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The Normal Distribution & Standard Normal Distribution

The Normal Distribution & Standard Normal Distribution

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The Normal Distribution & Standard Normal Distribution

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  1. The Normal Distribution & Standard Normal Distribution I The Normal Distribution A What is it? B Why is it everywhere? Probability Theory is why C The Skewed Normal Distribution D Kurtosis II The Standard Normal Distribution A Standardizing a Normal Distribution B Computing Proportions using Table B.1 Anthony J Greene

  2. A Normal Distribution:Chest Sizes of Scottish Militia Men Anthony J Greene

  3. A Normal Distribution:Histogram of Human Gestation Anthony J Greene

  4. The Normal Distribution: Height Anthony J Greene

  5. A Normal Distribution:Age At Retirement Anthony J Greene

  6. Normally Distributed Variables • The most common continuous (interval/ratio) variable type • Occurs predominantly in nature (biology, psychology, etc.) • Determined by the principles of Probability Anthony J Greene

  7. Probability and the Normal Distribution Probability is the Underlying Cause of the Normal Distribution Anthony J Greene

  8. Possible outcomes for four coin tosses There are 16 possibilities because there are 2 possible outcomes for each toss and 4 tosses: 24 In general the possible outcomes are mn where m is the number of outcomes per event and n is the number of events Anthony J Greene

  9. Probability distribution of the number of heads obtained in 4 coin tosses Anthony J Greene

  10. Probability distribution of the number of heads obtained in 4 coin tosses Anthony J Greene

  11. Frequencies for the numbers of heads obtained in 4 tosses for 1000 observations Anthony J Greene

  12. (a) Probability for 4 coin flips vs. (b) 1000 observations Anthony J Greene

  13. Interpretation of a Normal Distribution in terms of Probability Consider what would happen if there were only 4 genes for height (there are more), each of which has only 2 possible states (like heads versus tails for a coin), call the states T for tall and S for short. The distributions would be identical to that for the coin tosses (see left below) with the possibility of 0, 1, 2, 3, and 4 T’s. In reality height is controlled by many genes so that more than 5 outcomes are possible (see right below).

  14. And for 6 coins instead of 4? Anthony J Greene

  15. Another Example 2 Dice Possible outcomes: 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Anthony J Greene

  16. Another Example Anthony J Greene

  17. Age Height Weight I.Q. Sick Days per Year Hours Sleep per Night Words Read per Minute Calories Eaten per Day Hours of Work Done per Day Eyeblinks per Hour Insulting Remarks per Week Number of Pairs of Socks Owned Examples of the Normal Distribution Anthony J Greene

  18. The Skewed Normal Distribution Anthony J Greene

  19. Income Number of Empty Soda Cans in Car Drug Use per Week Car Accidents per Year Lifetime Hospitalizations Number of Guitars Owned Consecutive Days Unemployed Hand-Washings per Day Number of Languages Spoken Fluently Hours of T.V. per Day Examples of Skewed Normal Distributions Anthony J Greene

  20. Graph of a Normal Distribution 34.13% 13.59% 2.28% 34.13% 13.59% 2.28% Anthony J Greene

  21. Shapes of the Normal Distribution Kurtosis • Leptokurtic • Platokurtic Anthony J Greene

  22. Frequency and relative-frequency distributions for heights Anthony J Greene

  23. What do we do with Normal Distributions? • Determine the position of a given score relative to all other scores. • Compare distributions. Anthony J Greene

  24. Relative-frequency histogram for heights Anthony J Greene

  25. Comparing Two Distributions Two distributions of exam scores. For both distributions, µ = 70, but for one distribution, σ = 12. The position of X = 76 is very different for these two distributions. Anthony J Greene

  26. Data Transformations are Reversible and Do not Alter the Relations Among Items 1) Add or Subtract a Constant From Each Score 2) Multiply Each Score By a Constant • e.g., if you wanted to convert a group of Fahrenheit temperatures to Centigrade you would subtract 32 from each score then multiply by 5/9ths Anthony J Greene

  27. Transforming a distribution does not change the shape of the distribution, only its units Anthony J Greene

  28. Height a) in inches b) in centimetersinches X 2.54 = centimeters Anthony J Greene

  29. Transformations Anthony J Greene

  30. Standard Normal Distribution A normally distributed variable having mean 0 and standard deviation 1 is said to have the standard normal distribution. Its associated normal curve is called the standard normal curve. Anthony J Greene

  31. Transformation to Standard Units The idea is to transform (reversibly) any normal distribution into a STANDARD NORMALdistribution with μ = 0 and σ = 1 Anthony J Greene

  32. Standardized Normally Distributed Variable A normally distributed variable, x, is converted to a standard normal distribution, z, with the following formula Anthony J Greene

  33. Standardizing normal distributions Anthony J Greene

  34. Standard Normal Distribution • For a variable x, the variable (z-score) • is called the standardized version of x or the standardized variable corresponding to the variable x. • This transformation is standard for any variable and preserves the exact relationships among the scores Anthony J Greene

  35. Standard Normal Distributions • The z-score transformation is entirely reversible but allows any distribution to be compared (e.g., I.Q. and SAT score; does a top I.Q. score correspond to a top SAT score?) • z-scores all have a mean of zero and a standard deviation of 1, which gives them the simplest possible mathematical properties. Anthony J Greene

  36. Standard Normal Distributions An example of a z transformation from a variable (x) with mean 3 and standard deviation 2 Anthony J Greene

  37. Understanding x and z-scores Anthony J Greene

  38. Basic Properties of the Standard Normal Curve Property 1: The total area under the standard normal curve is equal to 1. Property 2: The standard normal curve extends indefinitely in both directions, approaching, but never touching, the horizontal axis as it does so. Property 3: The standard normal curve is symmetric about 0; that is, the left side of the curve should be a mirror image of the right side of the curve. Property 4: Most of the area under the standard normal curve lies between –3 and 3. Anthony J Greene

  39. Finding percentages for a normally distributed variable from areas under the standard normal curve Because the standard normal distribution is the same for all variables, it is an easy way to determine what proportion of scores is less than a, what proportion lies between a and b, and what proportion is greater than b (for any distribution and any desired points a and b). Anthony J Greene

  40. The relationship between z-score values and locations in a population distribution. Anthony J Greene

  41. The X-axis is relabeled in z-score units. The distance that is equivalent to σ corresponds to 1 point on the z-score scale. Anthony J Greene

  42. Table B.1p. 687

  43. Table B.1A Closer Look Anthony J Greene

  44. The Normal Distribution: why use a table? Anthony J Greene

  45. From x or z to PTo determine a percentage orprobability for a normally distributed variable Step 1 Sketch the normal curve associated with the variable Step 2 Shade the region of interest and mark the delimiting x-values Step 3 Compute the z-scores for the delimiting x-values found in Step 2 Step 4 Use Table B.1 to obtain the area under the standard normal curve delimited by the z-scores found in Step 3Use Geometry and remember that the total area under the curve is always 1.00. Anthony J Greene

  46. From x or z to PFinding percentages for a normally distributed variable from areas under the standard normal curve Anthony J Greene

  47. Finding percentages for a normally distributed variable from areas under the standard normal curve • ,  are given. • a and b are any two values of the variable x. • Compute z-scores for a and b. • Consult table B-1 • Use geometry to find desired area. Anthony J Greene

  48. Given that a quiz has a mean score of 14 and an s.d. of 3, what proportion of the class will score between 9 & 16? •  = 14 and  = 3. • a = 9 and b = 16. • za = -5/3 = -1.67, zb = 2/3 = 0.67. • In table B.1, we see that the area to the left of a is 0.0475 and that the area to the right of b is 0.2514. • The area between a and b is therefore 1 – (0.0475 + 0.2514) = 0.701 or 70.01% Anthony J Greene

  49. Finding the area under the standard normal curve to the left of z = 1.23 Anthony J Greene

  50. What if you start with x instead of z? What is the probability of selecting a random student who scored above 650 on the SAT? z = 1.50: Use Column C; P = 0.0668 Anthony J Greene