1. 1 The Normal Distribution William P. Wattles Psychology 302

2. Frequency distribution n A table or graph that indicates all the values a variable can take and how often each occurs.

3. Density curves A density curve is a mathematical model of a distribution. It is always on or above the horizontal axis. The total area under the curve, by definition, is equal to 1, or 100%. The area under the curve for a range of values is the proportion of all observations for that range. Histogram of a sample with the smoothed density curve theoretically describing the population

4. Normal Distribution Gaussian Distribution n Mean=Median=Mode

5. Normal Distribution Normal distributions have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails.

6. Here the means are different (  = 10, 15, and 20) while the standard deviations are the same (  = 3). Here the means are the same (  = 15) while the standard deviations are different (  = 2, 4, and 6). A family of density curves

7. 40 Z scores and the normal curve n The 68-95-99.7 rule n 68% fall within one standard deviation of the mean n 95% fall within two standard deviations of the mean n 99.7% of the observations fall with three standard deviations of the mean

9. Normal Curve

10. Because all Normal distributions share the same properties, we can standardize our data to transform any Normal curve N (  ) into the standard Normal curve N (0,1). The standard Normal distribution For each x we calculate a new value, z (called a z-score). N(0,1) => N(64.5, 2.5) Standardized height (no units)

11. 36 Standard Scores (Z-scores) n Can use appendix A (page 690) in back of book to determine the area under the curve cut off by any Z-score.

12. Using Table A (…).0082 is the area under N(0,1) left of z = -2.40.0080 is the area under N(0,1) left of z = -2.41 0.0069 is the area under N(0,1) left of z = -2.46

13. 47 Area under the curve n Height of young women – Mean = 64 – Standard deviation = 2.7 n What proportion of women are less than 70 inches tall?

14. 47 Area under the curve n Height of young women – Mean = 64 – Standard deviation = 2.7 n Z score for 5’10” +2.22 n Area to the left =.9868 n A woman 70 inches tall is taller than 99% of her peers.

15. n WAIS mean=100, SD=15 n What percent are retarded, I.e. less than 70?

16. n WAIS mean=100, SD=15 n What percent are MENSA eligible, I.e. greater than 130?

17. Area under the curve n WAIS mean=100, SD=15 n Z=X-mean/standard deviation n What percent are retarded, I.e. less than 70? n Z=70-100/15, Z=-2.00, 2.28% n What percent are MENSA eligible, I.e. greater than 130? n Z=130-100/15 Z=+2.00, 2.28%

18. 58 Percentile scores n The percent of all scores at or below a certain point. n The same procedure as with proportions n More commonly used than proportions

19. Sample Problem n SAT mean=1020, SD=207 n Division 1 athletes must have 820 to compete? Is this fair? n What percent score less than 820?

20. Normal Curve

21. n SAT mean=1020, SD=207 n What percent score less than 820?.1660

22. n SAT mean=1020, SD=207 n Division 1 athletes must have 720 to practice? Is this fair? n What percent score less than 720?

23. Normal Curve

24. n SAT mean=1020, SD=207 n What percent score less than 720?.0735

25. What is a Z score?

26. n A z-score tells how many standard deviations the score or observation falls from the mean and in which direction

27. Z-Score n A Z-score tells how many standard deviations an individual’s score lies above or below the mean.

29. Psy 302 Paper 1. Pick a subject that interests you. 2. Do some library research. 3. Collect data a.two groups (minimum 15 per group) b.measurement data 1. Analyze data with t-test a.SPSS b.Excel 2. Make a histogram of your results. 3. Write paper APA style per sample paper.

30. Houston’s G.M. Is a Revolutionary Spirit in a Risk- Averse Mind n Daryl Morey has charts and spreadsheets and clever formulas for evaluating basketball players, and a degree from M.I.T. to make sense of it all.

31. 60 The End