Measures of Central Tendency • Arithmetic Mean • Median • Mode

  5. Arithmetic Mean • Data (units produced by workers) 10, 20, 30 • Mean = • Ungrouped data (1) Mean = =20

  6. Arithmetic Mean • Ungrouped data (2) • Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 fx

  7. Arithmetic Mean • Grouped data • Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 fm Midpoint(m)

  8. Arithmetic Mean • Ungrouped data • Grouped data

  9. Features of Arithmetic Mean • Commonly used • Easily understood • Greatly affected by extreme values

  10. Median 1. Array 2. Median position 3. median

  11. Median • Ungrouped data (1) • Data (units produced by workers) 20, 10, 30 (odd) Array 10, 20, 30 Median position Median 20

  12. Median • Ungrouped data (1) • Data (units produced by workers) 20, 10, 40, 30 (even) Array 10, 20, 30, 40 Median position Median

  13. Median • Ungrouped data (2) • Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 c.f. Median position= √ Median= 25 units

  14. Median • Grouped data (2) • Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Median position = c.f. √ Median Class = 20-30 Median =

  15. Median • Ungrouped data • Grouped data

  16. Features of Median • Not affected by extreme values • When data is skewed, the median is often a better indicator of “average” than the mean. • Time consuming • Unfamiliar to most people

  17. Mode • Ungrouped data (1) • Data (units produced by workers) 10, 20, 20, 30 • Mode = 20

  18. Mode • Ungrouped data (2) • Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 25 √ Mode =

  19. Mode • Grouped data (2) • Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 The highest frequency: √ Modal group= 20-30 units Mode =

  20. Mode • Ungrouped data • Data with the highest frequency • Grouped data

  21. Features of Mode • Not affected by extreme values • Maybe more than one mode, or no mode • May not give a good indication of central values

  22. Skewness of Data Distribution • Normal • Mode = mean =median

  23. Skewness of Data Distribution • Positively skewed • Mode < median< mean

  24. Skewness of Data Distribution • Negatively skewed • Mean < median< mode

  25. Arithmetic Mean • ungrouped data • grouped data

  26. Median • ungrouped data • grouped data

  27. Mode • ungrouped data • Data with the highest frequency • grouped data

  28. Measures of Dispersion

  29. Measures of Dispersion • Definition • Measures of Dispersion(Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance, Coefficient of Variation)

  30. Dispersion • It describes the level of variation and also indicates the level of consistency in the distribution.

  31. Measures of Dispersion • Range • Quartile Deviation • Mean Deviation • Standard Deviation • Variance • Coefficient of Variation

  32. Range • It measures the difference between the highest and the lowest piece of data. Data1: Data2: 10, 20, 30 0, 20, 40 Range1 = xmax – xmin = 30 - 10 = 20 Range2 = xmax – xmin = 40 - 0 = 40

  33. Feature • It is easy to calculate and easy to understand. • It is distorted by extreme values.

  34. Quartile Deviation 1. Array 2.Quartile position 3. Quartile Value 4. IQR,QD

  35. Quartile Deviation • It excludes the first and last quarters of information and in doing so concentrates on the main core of data, ignoring extreme values. • 45 46 50 55 60 65 67 69 69 70 71 72 73 74 76 78 78 79 80 82 83 85 90 95 Q1 Q2 Q3 Interquartile Range = Q3 - Q1 Quartile Deviation =

  36. Quartile Deviation (ungrouped) Q1 position= Q3 position= Q1 value= Q3 value=

  37. Grouped data

  38. c. f. √

  39. c. f. √

  40. c. f. √

  41. c. f. √

  42. c. f.

  43. Feature • Not effected by extreme values. • Not widely used or understood.

  44. Quartile Deviation • Ungrouped: Q1 = Q3= • I.Q.R= Q3 value- Q1 value Quartile Deviation =

  45. Quartile Deviation • Grouped: Q1 = Q3= • I.Q.R= Q3 value- Q1 value Quartile Deviation =

  46. Mean Deviation • The absolute distance of each score away from the mean.

  47. Mean Deviation • Ungrouped data

  48. Mean Deviation • Ungrouped data • Team 1: 20 22 23 25 25 26 26 26 28 29 • Team 2: 12 14 18 24 28 30 30 30 31 33

  49. Mean Deviation • Ungrouped data • Team 1: 20 22 23 25 25 26 26 26 28 29 • Team 2: 12 14 18 24 28 30 30 30 31 33

  50. Mean Deviation • Ungrouped data • Team 1: 20 22 23 25 25 26 26 26 28 29 • Team 2: 12 14 18 24 28 30 30 30 31 33

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