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STATISTICS EXERCISE

STATISTICS EXERCISE. EDUCATIONAL RESEARCH. Organizing Data: An Array. 19 23 71 56 17 32 95 23 17 95 71 56 32 23 23 19 17 17. Array: Quiz Scores. 14.75 12 11.5 13.5 14.75 14.75 13 12.5 13.5 14.75 14.75 14.75 13.50 13.50 13.00

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STATISTICS EXERCISE

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  1. STATISTICS EXERCISE • EDUCATIONAL RESEARCH

  2. Organizing Data: An Array 19 23 71 56 17 32 95 23 17 • 95 • 71 • 56 • 32 • 23 • 23 • 19 • 17 • 17

  3. Array: Quiz Scores • 14.75 12 11.5 13.5 14.75 14.75 13 12.5 13.5 • 14.75 • 14.75 • 14.75 • 13.50 • 13.50 • 13.00 • 12.50 • 12.00 • 11.50

  4. 2. Frequency (f) - • Cumulative frequency (cf) -

  5. Frequency Example 1: Weights • X Tallies f cf • 95 1 1 9 • 71 1 1 8 • 56 1 1 7 • 32 1 1 6 • 23 1 1 2 5 • 19 1 1 3 • 17 1 1 2 2

  6. Frequency Example 2: Quiz scores • X Tallies f cf • 14.75 1 1 1 3 9 • 13.50 1 1 2 6 • 13.00 1 1 4 • 12.50 1 1 3 • 12.00 1 1 2 • 11.5 1 1 1

  7. SUMMATION • Weights Quiz scores • 95 14.75 • 71 14.75 • 56 14.75 • 32 13.50 • 23 13.50 • 23 13.00 • 19 12.50 • 17 12.00 • 17 11.50 • 353 120.25

  8. Putting it All Together Weights • X f cf fx • 95 1 9 95 • 71 1 8 71 • 56 1 7 56 • 32 1 6 32 • 23 2 5 46 • 19 1 3 19 • 17 2 2 34 • n = 9 = 353

  9. Putting it All Together Quiz scores • X f cf fx • 14.75 3 9 44.25 • 13.50 2 6 27.00 • 13.00 1 4 13.00 • 12.50 1 3 12.50 • 12.00 1 2 12.00 • 11.50 1 1 11.50 • n = 9 = 120.25

  10. Mean Weights • = 353 / 9 • = 39.22 • Mean Quiz Scores • = 120.25 / 9 • = 13.36

  11. Mode (Mo) Weights • 95 • 71 • 56 • 32 • 23 • 23 • 19 • 17 • 17 • Mo = 17 & 23 -- bimodal

  12. Mode Quiz Scores • 14.75 • 14.75 • 14.75 • 13.50 • 13.50 • 13.00 • 12.50 • 12.00 • 11.50 • Mo = 14.75

  13. 3. Median (Mdn) • Group A Group B • X X • 7 50 • 6 6 • 5 5 • 4 -- Mdn 4 -- Mdn • 3 3 • 2 2 • 1 0

  14. Situations where calculating the median will NOT be so easy. Consider: • 7 7 7 8 8 8 9 9 10 10 • Mdn = L +[ ( n / 2 - cfb) / fw) } i • 7.5 + { ( 10 / 2 - 3 ) / 3 } 1 • = 7.5 + (5 - 3) / 3} 1 • = 7.5 + (2 / 3) 1 • = 8.17

  15. E. Measures of Variability • 1. Range • R = Xh - Xl • Example 1: Weights • R = 95 - 17 = 78 • Example 2: Quiz Scores • R = 14.75 - 11.5 = 3.25

  16. Deviation Scores • x (little x) = X (test score) - Mean • Example 1: Weights

  17. Score X X - Mean x2 • 95 55.78 3111.41 • 71 31.78 1009.97 • 56 16.78 281.57 • 32 -7.22 52.13 • 23 -16.22 263.09 • 23 -16.22 263.09 • 19 -20.22 408.85 • 17 -22.22 493.73 • 17 -22.22 493.73 • n = 9 x2=6377.57 • Sum = 353 Mean = 39.22

  18. Example 2: Quiz Scores • X X - Mean x2 • 14.75 1.39 1.93 • 14.75 1.39 1.93 • 14.75 1.39 1.93 • 13.50 0.14 0.02 • 13.50 0.14 0.02 • 13.00 -0.36 0.13 • 12.50 -0.86 0.74 • 12.00 -1.36 1.85 • 11.50 -1.86 3.46 • n = 9 = 0 x2 = 12.01 • Mean = 13.36

  19. Example 1: Weights • sigma 2 = 6377.63 / 9 • = 708.63 • Example 2: Quiz Scores • sigma 2 = 12.01 / 9 • = 1.33

  20. Example 1: Weights • sigma = square root { 63377.63 / 9} • = square root {708. 63} • = 26.62 • Example 2: Quiz Scores • sigma = square root {1.33} • = 1.15

  21. Standard Scores • z-scores • Mean = 0 Standard Deviation = 1 • Equation: z = (X - Mean) / sigma • Mean of raw score distribution • sigma = SD of raw score distribution

  22. b. T-scores • Mean = 50 SD = 10 • Equation: T = 50 + 10 (z) • Example: Let's suppose that a teacher wants to compare the results of an English and of an Algebra test: • Test Score Mean Highest SD • English 84 110 180 26 • Algebra 40 47 60 5

  23. English z-score = ( 84 - 110) / 26 • = - 26 / 26 • z = - 1.00 • Algebra z-score = ( 40 - 47) / 5 • = -7 / 5 • z = -1.4

  24. English T-score = 50 + 10 (-1.00) • T = 50 + -10 • T = 40.00 • Algebra T-score = 50 + 10 (-1.4) • T = 50 + -14.00 • T = 36.00

  25. FINISHED--DONE--COMPLETED • AT LONG, LONG LAST

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