1. THE NORMAL DISTRIBUTION AND Z- SCORES Areas Under the Curve

2. Let’s Practice! x: { 3, 8, 1} Find  : x381x381 444444  x  4 -3  x   1 16 9  x    SS  x    26  x    √ =√(26/3) = 2.94 OR SS =  x 2 (  x) 2 N __ x381x381 x  9 64 1 74 - (144/3) = 26 Then √(26/3) = 2.94  x = 12  x  = 74

3. The Philosophy of Statistics & Standard Deviation N=50

4. The Philosophy of Statistics & Standard Deviation.24.20.16.12.08.04 Proportion

5. The Philosophy of Statistics & Standard Deviation.24.20.16.12.08.04 Proportion

6. Standard Deviation and Distribution Shape IQ

7. Example: IQs of Sample of Psychologists ID IQ 1128 2155 3135 4134 5144 6101 7167 8198 994 10128 11155 12145 = +0.13, “normal” = +2.07, abnormally high = -1.66, low side of normal z(144) = [ 144 – 140.33]/ 27.91 z(198) = [ 198 – 140.33]/ 27.91 z(94) = [ 94 – 140.33]/ 27.91 x - x z = s x =140.33 s = 27.91 With some simple calculation we find:

8. Forward and reverse transforms Example: If population μ = 120 and σ =20 Find the raw score associated with a z-score of 2.5 x = 120 + 2.5(20) x = 120 + 50 x = 170 “forward” x -  z =  x - x z = s population sample Raw score  Z-score “reverse” x =   z  x = x  z s Z- score  Raw Score

9. Why are z-scores important? z-scores can be used to describe how normal/abnormal scores within a distribution are With a normal distribution, there are certain relationships between z-scores and the proportion of scores contained in the distribution that are ALWAYS true. 1. The entire distribution contains 100% of the scores 2. 68% of the scores are contained within 1 standard deviation below and above the mean 3. 95% of the scores are contained within 2 standard deviations below and above the mean

10. What percentage of scores are contained between 96 and 160? What percentage of scores are between 128 and 160? If I have a total of 200 scores, how many of them are less than 128? Z-score -4 -3 -2 -1 0 1 2 3 4  = 128  = 32 68% 95%

11. Z-score -4 -3 -2 -1 0 1 2 3 4  = 128  = 32 Table A in appendix D contains the areas under the normal curve indexed by Z-score. From these tables you can determine the number of individuals on either side of any z-score. But how do we find areas associated with z-scores that are not simply 0, 1, or 2? What proportion of people got a z score of 1.5 or higher?

12. z-score 1.5

13. Examples of AREA C 2.3 -1.7

14. What percentage of people have a z-score of 0 or greater? What percentage of people have a z-score of 1 or greater? What percentage of people have a z-score of -2.5 or less? What percentage of people have a z-score of 2.3 or greater? What percentage of people have a z-score of -1.7 or less? 50% 15.87%.62% 1.07% 4.46%

15. Examples of AREA B

16. What percentage of people have a z-score between 0 and 1? What percentage of people have a z-score between 0 and 2.3? What percentage of people have a z-score between 0 and -2.4? What percentage of people have a z-score between 0 and 1.27? What percentage of people have a z-score between 0 and 1.79? What percentage of people have a z-score between 0 and -3.24? 34.13% 48.93% 49.18% 39.80% 46.33% 49.94%

17. What percentage of people have a z-score of 1 or less?84.13% Areas which require a COMBINATION of z-scores

18. What percentage of people have a z-score between -1 and 2.3?84.13% Areas which require a COMBINATION of z-scores

19. What percentage of people have a z-score of 1 or less?84.13% Areas which require a COMBINATION of z-scores

20. What percentage of people have a z-score of 1 or less?84.13% Areas which require a COMBINATION of z-scores

21. What percentage of people have a z-score of 1 or less?84.13% Areas which require a COMBINATION of z-scores

22. What percentage of people have a z-score of 1 or less?84.13% Areas which require a COMBINATION of z-scores

23. What percentage of people have a z-score of -1.7 or less?4.46% What z-score is required for someone to be in the bottom 4.46%? -1.7 What percentage of people have a score of 73.6 or less? 4.46%? 128 + (-1.7)32  = 128  = 32 128 - 54.4 73.6 or below Raw Score 0 32 64 96 128 160 192 224 256 What score is required for someone to be in the bottom 4.46%?

24. What z-score is required for someone to be in the top 25%? What z-score is required for someone to be in the top 5%? What z-score is required for someone to be in the bottom 10%? What z-score is required for someone to be in the bottom 70%? What z-score is required for someone to be in the top 50%? What z-score is required for someone to be in the bottom 30%?.68 1.65 -1.29.52 0 -.53

25. What percentage of scores fall between the mean and a score of 132? Here, we must first convert this raw score to a z-score in order to be able to use what we know about the normal distribution. (132-128)/32 = 0.125, or rounded, 0.13. Area B in the z-table indicates that the area contained between the mean and a z-score of.13 is.0517, which is 5.17%  = 128  = 32

26. What percentage of scores fall between a z-score of -1 and 1.5? If we refer to the illustration above, it will require two separate areas added together in order to obtain the total area: Area B for a z-score of -1:.3413 Area B for a z-score of 1.5:.4332 Added together, we get.7745, or 77.45%  = 128  = 32

27. What percentage of scores fall between a z-score of 1.2 and 2.4? Notice that this area is not directly defined in the z-table. Again, we must use two different areas to come up with the area we need. This time, however, we will use subtraction. Area B for a z-score of 2.4:.4918 Area B for a z-score of 1.2:.3849 When we subtract, we get.1069, which is 10.69%  = 128  = 32

28. If my population has 200 people in it, how many people have an IQ below a 65? First, we must convert 65 into a z-score: (65-128)/32 = -1.96875, rounded = -1.97 Since we want the proportion BELOW -1.97, we are looking for Area C of a z- score of 1.97 (remember, the distribution is symmetrical!) :.0244 = 2.44% Last step: What is 2.44% of 200? 200(.0244) = 4.88  = 128  = 32

29. What IQ score would I need to have in order to make it to the top 5%? Since we’re interested in the ‘top’ or the high end of the distribution, we want to find an Area C that is closest to.0500, then find the z-score associated with it. The closest we can come is.0495 (always better to go under). The z-score associated with this area is 1.65. Let’s turn this z-score into a raw score: 128 + 1.65(32) = 180.8  = 128  = 32

30. A possible type of test question: A class of 30 students takes a difficult statistics exam. The average grade turns out to be 65. Michael is a student in this class. His grade on the exam is 80. The following is known: SS = 2883.2 Assuming that these 30 students make up the population of interest, what is the approximate number of people that did better than Michael on the exam? SS= 2883.2  = 65 N = 30 80 - 65  = √ SS N = √ 2883.2 30 = √96.11 = 9.80 9.80 z(80) = (80-65)/9.80 = 1.53 Area C for a z score of 1.53 =.0630, so about 6.3%, or 1.89 people