Understanding the Normal Distribution and Density Curves

1. Chapter 3 The Normal Distributions Chapter 3

2. Density Curves • Here is a histogram of vocabulary scores of 947 seventh graders • The smooth curve drawn over the histogram is a mathematical model for the distribution. • The mathematical model is called the density function. Chapter 3

3. Density Curves • The areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0. • This proportion is equal to 0.303. Chapter 3

4. Areas under density curves • The scale of the Y-axis of the density curve is adjusted so the total area under the curve is 1 • The area under the curve to the left of 6.0 is shaded, and is equal to 0.293 • This similar to the areas of the shaded bars in the prior slide! Chapter 3

5. Density Curves Chapter 3

6. Density Curves • There are many types of density curves • We are going to focus on a family of curves called Normal curves • Normal curves are • bell-shaped • not too steep, not too fat • defined means & standard deviations Chapter 3

7. Normal Density Curves • The mean and standard deviation computed from actual observations (data) are denoted by and s,respectively. • The mean and standard deviation of the distribution represented by the density curve are denoted by µ (“mu”) and  (“sigma”),respectively. Chapter 3

8. standard deviation mean Bell-Shaped Curve:The Normal Distribution Chapter 3

9. The Normal Distribution • Mean µ defines the center of the curve • Standard deviation  defines the spread • Notation is N(µ,). Chapter 3

10. Practice Drawing Curves! • Symmetrical around μ • Infections points (change in slope, blue arrows) at ± σ Chapter 3

11. 68-95-99.7 Rule forAny Normal Curve • 68% of the observations fall within one standard deviation of the mean • 95% of the observations fall within two standard deviations of the mean • 99.7% of the observations fall within three standard deviations of the mean Chapter 3

12. 68% 95% - µ + -2 µ +2 99.7% -3 +3 µ 68-95-99.7 Rule forAny Normal Curve Chapter 3

13. 68-95-99.7 Rule forAny Normal Curve Chapter 3

14. Men’s Height Example (NHANES, 1980) • Suppose heights of men follow a Normal distribution with mean = 70.0 inches and standard deviation = 2.8 inches • Shorthand: X ~ N(70, 2.8) X  the variable ~  “distributed as” N(μ, σ)  Normal with mean μ and standard deviation σ Chapter 3

15. Men’s Height Example 68-95-99.7 rule If X~N(70, 2.8) • 68% between µ = 70.0  2.8 = 67.2 to 72.8 • 95% between µ 2 = 70.0  2(2.8) = 70.0  5.6 = 64.4 to 75.6 inches • 99.7% between µ 3 = 70.0  3(2.8) = 70.0  8.4 = 61.6 and 78.4 inches Chapter 3

16. 68% (by 68-95-99.7 Rule) ? 16% -1 +1 70 72.8 (height) 84% NHANES (1980) Height Example What proportion of men are less than 72.8 inches tall? (Note: 72.8 is one σ above μ on this distribution) Chapter 3

17. ? 68 70 (height values) NHANES Height Example What proportion of men are less than 68 inches tall? How many standard deviations is 68 from 70? Chapter 3

18. Standard Normal (Z) Distribution • The Standard Normal distribution has mean 0 and standard deviation 1 • We call this a Z distribution: Z~N(0,1) • Any Normal variable x can be turned into a Z variable (standardized) by subtracting μ and dividing by σ: Chapter 3

19. Standardized Scores • How many standard deviations is 68 from μ on X~N(70,2.8)? • z = (x – μ) / σ = (68 - 70) / 2.8 = -0.71 • The value 68 is 0.71 standard deviations below the mean 70 Chapter 3

20. ? 68 70 (height values) Men’s Height Example (NHANES, 1980) • What proportion of men are less than 68 inches tall? -0.71 0 (standardized values) Chapter 3

21. Table A in text:Standard Normal Table Chapter 3

22. Table A:Standard Normal Probabilities .01 0.7 .2389 Chapter 3

23. 68 70 (height values) -0.71 0 (standardized values) Men’s Height Example (NHANES, 1980) • What proportion of men are less than 68 inches tall? .2389 Chapter 3

24. 68 70 (height values) -0.71 0 (standardized values) Men’s Height Example (NHANES, 1980) • What proportion of men are greater than 68 inches tall? • Area under curve sums to 1, so Pr(X > x) = 1 – Pr(X < x), as shown below: 1.2389 = .7611 .2389 Chapter 3

25. .10 ? 70 (height values) Men’s Height Example (NHANES, 1980) • How tall must a man be to place in the lower 10% for men aged 18 to 24? Chapter 3

26. Table A:Standard Normal Table • Use Table A • Look up the closest proportion in the table • Find corresponding standardized score • Solve for X (“un-standardize score”) Chapter 3

27. Table A:Standard Normal Proportion .08 1.2 .1003 Pr(Z < -1.28) = .1003 Chapter 3

28. .10 ? 70 (height values) Men’s Height Example (NHANES, 1980) • How tall must a man be to place in the lower 10% for men aged 18 to 24? -1.28 0 (standardized values) Chapter 3

29. Observed Value for a Standardized Score • “Unstandardize” z-score to find associated x : Chapter 3

30. Observed Value for a Standardized Score • x = μ + zσ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42 • A man would have to be approximately 66.42 inches tall or less to place in the lower 10% of the population Chapter 3