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Lab 3

Lab 3 . z-Scores, the Normal Curve, and Computer Graphs of Distributions. Learning Objectives. Define a z-score. Calculate z-scores. State the mean and SD of the Standard Normal Curve. Use tabled values of the normal curve to estimate percentages of a distribution. Graph Distributions. .

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Lab 3

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  1. Lab 3 • z-Scores, the Normal Curve, and Computer Graphs of Distributions

  2. Learning Objectives • Define a z-score. • Calculate z-scores. • State the mean and SD of the Standard Normal Curve. • Use tabled values of the normal curve to estimate percentages of a distribution. • Graph Distributions.

  3. What is a z-score? Z-score defined. • A z-score tells the location of an observation in terms of standard deviations from the mean. • If a z-score is zero, it’s on the mean. • If a z-score is positive, it’s above the mean. • If a z-score is negative, it’s below the mean. • The value of the z-score tells how many standard deviations above or below the mean it is. A z-score of 2 is 2 SDs above; -1 is 1 SD below.

  4. Z-score formulas • To find a z-score, subtract the mean from the score and divide by the SD. • A z-score for a population: • A z-score for a sample:

  5. Computing z-scores from raw scores

  6. Computing z-scores • What is z? • =10, =8, s=4, z = ? • A: .5 • =8, =10, s=2, z=? • A: -1 • =20, =15, s=5, z =? • A: 1

  7. Computing raw scores from z-scores -- Formulas • To find a raw score, multiply the z-score by the SD and then add the mean. • Population formula: • Sample formula:

  8. Computing raw scores from z-scores -- Examples

  9. Computing raw scores • What is X? • z =1, =10, s=2, X=? • A: 12 • Z = .5, =8, s=4, X=? • A: 10 • z = -1, =20, s=5, X=? • A: 15

  10. The Standard Normal Curve • The shape of the normal curve is known (you know, that bell shape). • The standard normalcurve has a mean of 0 and a standard deviation of 1.

  11. Standard Normal Curve • If a distribution is normal, about what percent of the scores fall below a z-score of 1? Is it 15, 50, 85 or 99? • A: 85. Bottom 50 percent plus 34.13 is 84.13 percent.

  12. Relating Data to the Normal Curve • If your data are approximately normally distributed, you can use the Standard Normal to estimate percentages. • Can relate your mean and SD to the Normal.

  13. Estimating Percentages using Tabled Values of the Normal

  14. What z-score separates the bottom 70 percent from the top 30 percent of scores?

  15. What z-score separates the bottom 70 percent from the top 30 percent of scores? A: About z=.50. Top 30.85 percent are cut off by a z-score of .50. Or add 50 to 19.15 to get 69.15 (almost 70). Estimating Percentages with the Normal

  16. What z score separates the top 30 from the bottom 70? • We can represent the problem graphically. • Note how the lines in the graph correspond to the entries in the table.

  17. Another Example • What z-score separates the top 10 percent from the bottom 90 percent?

  18. Another Example

  19. Boxplots andStem-and-Leaf graphs Graphing and Frequency Distributions

  20. The Normal Curve Frequency Score

  21. Boxplots Example of a Boxplot for a Normal Distribution

  22. Example of a Boxplot with Outliers & Extreme Outliers

  23. Example of a Skewed Distribution

  24. Stem-and-Leaf Plots Example of a Stem-and-Leaf Plot for a Normal Distribution Frequency Stem & Leaf 1.00 2 . 0 2.00 3 . 00 3.00 4 . 000 4.00 5 . 0000 3.00 6 . 000 2.00 7 . 00 1.00 8 . 0 Stem width: 1.00 Each leaf: 1 case(s)

  25. Volcano Heights Boxplot Frequency Stem & Leaf 8.00 0 . 25666789 8.00 1 . 01367799 23.00 2 . 00011222444556667788999 21.00 3 . 011224445555566677899 21.00 4 . 011123333344678899999 24.00 5 . 001122234455666666677799 18.00 6 . 001144556666777889 26.00 7 . 00000011112233455556678889 12.00 8 . 122223335679 14.00 9 . 00012334455679 13.00 10 . 0112233445689 10.00 11 . 0112334669 9.00 12 . 111234456 5.00 13 . 03478 2.00 14 . 00 3.00 15 . 667 2.00 16 . 25 2.00 17 . 29 6.00 Extremes (>=18500) Stem width: 1000.00 Each leaf: 1 case(s)

  26. Height in Inches for Male and Female Students HT_INCH Stem-and-Leaf Plot Frequency Stem & Leaf 1.00 61 . 0 1.00 62 . 0 4.00 63 . 0000 15.00 64 . 000000000000000 16.00 65 . 0000000000000000 22.00 66 . 0000000000000000000000 38.00 67 . 00000000000000000000000000000000000000 29.00 68 . 00000000000000000000000000000 22.00 69 . 0000000000000000000000 29.00 70 . 00000000000000000000000000000 14.00 71 . 00000000000000 4.00 72 . 0000 5.00 73 . 00000 Stem width: 1.00 Each leaf: 1 case(s)

  27. Separte Male & Female Plots of Height

  28. Homework • Compute and describe uses of z-scores.

  29. SPSS • Computing z scores • Graphing Distributions

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