Theoretical Normal Curve  Normality frequently occurs in many situations of psychology, and other sciences

Putting it together Remember that many empirical distributions are approximately normal

Putting it together Thus you can compute z scores from raw scores and use the theoretical normal distribution (Table C) to estimate the probability of that score!

Remember Remember how to convert raw scores to Z scores

Z-score Z scores have a mean of 0 Z scores have a standard deviation of 1

Example: IQ Mean IQ = 100 Standard deviation = 15 What proportion of people have an IQ of 120 or higher?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 120 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (120 - 100) / 15 = 1.33 120 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = 1.33; Column C = .0918 120 .0918 -3 -2 -1  1 2  3 

Example: IQ A proportion of .0918 or 9.18 percent of the population have an IQ above 120.

Practice The Neuroticism Measure = 23.32 S = 6.24 n = 54 If your neuroticism score was 36 how many people are likely more neurotic than you in this room?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 2: Calculate Z score (36 - 23.32) / 6.24 = 2.03 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = 2.03; Column C = .0212 -3 -2 -1  1 2  3 

Practice A proportion of .0212 or 2.12 percent of the population is more neurotic. In a class with 54 people 1.14 or 1person is probably more neurotic (.0212) * 54 = 1.14 or 1 person

Example: IQ Mean IQ = 100 SD = 15 What proportion of the population have an IQ below 110?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 110 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (110 - 100) / 15 = .67 110 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = .67 ; Column B = .2486 110 .2486 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table .2486 + .50 = .7486 110 .50 .2486 -3 -2 -1  1 2  3 

Example: IQ A proportion of .7486 or 74.86 percent of the population have an IQ below 110. In a class with 600 children how many probably have an IQ below 110? (.7486) * 600 = 449.16 or 449 children

Practice Mean IQ = 100 SD = 15 What is the probability of randomly selecting someone with an IQ over 80?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 80 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (80 - 100) / 15 = -1.33 80 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = -1.33; Column B = .4082 80 .4082 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table .4082 + .50 = .9082 80 .4082 .50 -3 -2 -1  1 2  3 

Example: IQ The probability of randomly selecting someone with an IQ over 80 is .9082

Finding the Proportion of the Population Between Two Scores What proportion of the population have IQ scores between 90 and 110?

Step 1: Sketch out question 90 110 ? -3 -2 -1  1 2  3 

Step 2: Calculate Z scores for both values Z = (X -  ) /  Z = (90 - 100) / 15 = -.67 Z = (110 - 100) / 15 = .67

Step 3: Look in Table C -.67 .67 .2486 .2486 -3 -2 -1  1 2  3 

Step 4: Add together the two values -.67 .67 .4972 -3 -2 -1  1 2  3 

A proportion of .4972 or 49.72 percent of the population have an IQ between 90 and 110.

What proportion of the population have an IQ between 110 and 130?

Step 1: Sketch out question 110 130 ? -3 -2 -1  1 2  3 

Step 2: Calculate Z scores for both values Z = (X -  ) /  Z = (110 - 100) / 15 = .67 Z = (130 - 100) / 15 = 2.0

Step 3: Look in Table C .67 2.0 .4772 -3 -2 -1  1 2  3 

Step 3: Look in Table C .67 2.0 .4772 .2486 -3 -2 -1  1 2  3 

Step 4: Subtract .4772 - .2486 = .2286 .67 2.0 .2286 -3 -2 -1  1 2  3 

A proportion of .2286 or 22.86 percent of the population have an IQ between 110 and 130.

Practice The Neuroticism Measure = 23.32 S = 6.24 n = 54 How many people likely have a neuroticism score between 29 and 34?

Practice (29-23.32) /6.24 = .91 B = .3186 ( 34-23.32)/6.26 = 1.71 .4564-.3186 = .1378 .1378*54 = 7.44 or 7 people

Finding a score when given a probability What IQ score is required to fall in the top 20 percent of the population?

Step 1: Sketch out question .20 100 ?

Step 2: Look in Table C In column C get as close to .20 as you can and find the corresponding Z score = .84 .20 100 ?

Step 3: Find the X score that goes with the Z score Z = (X -  ) /  .84 = (X - 100) / 15 Must solve for X X =  + (z)() X = 100 + (.84)(15)

Step 3: Find the X score that goes with the Z score Z = (X -  ) /  .84 = (X - 100) / 15 Must solve for X X =  + (z)() X = 100 + (.84)(15) = 112.6 A score of 112.6 is needed to be in the top 20 percent!